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Psychological Review Copyright 1996 by the American Psychological Association, Inc. 1996, Vol. 103. No. 4, 650-669 0033-295X/96/$3.00

Reasoning the Fast and Frugal Way: Models of Bounded Rationality

Gerd Gigerenzer and Daniel G. Goldstein Max Planck Institute for Psychological Research and University of Chicago

Humans and animals make inferences about the world under limited time and knowledge. In con- trast, many models of rational inference treat the mind as a Laplacean Demon, equipped with un- limited time, knowledge, and computational might. Following H. Simon's notion of satisficing, the authors have proposed a family of algorithms based on a simple psychological mechanism: one- reason decision making. These fast and frugal algorithms violate fundamental tenets of classical rationality: They neither look up nor integrate all information. By computer simulation, the authors held a competition between the satisficing "Take The Best" algorithm and various "rational" infer- ence procedures (e.g., multiple regression). The Take The Best algorithm matched or outperformed all competitors in inferential speed and accuracy. This result is an existence proof that cognitive mechanisms capable of successful performance in the real world do not need to satisfy the classical norms of rational inference.

Organisms make inductive inferences. Darwin ( 1872/1965 ) observed that people use facial cues, such as eyes that waver and lids that hang low, to infer a person's guilt. Male toads, roaming through swamps at night, use the pitch of a rival's croak to infer its size when deciding whether to fight (Krebs & Davies, 1987). Stock brokers must make fast decisions about which of several stocks to trade or invest when only limited information is avail- able. The list goes on. Inductive inferences are typically based on uncertain cues: The eyes can deceive, and so can a tiny toad with a deep croak in the darkness.

How does an organism make inferences about u n k n o w n as- pects of the environment? There are three directions in which to look for an answer. From Pierre Laplace to George Boole to Jean Piaget, m a n y scholars have defended the now classical view that the laws of h u m a n inference are the laws of probability and statistics (and to a lesser degree logic, which does n o t deal as easily with uncertainty). Indeed, the Enlightenment probabi- lists derived the laws of probability from what they believed to be the laws of h u m a n reasoning (Daston, 1988). Following this time-honored tradition, much contemporary research in psy- chology, behavioral ecology, and economics assumes standard

Gerd Gigerenzer and Daniel G. Goldstein, Center for Adaptive Be- havior and Cognition, Max Planck Institute for Psychological Research, Munich, Germany, and Department of Psychology, University of Chicago.

This research was funded by National Science Foundation Grant SBR-9320797/GG.

We are deeply grateful to the many people who have contributed to this article, including Hal Arkes, Leda Cosmides, Jean Czerlinski, Lor- raine Daston, Ken Hammond, Reid Hastie, Wolfgang Hell, Ralph Her- twig, Ulrich Hoffrage, Albert Madansky, Laura Martignon, Geoffrey Miller, Silvia Papal, John Payne, Terry Regier, Werner Schub6, Peter Sedlmeier, Herbert Simon, Stephen Stigler, Gerhard Strube, Zeno Swi- jtink, John Tooby, William Wimsatt, and Werner Wittmann.

Correspondence concerning this article should be addressed to Gerd Gigerenzer or Daniel G. Goldstein, Center for Adaptive Behavior and Cognition, Max Planck Institute for Psychological Research, Leo- poldstrasse 24, 80802 Munich, Germany. Electronic mail may be sent via Internet to [email protected].

statistical tools to be the normative and descriptive models of inference and decision making. Multiple regression, for in- stance, is both the economist's universal tool (McCloskey, 1985 ) and a model of inductive inference in multiple-cue learn- ing ( H a m m o n d , 1990) and clinical j u d g m e n t (B. Brehmer, 1994); Bayes's theorem is a model of how animals infer the presence of predators or prey (Stephens & Krebs, 1986) as well as of h u m a n reasoning and memory (Anderson, 1990). This Enlightenment view that probability theory and h u m a n reason- ing are two sides of the same coin crumbled in the early nine- teenth century but has remained strong in psychology and economics.

In the past 25 years, this stronghold came under attack by proponents of the heuristics and biases program, who con- cluded that h u m a n inference is systematically biased and error prone, suggesting that the laws of inference are quick-and-dirty heuristics and not the laws of probability ( K a h n e m a n , Slovic, & Tversky, 1982). This second perspective appears diametrically opposed to the classical rationality of the Enlightenment, but this appearance is misleading. It has retained the normative kernel of the classical view. For example, a discrepancy between the dictates of classical rationality and actual reasoning is what defines a reasoning error in this program. Both views accept the laws of probability and statistics as normative, b u t they disagree about whether h u m a n s can stand up to these norms.

Many experiments have been conducted to test the validity of these two views, identifying a host of conditions under which the h u m a n m i n d appears more rational or irrational. But most of this work has dealt with simple situations, such as Bayesian inference with b i n a r y hypotheses, one single piece of binary data, and all the necessary information conveniently laid out for the participant (Gigerenzer & Hoffrage, 1995). In many real-world situations, however, there are multiple pieces of in- formation, which are not independent, b u t redundant. Here, Bayes's theorem and other "rational" algorithms quickly be- come mathematically complex and computationally intracta- ble, at least for ordinary h u m a n minds. These situations make neither of the two views look promising. If one would apply the classical view to such complex real-world environments, this

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REASONING THE FAST AND FRUGAL WAY 651

would suggest that the m i n d is a supercalculator like a Lapla- cean D e m o n (Wimsatt, 1976 )---carrying a r o u n d the collected works o f Kolmogoroff, Fisher, or N e y m a n - - a n d simply needs a m e m o r y jog, like the slave in Plato's Meno. O n the other hand, the heuristics-and-biases view o f h u m a n irrationality would lead us to believe that h u m a n s are hopelessly lost in the face o f real-world complexity, given their supposed inability to reason according to the canon o f classical rationality, even in simple laboratory experiments.

There is a third way to look at inference, focusing on the psy- chological and ecological rather than on logic and probability theory. This view questions classical rationality as a universal n o r m and thereby questions the very definition o f " g o o d " rea- soning on which both the Enlightenment and the heuristics- and-biases views were built. Herbert Simon, possibly the best- known proponent o f this third view, proposed looking for models o f bounded rationality instead o f classical rationality. Simon (1956, 1982) argued that information-processing sys- tems typically need to satisfice rather than optimize. Satisficing, a blend o f sufficing and satisfying, is a word o f Scottish origin, which Simon uses to characterize algorithms that successfully deal with conditions o f limited time, knowledge, or computa- tional capacities. His concept o f satisficing postulates, for in- stance, that an organism would choose the first object (a mate, perhaps) that satisfies its aspiration level--instead o f the intrac- table sequence o f taking the time to survey all possible alterna- tives, estimating probabilities and utilities for the possible out- comes associated with each alternative, calculating expected utilities, and choosing the alternative that scores highest.

Let us stress that Simon's notion o f bounded rationality has two sides, one cognitive and one ecological. As early as in Ad- ministrative Behavior ( 1945 ), he emphasized the cognitive lim- itations o f real minds as opposed to the omniscient Laplacean D e m o n s o f classical rationality. As early as in his Psychological Review article titled "Rational Choice and the Structure o f the Environment" (1956), Simon emphasized that minds are adapted to real-world environments. The two go in tandem: " H u m a n rational behavior is shaped by a scissors whose two blades are the structure o f task environments and the computa- tional capabilities o f the actor" (Simon, 1990, p. 7). For the most part, however, theories o f h u m a n inference have focused exclusively on the cognitive side, equating the notion o f bounded rationality with the statement that h u m a n s are limited information processors, period. In a Procrustean-bed fashion, bounded rationality became almost synonymous with heuris- tics and biases, thus paradoxically reassuring classical rational- ity as the normative standard for both biases and bounded ra- tionality (for a discussion o f this confusion see Lopes, 1992). Simon's insight that the minds o f living systems should be un- derstood relative to the environment in which they evolved, rather than to the tenets o f classical rationality, has had little impact so far in research on h u m a n inference. Simple psycho- logical algorithms that were observed in h u m a n inference, rea- soning, or decision making were often discredited without a fair trial, because they looked so stupid by the n o r m s o f classical rationality. For instance, when Keeney and Raiffa (1993) dis- cussed the lexicographic ordering procedure they had observed in p r a c t i c e - - a procedure related to the class o f satisficing algo- rithms we propose in this a r t i c l e - - t h e y concluded that this pro- cedure "is naively simple" and "will rarely pass a test o f

'reasonableness' " ( p . 78 ). They did not report such a test. We shall.

Initially, the concept o f bounded rationality was only vaguely defined, often as that which is not classical economics, and one could "fit a lot o f things into it by foresight and hindsight," as Simon ( 1992, p. 18) himself put it. We wish to do more than oppose the Laplacean D e m o n view. We strive to come up with something positive that could replace this unrealistic view o f mind. What are these simple, intelligent algorithms capable o f making near-optimal inferences? How fast and how accurate are they? In this article, we propose a class o f models that exhibit bounded rationality in both o f Simon's senses. These satisficing algorithms operate with simple psychological principles that satisfy the constraints o f limited time, knowledge, and c o m p u - tational might, rather than those o f classical rationality. At the same time, they are designe d to be fast and frugal without a significant loss o f inferential accuracy, because the algorithms can exploit the structure o f environments.

The article is organized as follows. We begin by describing the task the cognitive algorithms are designed to address, the basic algorithm itself, and the real-world environment on which the performance o f the algorithm will be tested. Next, we report on a competition in which a satisficing algorithm competes with "rational" algorithms in making inferences about a real-world environment. The "rational" algorithms start with an advan- tage: They use more time, information, and computational might to make inferences. Finally, we study variants o f the sati- sficing algorithm that make faster inferences and get by with even less knowledge.

T h e T a s k

We deal with inferential tasks in which a choice must be made between two alternatives on a quantitative dimension. Consider the following example:

Which city has a larger population? (a) Hamburg (b) Cologne.

Two-alternative-choice tasks occur in various contexts in which inferences need to be made with limited time and knowledge, such as in decision making and risk assessment during driving (e.g., exit the highway now or stay on ); treatment-allocation de- cisions (e.g., who to treat first in the emergency room: the 80- year-old heart attack victim or the 16-year-old car accident victim); and financial decisions (e.g., whether to b u y or sell in the trading pit). Inference concerning population demograph- ics, such as city populations o f the past, present, and future (e.g., Brown & Siegler, 1993), is o f importance to people work- ing in urban planning, industrial development, and marketing. Population demographics, which is better understood than, say, the stock market, will serve us later as a "drosophila" environ- ment that allows us to analyze the behavior o f satisficing algorithms.

We study two-alternative-choice tasks in situations where a person has to make an inference based solely on knowledge re- trieved from memory. We refer to this as inference from mem- ory as opposed to inference from givens. Inference from m e m - ory involves search in declarative knowledge and has been in- vestigated in studies of, inter alia, confidence in general knowledge (e.g., Juslin, 1994; Sniezek & Buckley, 1993); the

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effect o f repetition on belief (e.g., Hertwig, Gigerenzer, & Hoffrage, in press); hindsight bias (e.g., Fischhoff, 1977); quan- titative estimates o f a r e a and population o f nations (Brown & Siegler, 1993); and a u t o b i o g r a p h i c m e m o r y o f t i m e (Huttenlocher, Hedges, & Prohaska, 1988). Studies o f infer- ence from givens, on the other hand, involve making inferences from information presented by an e x p e r i m e n t e r (e.g., H a m - mond, Hursch, & Todd, 1964). In the t r a d i t i o n o f Ebbinghaus's nonsense syllables, a t t e m p t s a r e often m a d e here to prevent in- dividual knowledge from i m p a c t i n g on the results by using p r o b l e m s a b o u t hypothetical referents instead o f actual ones. For instance, in celebrated j u d g m e n t and decision-making tasks, such as the " c a b " p r o b l e m and the " L i n d a " p r o b l e m , all the relevant information is provided by the experimenter, and individual knowledge a b o u t cabs and h i t - a n d - r u n accidents, o r feminist b a n k tellers, is considered o f no relevance (Gigerenzer & Murray, 1987). As a consequence, l i m i t e d knowledge or in- dividual differences in knowledge play a small role in inference from givens. In contrast, the satisflcing algorithms p r o p o s e d in this article p e r f o r m inference from memory, they use limited knowledge as input, and as we will show, they can actually profit from a lack o f knowledge.

Assume that a person does not know or c a n n o t deduce the answer to the H a m b u r g - C o l o g n e question but needs to m a k e an inductive inference from related real-world knowledge. How is this inference derived? How can we predict choice ( H a m b u r g or Cologne) from a p e r s o n ' s state o f knowledge?

T h e o r y

The cognitive algorithms we propose are realizations o f a framework for modeling inferences from memory, the theory o f probabilistic mental models (PMM theory; see Gigerenzer, 1993; Gigerenzer, Hoffrage, & Kleinb61ting, 1991 ). The theory o f probabilistic mental models assumes t h a t inferences a b o u t u n k n o w n states o f the world are based on p r o b a b i l i t y cues (Brunswik, 1955). The theory relates three visions: ( a ) Induc- tive inference needs to be studied with respect to natural envi- ronments, as e m p h a s i z e d by Brunswik a n d Simon; ( b ) induc- tive inference is c a r r i e d o u t by satisficing algorithms, as e m p h a - sized by Simon; a n d ( c ) inductive inferences are based on frequencies o f events in a reference class, as p r o p o s e d b y Rei- chenbach and other frequentist statisticians. The t h e o r y o f probabilistic mental models accounts for choice a n d confi- dence, b u t only choice is addressed in this article.

The m a j o r t h r u s t o f the theory is t h a t it replaces the c a n o n o f classical rationality with simple, plausible psychological mech- anisms o f i n f e r e n c e - - m e c h a n i s m s t h a t a m i n d can actually c a r r y out under limited t i m e and knowledge and t h a t could have possibly arisen through evolution. Most traditional models o f inference, from linear multiple regression models to Bayesian models to neural networks, try to find some optimal integration o f all i n f o r m a t i o n available: Every bit o f information is taken into account, weighted, and c o m b i n e d in a c o m p u t a t i o n a l l y ex- pensive way. T h e family o f a l g o r i t h m s in P M M t h e o r y does not i m p l e m e n t this classical ideal. Search in m e m o r y for relevant information is r e d u c e d t o a m i n i m u m , and there is no integra- tion ( b u t rather a substitution) o f pieces o f information. These satisficing algorithms dispense with the fiction o f the o m n i - scient Laplacean D e m o n , who has all the t i m e and knowledge

Figure 1. Illustration of bounded search through limited knowledge. Objects a, b, and c are recognized; object d i s not. Cue values are posi- tive ( + ) or negative ( - ) ; missing knowledge is shown by question marks. Cues are ordered according to their validities. To infer whether a > b, the Take The Best algorithm looks up only the cue values in the shaded space; to infer whether b > c, search is bounded to the dotted space. The other cue values are not looked up.

to search for all relevant information, to c o m p u t e the weights and covariances, and then to integrate all this information into an inference.

Limited Knowledge

A P M M is an inductive device t h a t uses l i m i t e d knowledge to make fast inferences. Different from mental models o f syllo- gisms and deductive inference ( J o h n s o n - L a i r d , 1983), which focus on the logical task o f t r u t h preservation and where knowl- edge is irrelevant (except for the meaning o f connectives and other logical t e r m s ) , P M M s p e r f o r m intelligent guesses a b o u t unknown features o f the world, based on u n c e r t a i n indicators. To m a k e an inference about which o f two objects, a or b, has a higher value, knowledge a b o u t a reference class R is searched, with a , b E R . In our example, knowledge a b o u t the reference class "cities in G e r m a n y " could be searched. The knowledge consists o f p r o b a b i l i t y cues Ci ( i = 1 , . . . , n), and the cue values ai and bi o f the objects for the i t h cue. For instance, when m a k - ing inferences a b o u t populations o f G e r m a n cities, the fact that a city has a professional soccer t e a m in the m a j o r league (Bundesliga) may c o m e to a person's m i n d as a potential cue. T h a t is, when considering pairs o f G e r m a n cities, i f one city has a soccer t e a m in the m a j o r league and the other does not, then the city with the t e a m is likely, b u t n o t certain, t o have the larger population.

Limited knowledge means that the m a t r i x o f objects by cues has missing entries (i.e., objects, cues, or cue values may be u n k n o w n ) . Figure 1 models the l i m i t e d knowledge o f a person. She has heard o f three G e r m a n cities, a , b, and c, but not o f d (represented by three positive and one negative recognition values). She knows some facts ( c u e values) a b o u t these cities with respect to five b i n a r y cues. For a b i n a r y cue, there are two cue values, positive (e.g., the city has a soccer t e a m ) or negative ( i t does n o t ) . Positive refers to a cue value t h a t signals a higher value on the target variable (e.g., having a soccer t e a m is corre- lated with high p o p u l a t i o n ) . U n k n o w n cue values are shown by a question mark. Because she has never heard o f d , all cue val- ues for object d are, b y definition, unknown.

People rarely know all information on which an inference

REASONING THE FAST AND FRUGAL WAY 6 5 3

could be based, t h a t is, knowledge is limited. We m o d e l limited knowledge in two respects: A person can have ( a ) i n c o m p l e t e knowledge o f the objects in the reference class (e.g., she recog- nizes only some o f the cities ), ( b ) l i m i t e d knowledge o f the cue values (facts a b o u t cities), o r ( c ) both. For instance, a person who does n o t know all o f the cities with soccer t e a m s may know some cities with positive cue values (e.g., M u n i c h and H a m b u r g certainly have t e a m s ) , m a n y with negative cue values ( e.g., Hei- delberg and P o t s d a m certainly do n o t have t e a m s ) , and several cities for which cue values will n o t be known.

The Take The Best Algorithm

The first satisficing a l g o r i t h m presented is called the Take The Best algorithm, because its policy is " t a k e the best, ignore the r e s t " It is the basic a l g o r i t h m in the P M M framework. Vari- ants t h a t work faster or with less knowledge are described later. We explain the steps o f the Take The Best a l g o r i t h m for b i n a r y cues (the algorithm can be easily generalized to m a n y valued c u e s ) , using Figure 1 for illustration.

The Take The Best a l g o r i t h m assumes a subjective r a n k order o f cues according to their validities (as in Figure 1 ). We call the highest ranking cue ( t h a t d i s c r i m i n a t e s between the two alternatives) the best cue. The a l g o r i t h m is shown in the form o f a flow d i a g r a m in Figure 2.

Figure 3. Discrimination rule. A cue discriminates between two al- ternatives if one has a positive cue value and the other does not. The four discriminating cases are shaded.

1 is asked to infer which o f city a and city d has m o r e inhabi- tants, the inference will be city a , because the person has never heard o f city d before.

Step 2: Search for Cue Values

For the two objects, retrieve the cue values o f the highest ranking cue from memory.

Step 1: Recognition Principle

The recognition p r i n c i p l e is invoked when the m e r e recogni- tion o f an object is a p r e d i c t o r o f the target variable (e.g., p o p u l a t i o n ) . The recognition p r i n c i p l e states the following: I f only one o f the two objects is recognized, then choose the rec- ognized object. I f neither o f the two objects is recognized, then choose r a n d o m l y between them. I f b o t h o f the objects are rec- ognized, then proceed to Step 2.

Example: I f a person in the knowledge state shown in Figure

Start

Choose the ] I alternative [ to which the I cue points J

I Choose the best cue I

+

No ~ Y e s

Figure 2. Flow diagram of the Take The Best algorithm.

Step 3." Discrimination Rule

Decide whether the cue discriminates. The cue is said to dis- c r i m i n a t e between two objects i f one has a positive cue value and the other does not. The four shaded knowledge states in Figure 3 are those in which a cue discriminates.

Step 4: Cue-Substitution Principle

I f the cue discriminates, then stop searching for cue values. I f the cue does not d i s c r i m i n a t e , go b a c k to Step 2 and continue with the next cue until a cue t h a t d i s c r i m i n a t e s is found.

Step 5: Maximizing Rule for Choice

Choose the object with the positive cue value. I f no cue dis- criminates, then choose randomly.

Examples: Suppose the task is judging which o f city a or b is larger ( F i g u r e 1). Both cities are recognized (Step 1 ), and search for the best cue results with a positive and a negative cue value for Cue 1 ( S t e p 2). The cue d i s c r i m i n a t e s ( S t e p 3), a n d search is t e r m i n a t e d ( S t e p 4). The person makes the inference t h a t city a is larger ( S t e p 5 ).

Suppose now the task is judging which o f city b or e is larger. Both cities are recognized ( S t e p 1 ), and search for the cue val- ues cue results in negative cue value on object b for Cue 1, b u t the corresponding cue value for object e is u n k n o w n ( S t e p 2 ) . The cue does n o t d i s c r i m i n a t e ( S t e p 3 ), so search is c o n t i n u e d ( S t e p 4). Search for the next cue results with positive and a negative cue values for Cue 2 ( S t e p 2). This cue d i s c r i m i n a t e s ( S t e p 3), and search is t e r m i n a t e d ( S t e p 4). The person makes the inference that city b is larger ( S t e p 5 ).

The features o f this a l g o r i t h m are ( a ) search extends through only a p o r t i o n o f the total knowledge in m e m o r y (as shown b y the shaded and d o t t e d parts o f Figure 1 ) and is stopped i m m e -

6 5 4 GIGERENZER AND GOLDSTEIN

diately when the first d i s c r i m i n a t i n g cue is found, ( b ) the algo- r i t h m does not a t t e m p t to integrate i n f o r m a t i o n but uses cue substitution instead, and ( c ) the total a m o u n t o f i n f o r m a t i o n processed is contingent on each task ( p a i r o f objects) and varies in a predictable way a m o n g individuals with different knowl- edge. This fast and c o m p u t a t i o n a l l y simple algorithm is a model o f b o u n d e d rationality rather t h a n o f classical rationality. There is a close parallel with S i m o n ' s concept o f "satisficing": The Take The Best algorithm stops search after the first d i s c r i m i n a t - ing cue is found, j u s t as S i m o n ' s satisficing algorithm stops search after the first option t h a t meets an aspiration level.

The algorithm is h a r d l y a s t a n d a r d statistical tool for induc- tive inference: It does not use all available information, it is non- c o m p e n s a t o r y and nonlinear, and variants o f it can violate tran- sitivity. Thus, it differs from s t a n d a r d linear tools for inference such as multiple regression, as well as from n o n l i n e a r neural networks t h a t are c o m p e n s a t o r y in nature. The Take The Best algorithm is n o n c o m p e n s a t o r y because only the best discrimi- nating cue d e t e r m i n e s the inference or decision; no c o m b i n a - tion o f other cue values can override this decision. In this way, the algorithm does n o t conform to the classical economic view o f h u m a n behavior (e.g., Becker, 1976), where, under the as- s u m p t i o n that all aspects can be r e d u c e d to one dimension (e.g., m o n e y ) , there exists always a trade-offbetween c o m m o d i t i e s or pieces o f information. T h a t is, the a l g o r i t h m violates the Archi- median axiom, which implies that for any multidimensional object a ( a j , a2 . . . . . a~) preferred to b ( b l , b2 . . . . . b~), where al d o m i n a t e s b~, this preference can be reversed by taking multiples o f any one or a c o m b i n a t i o n o f b2, b3 . . . . . bn. As we discuss, variants o f this a l g o r i t h m also violate transitivity, one o f the cornerstones o f classical rationality (McClennen, 1990).

E m p i r i c a l E v i d e n c e

Despite their flagrant violation o f the t r a d i t i o n a l standards o f rationality, the Take The Best a l g o r i t h m and o t h e r models from the framework o f P M M t h e o r y have been successful in integrat- ing various striking p h e n o m e n a in inference from m e m o r y and predicting novel phenomena, such as the confidence-frequency effect (Gigerenzer et al., 1991) and the less-is-more effect ( G o l d s t e i n , 1994; G o l d s t e i n & Gigerenzer, 1996). The theory o f probabilistic mental models seems to be the only existing process theory o f the overconfidence bias that successfully pre- dicts conditions under which overestimation occurs, disappears, and inverts to underestimation (Gigerenzer, 1993; Gigerenzer et al., 1991; Juslin, 1993, 1994; Juslin, W i n m a n , & Persson, t995; b u t see Griffin & Tversk); 1992). Similarly, the theory predicts when the h a r d - e a s y effect occurs, disappears, and in- v e r t s w p r e d i c t i o n s t h a t have been experimentally confirmed by Hoffrage (1994) and b y Juslin ( 1993 ). The Take The Best algo- r i t h m explains also why the popular confirmation-bias expla- nation o f the overconfidence bias ( K o r i a t , Lichtenstein, & Fischhoff, 1980) is n o t s u p p o r t e d by experimental d a t a (Gigerenzer et al., 1991, pp. 521-522 ).

Unlike earlier accounts o f these striking p h e n o m e n a in con- fidence a n d choice, the algorithms in the P M M framework al- low for predictions o f choice based on each individual's knowl- edge. Goldstein a n d Gigerenzer ( 1 9 9 6 ) showed t h a t the recog- nition p r i n c i p l e p r e d i c t e d individual participants" choices in a b o u t 90% to 100% o f all cases, even when p a r t i c i p a n t s were

taught i n f o r m a t i o n t h a t suggested doing otherwise (negative cue values for the recognized objects). A m o n g the evidence for the e m p i r i c a l validity o f the Take-The-Best algorithm are the tests o f a b o l d prediction, the less-is-more effect, which postu- lates conditions under which people with little knowledge make better inferences t h a n those who k n o w more. This surprising prediction has been experimentally confirmed. For instance, U.S. students m a k e slightly m o r e correct inferences a b o u t G e r - m a n city populations ( a b o u t which they know little) t h a n a b o u t U.S. cities, and vice versa for G e r m a n students (Gigerenzer, 1993; G o l d s t e i n 1994; G o l d s t e i n & Gigerenzer, 1995; Hoffrage, 1994). The theory o f probabilistic mental models has been ap- plied to other situations in which inferences have to be m a d e u n d e r limited t i m e and knowledge, such as rumor-based stock m a r k e t t r a d i n g ( D i F o n z o , 1994). A general review o f the theory and its evidence is presented in McClelland a n d Bolger ( 1 9 9 4 ) .

The reader f a m i l i a r with the original a l g o r i t h m presented in Gigerenzer et al. ( 1991 ) will have noticed t h a t we simplified the d i s c r i m i n a t i o n rule.~ In the present version, search is a l r e a d y t e r m i n a t e d if one object has a positive cue value and the other does not, whereas in the earlier version, search was t e r m i n a t e d only when one object h a d a positive value a n d the other a nega- tive one (cf. Figure 3 in Gigerenzer et al. with Figure 3 in this article). This change follows e m p i r i c a l evidence that p a r t i c i - p a n t s tend to use this faster, simpler d i s c r i m i n a t i o n r u l e (Hoffrage, 1994).

This article does not a t t e m p t to provide further e m p i r i c a l ev- idence. For the moment, we assume t h a t the m o d e l is descrip- tively valid and investigate how accurate this satisficing algo- r i t h m is in drawing inferences a b o u t u n k n o w n aspects o f a real-world environment. Can an a l g o r i t h m based on simple psychological principles that violate the n o r m s o f classical ra- tionality m a k e a fair n u m b e r o f a c c u r a t e inferences?

T h e E n v i r o n m e n t

We tested the p e r f o r m a n c e o f the Take The Best algorithm on how accurately it m a d e inferences a b o u t a real-world environ- ment. The e n v i r o n m e n t was the set o f all cities in G e r m a n y with more than 100,000 inhabitants (83 cities after G e r m a n reunification), with p o p u l a t i o n as the target variable. The model o f the e n v i r o n m e n t consisted o f 9 b i n a r y ecological cues and the actual 9 × 83 cue values. The full model o f the environ- m e n t is shown in the Appendix.

Each cue has an associated validity, which is indicative o f its predictive power. The ecological validity o f a cue is the relative frequency with which the cue correctly predicts the target, de- fined with respect to the reference class (e.g., all G e r m a n cities with m o r e t h a n 100,000 i n h a b i t a n t s ) . For instance, i f one checks all pairs in which one city has a soccer t e a m but the other city does not, one finds that in 87% o f these cases, the city with the t e a m also has the higher population. This value is the eco- logical validity o f the soccer t e a m cue. T h e validity vi o f the i t h cue is

v, = p[ t ( a ) > t ( b ) l ai is positive and b~ is negative ],

Also, we now use the term discrimination rule instead of activation rule.

REASONING T H E FAST A N D FRUGAL WAY 655

Table 1 Cues, Ecological Validities, and Discrimination Rates

Ecological Discrimination Cue validity rate

National capital (Is the city the national capital?) 1.00 .02

Exposition site (Was the city once an exposition site?) .91 .25

Soccer team (Does the city have a team in the major league?) .87 .30

Intercity train (Is the city on the Intercity line?) .78 .38

State capital (Is the city a state capital?) .77 .30 License plate (Is the abbreviation only

one letter long?) .75 .34 University (Is the city home to a

university?) .71 .51 Industrial belt (Is the city in the

industrial belt?) .56. .30 East Germany (Was the city formerly

in East Germany?) .51 .27

w h e r e t(a) a n d t(b) a r e t h e v a l u e s o f objects a a n d b o n t h e target v a r i a b l e t a n d p is a p r o b a b i l i t y m e a s u r e d as a relative f r e q u e n c y in R .

T h e ecological v a l i d i t y o f t h e n i n e c u e s r a n g e d over t h e w h o l e s p e c t r u m : f r o m .51 ( o n l y slightly b e t t e r t h a n c h a n c e ) t o 1.0 ( c e r t a i n t y ) , as s h o w n in Table 1. A c u e w i t h a h i g h ecological validity, however, is n o t o f t e n useful i f its d i s c r i m i n a t i o n r a t e is small.

Table i shows also t h e discrimination rates for e a c h cue. T h e d i s c r i m i n a t i o n rate o f a c u e is t h e relative f r e q u e n c y w i t h w h i c h t h e c u e d i s c r i m i n a t e s b e t w e e n a n y t w o o b j e c t s f r o m t h e refer- e n c e class. T h e d i s c r i m i n a t i o n r a t e is a f u n c t i o n o f t h e d i s t r i b u - t i o n o f t h e c u e v a l u e s a n d t h e n u m b e r N o f o b j e c t s in t h e refer- e n c e class. L e t t h e relative f r e q u e n c i e s o f t h e positive a n d nega- tive c u e v a l u e s b e x a n d y , respectively. T h e n t h e d i s c r i m i n a t i o n r a t e d~ o f t h e i t h c u e is

2xiyi 4 = 1 '

1 - - - -

N

as a n e l e m e n t a r y c a l c u l a t i o n shows. T h u s , i f N is v e r y large, t h e d i s c r i m i n a t i o n r a t e is a p p r o x i m a t e l y 2xiyi .2 T h e larger t h e ecological v a l i d i t y o f a cue, t h e b e t t e r t h e inference. T h e larger t h e d i s c r i m i n a t i o n rate, t h e m o r e o f t e n a c u e c a n b e u s e d t o m a k e a n inference. I n t h e p r e s e n t e n v i r o n m e n t , ecological va- lidities a n d d i s c r i m i n a t i o n rates a r e negatively c o r r e l a t e d . T h e r e d u n d a n c y o f c u e s i n t h e e n v i r o n m e n t , as m e a s u r e d b y pair- wise c o r r e l a t i o n s b e t w e e n cues, ranges b e t w e e n - . 2 5 a n d .54, w i t h a n average a b s o l u t e v a l u e o f . 19. 3

T h e C o m p e t i t i o n

T h e q u e s t i o n o f h o w well a satisficing a l g o r i t h m p e r f o r m s in a r e a l - w o r l d e n v i r o n m e n t has r a r e l y b e e n p o s e d i n r e s e a r c h o n i n d u c t i v e inference. T h e p r e s e n t s i m u l a t i o n s s e e m t o b e t h e first t o test h o w well s i m p l e satisficing a l g o r i t h m s d o c o m p a r e d w i t h s t a n d a r d i n t e g r a t i o n a l g o r i t h m s , w h i c h r e q u i r e m o r e k n o w l -

edge, t i m e , a n d c o m p u t a t i o n a l power. T h i s q u e s t i o n is i m p o r - t a n t for S i m o n ' s p o s t u l a t e d l i n k b e t w e e n t h e c o g n i t i v e a n d t h e ecological: I f t h e s i m p l e psychological p r i n c i p l e s i n satisficing a l g o r i t h m s a r e t u n e d t o ecological s t r u c t u r e s , t h e s e a l g o r i t h m s s h o u l d n o t fail o u t r i g h t . We p r o p o s e a c o m p e t i t i o n b e t w e e n var- i o u s i n f e r e n t i a l a l g o r i t h m s . T h e c o n t e s t will g o t o t h e a l g o r i t h m t h a t scores t h e highest p r o p o r t i o n o f c o r r e c t i n f e r e n c e s in t h e shortest t i m e .

Simulating Limited Knowledge

We s i m u l a t e d p e o p l e w i t h v a r y i n g degrees o f k n o w l e d g e a b o u t cities in G e r m a n y . L i m i t e d k n o w l e d g e c a n t a k e t w o f o r m s . O n e is l i m i t e d r e c o g n i t i o n o f objects i n t h e r e f e r e n c e class. T h e o t h e r is l i m i t e d k n o w l e d g e a b o u t t h e c u e v a l u e s o f r e c o g n i z e d objects. To m o d e l l i m i t e d r e c o g n i t i o n kno wle dge , we s i m u l a t e d p e o p l e w h o r e c o g n i z e d b e t w e e n 0 a n d 83 G e r m a n cities. To m o d e l l i m i t e d k n o w l e d g e o f c u e values, we s i m u l a t e d 6 basic classes o f people, w h o k n e w 0%, 10%, 20%, 50%, 75%, o r 100% o f t h e c u e v a l u e s a s s o c i a t e d w i t h t h e objects t h e y rec- o g n i z e d . C o m b i n i n g t h e t w o s o u r c e s o f l i m i t e d k n o w l e d g e re- sulted i n 6 x 84 t y p e s o f people, e a c h h a v i n g different de gr e e s a n d k i n d s o f l i m i t e d knowledge. W i t h i n e a c h t y p e o f people, we c r e a t e d 500 s i m u l a t e d individuals, w h o differed r a n d o m l y f r o m o n e a n o t h e r in t h e p a r t i c u l a r objects a n d c u e v a l u e s t h e y knew. All objects a n d c u e v a l u e s k n o w n were d e t e r m i n e d r a n d o m l y w i t h i n t h e a p p r o p r i a t e c o n s t r a i n t s , t h a t is, a c e r t a i n n u m b e r o f objects k n o w n , a c e r t a i n t o t a l p e r c e n t a g e o f c u e v a l u e s k n o w n , a n d t h e v a l i d i t y o f t h e r e c o g n i t i o n p r i n c i p l e (as e x p l a i n e d in t h e following p a r a g r a p h ) .

T h e s i m u l a t i o n n e e d e d t o b e realistic in t h e sense t h a t t h e s i m u l a t e d p e o p l e c o u l d i n v o k e t h e r e c o g n i t i o n p r i n c i p l e . T h e r e - fore, t h e sets o f cities t h e s i m u l a t e d p e o p l e k n e w h a d t o b e c a r e - fully c h o s e n so t h a t t h e r e c o g n i z e d cities were larger t h a n t h e u n r e c o g n i z e d o n e s a c e r t a i n p e r c e n t a g e o f t h e t i m e . We per- f o r m e d a s u r v e y t o get a n e m p i r i c a l e s t i m a t e o f t h e a c t u a l c o -

2 For instance, if N = 2 and one cue value is positive and the other negative (xr = Yr = .5), dr = 1.0. If Nincreases, with xr and Yi held constant, then dr decreases and converges to 2xr Yr.

3 There are various other measures o f redundancy besides pairwise correlation. The important point is that whatever measure o f redun- dancy one uses, the resultant value does not have the same meaning for all algorithms. For instance, all that counts for the Take The Best algorithm is what proportion o f correct inferences the second cue adds to the first in the cases where the first cue does not discriminate, how much the third cue adds to the first two in the cases where they do not discriminate, and so on. If a cue discriminates, search is terminated, and the degree of redundancy in the cues that were not included in the search is irrelevant. Integration algorithms, in contrast, integrate all information and, thus, always work with the total redundancy in the environment (or knowledge base). For instance, when deciding among objects a, b, c, and d i n Figure 1, the cue values of Cues 3, 4, and 5 do not matter from the point of view o f the Take The Best algorithm (because search is terminated before reaching Cue 3). However, the values of Cues 3, 4, and 5 affect the redundancy of the ecological system, from the point of view of all integration algorithms. The lesson is that the degree o f redundancy in an environment depends on the kind o f algorithm that operates on the environment. One needs to be cautious in interpreting measures o f redundancy without reference to an algorithm.

6 5 6 G1GERENZER AND GOLDSTEIN

variation between recognition of cities and city populations. Let us define the validity c~ of the recognition principle to be the probability; in a reference class, that one object has a greater value on the target variable t h a n another, in the cases where the one object is recognized and the other is not:

a = p[ t ( a ) > t ( b ) t a, is positive and b, is negative ],

where t(a) and t(b) are the values of objects a and b on the target variable t, a, and br are the recognition values of a and b, and p is a probability measured as a relative frequency in R.

In a pilot study of 26 undergraduates at the University of Chi- cago, we found that the cities they recognized (within the 83 largest in G e r m a n y ) were larger than the cities they did not rec- ognize in about 80% of all possible comparisons. We incorpo- rated this value into o u r simulations by choosing sets of cities (for each knowledge state, i.e., for each n u m b e r of cities recognized) where the known cities were larger than the u n - known cities in about 80% of all cases. Thus, the cities known by the simulated individuals had the same relationship between recognition and population as did those of the h u m a n individu- als. Let us first look at the performance of the Take The Best algorithm.

Testing the T a k e T h e B e s t A l g o r i t h m ~

We tested how well individuals using the Take The Best algo- rithm did at answering real-world questions such as, Which city has more inhabitants: ( a ) Heidelberg or ( b ) Bonn? Each of the 500 simulated individuals in each of the 6 X 84 types was tested on the exhaustive set of 3,403 city pairs, resulting in a total of 500 x 6 X 84 x 3,403 tests, that is, about 858 million.

The curves in Figure 4 show the average proportion of correct inferences for each proportion of objects and cue values known. The x axis represents the n u m b e r of cities recognized, and the y axis shows the proportion of correct inferences that the Take The Best algorithm drew. Each of the 6 x 84 points that make up the six curves is an average proportion of correct inferences taken from 500 simulated individuals, who each made 3,403 inferences.

When the proportion of cities recognized was zero, the pro- portion of correct inferences was at chance level (.5). When up to half of all cities were recognized, performance increased at all levels of knowledge about cue values. The m a x i m u m per- centage of correct inferences was a r o u n d 77%. The striking re- sult was that this m a x i m u m was not achieved when individuals knew all cue values of all cities, b u t rather when they knew less. This result shows the ability of the algorithm to exploit limited knowledge, that is, to do best when not everything is known. Thus, the Take The Best algorithm produces the less-is-more effect. At any level of limited knowledge of cue values, learning more G e r m a n cities will eventually cause a decrease in propor- tion correct. Take, for instance, the curve where 75% of the cue values were known and the point where the simulated partici- pants recognized about 60 G e r m a n cities. If these individuals learned about the remaining G e r m a n cities, their proportion correct would decrease. The rationale behind the less-is-more effect is the recognition principle, and it can be understood best from the curve that reflects 0% of total cue values known. Here, all decisions are made on the basis of the recognition principle,

.8 .8

f f

7 L / / /

f / \ 0 ;

Lt/ '

0 10 20 30 40 50 60 70 80 Number of Objects Recognized

Figure 4. Correct inferences about the population of German cities (two-alternative-choice tasks) by the Take The Best algorithm. Infer- ences are based on actual information about the 83 largest cities and nine cues for population (see the Appendix). Limited knowledge of the simulated individuals is varied across two dimensions: (a) the number of cities recognized (x axis ) and (b) the percentage of cue values known (the six curves).

or by guessing. O n this curve, the recognition principle comes into play most when half of the cities are known, so it takes on an inverted-U shape. When half the cities are known, the recognition principle can be activated most often, that is, for roughly 50% of the questions. Because we set the recognition validity in advance, 80% of these inferences will be correct. In the remaining half of the questions, when recognition c a n n o t be used (either both cities are recognized or both cities are unrecognized), then the organism is forced to guess and only 50% of the guesses will be correct. Using the 80% effective rec- ognition validity half of the time and guessing the other half of the time, the organism scores 65% correct, which is the peak of the bottom curve. The mode of this curve moves to the right with increasing knowledge about cue values. Note that even when a person knows everything, all cue values of all cities, there are states of limited knowledge in which the person would make more accurate inferences. We are n o t going to discuss the conditions of this counterintuitive effect and the supporting experimental evidence here (see Goldstein & Gigerenzer, 1996). O u r focus is on how much better integration algorithms can do in making inferences.

Integration A l g o r i t h m s

We asked several colleagues in the fields of statistics and eco- nomics to devise decision algorithms that would do better than the Take The Best algorithm. The five integration algorithms we simulated and pitted against the Take The Best algorithm in a competition were among those suggested by our colleagues.

REASONING THE FAST AND FRUGAL WAY 6 5 7

These competitors include " p r o p e r " and " i m p r o p e r " linear models (Dawes, 1979; Lovie & Lovie, 1986). These algorithms, in contrast to the Take The Best algorithm, e m b o d y two classi- cal principles o f rational inference: ( a ) c o m p l e t e s e a r c h - - t h e y use all available i n f o r m a t i o n ( c u e values ) - - a n d ( b ) c o m p l e t e i n t e g r a t i o n - - t h e y c o m b i n e all these pieces o f information into a single value. In short, we refer in this article to algorithms t h a t satisfy these principles as " r a t i o n a l " ( i n q u o t a t i o n m a r k s ) algorithms.

C o n t e s t a n t 1: T a l l y i n g

Let us start with a simple integration algorithm: tallying o f positive evidence ( G o l d s t e i n , 1994). In this algorithm, the n u m b e r o f positive cue values for each object is tallied across all cues ( i = 1 , . . . , n ) , and the object with the largest n u m b e r o f positive cue values is chosen. Integration algorithms are not based (at least explicitly) on the recognition principle. For this reason, and to m a k e the integration algorithms as strong as pos- sible, we allow all the integration a l g o r i t h m s to m a k e use o f rec- ognition i n f o r m a t i o n (the positive and negative recognition val- ues, see Figure 1 ). Integration algorithms t r e a t recognition as a cue, like the nine ecological cues in Table 1. T h a t is, in the competition, the n u m b e r o f cues ( n ) is thus equal to 10 (because recognition is i n c l u d e d ) . The decision criterion for tallying is the following:

I f ~ a~ > b~, then choose city a . i = 1 i = 1

n n

I f ~ ai < ~, bi, then choose city b. i = 1 iffil

I f ~ ai = bi, then guess. i = l i = l

The assignments o f ai and b~ are the following:

1 i f the i t h cue value is positive

a~, b~ = 0 i f the i t h cue value is negative

0 i f the i t h cue value is unknown.

Let us c o m p a r e cities a and b, from Figure 1. By tallying the positive cue values, a would score 2 p o i n t s and b would score 3. Thus, tallying would choose b to be the larger, in opposition to the Take The Best algorithm, which would infer t h a t a is larger. Variants o f tallying, such as the frequency-of-good-features heuristic, have been discussed in the decision literature ( A l b a & Marmorstein, 1987; Payne, Bettman, & Johnson, 1993).

C o n t e s t a n t 2. W e i g h t e d T a l l y i n g

Tallying treats all cues alike, independent o f cue validity. Weighted tallying o f positive evidence is identical with tallying, except that it weights each cue according to its ecological valid- ity, vt. The ecological validities o f the cues a p p e a r in Table 1. We set the validity o f the recognition cue to .8, which is the e m p i r i c a l average d e t e r m i n e d by the pilot study. The decision rule is as follows:

I f Z aivi > ~, bivi, then choose city a . i = 1 i = 1

I f ~ aivi < bivi, then choose city b. i = 1 i = 1

I f ~ aivi = bivi, then guess. i = 1 i = 1

N o t e that weighted tallying needs m o r e i n f o r m a t i o n t h a n either tallying or the Take The Best algorithm, namely, quantitative i n f o r m a t i o n a b o u t ecological validities. In the simulation, we provided the real ecological validities to give this a l g o r i t h m a good chance.

Calling again on the c o m p a r i s o n o f objects a and b from Fig- ure 1, let us assume that the validities would be .8 for recogni- tion and .9, .8, .7, .6, .51 for Cues 1 through 5. Weighted tallying would thus assign 1.7 points to a and 2.3 points to b. Thus, weighted tallying would also choose b to be the larger.

Both tallying algorithms treat negative i n f o r m a t i o n and miss- ing i n f o r m a t i o n identically. T h a t is, they consider only positive evidence. The following algorithms distinguish between nega- tive and missing i n f o r m a t i o n and integrate both positive and negative in formation.

C o n t e s t a n t 3. Unit- W e i g h t L i n e a r M o d e l

The unit-weight linear model is a special case o f the equal- weight linear model (Huber, 1989) and has been a d v o c a t e d as a good a p p r o x i m a t i o n o f weighted linear models (Dawes, 1979; Einhorn & Hogarth, 1975). The decision criterion for unit- weight integration is the same as for tallying, only the assign- m e n t ofa~ and bi differs:

1 if the i t h cue value is positive

a i , b~ = - 1 i f the i t h cue value is negative

0 if the i t h cue value is unknown.

C o m p a r i n g objects a and b from Figure 1 would involve as- signing 1.0 points to a and 1.0 p o i n t s to b and, thus, choosing randomly. This simple linear model corresponds to Model 2 in Einhorn and Hogarth ( 1975, p. 177 ) with the weight p a r a m e t e r set equal to 1.

C o n t e s t a n t 4: W e i g h t e d L i n e a r M o d e l

This model is like the unit-weight linear model except t h a t the values o f a i and bi are multiplied by their respective ecolog- ical validities. The decision c r i t e r i o n is the same as with weighted tallying. The weighted linear model ( o r some variant o f it) is often viewed as an optimal rule for preferential choice, under the idealization o f independent dimensions or cues (e.g., Keeney & Raiffa, 1993; Payne et al., 1993). C o m p a r i n g objects a and b from Figure 1 would involve assigning 1.0 points to a and 0.8 p o i n t s to b and, thus, choosing a to be the larger.

C o n t e s t a n t 5: M u l t i p l e R e g r e s s i o n

The weighted linear model reflects the different validities o f the cues, b u t not the dependencies between cues. M u l t i p l e re- gression creates weights that reflect the covariances between

658 GIGERENZER AND GOLDSTEIN

predictors or cues and is commonly seen as an "optimal" way to integrate various pieces of information into an estimate (e.g., Brunswik, 1955; H a m m o n d , 1966). Neural networks using the delta rule determine their "optimal" weights by the same prin- ciples as multiple regression does (Stone, 1986). The delta rule carries out the equivalent of a multiple linear regression from the i n p u t patterns to the targets.

The weights for the multiple regression could simply be cal- culated from the full information about the nine. ecological cues, as given in the Appendix. To make multiple regression an even stronger competitor, we also provided information about which cities the simulated individuals recognized. Thus, the multiple regression used n i n e ecological cues and the recogni- tion cue to generate its weights. Because the weights for the rec- ognition cue depend on which cities are recognized, we calcu- lated 6 × 500 × 84 sets of weights: one for each simulated indi- vidual. Unlike any of the other algorithms, regression had access to the actual city populations (even for those cities not recognized by the hypothetical person) in the calculation of the weights. 4 D u r i n g the quiz, each simulated person used the set of weights provided to it by multiple regression to estimate the populations of the cities in the comparison.

There was a missing-values problem in computing these 6 X 84 × 500 sets of regression coefficients, because most simulated individuals did not know certain cue values, for instance, the cue values of the cities they did n o t recognize. We strengthened the performance of multiple regression by substituting u n - known cue values with the average of the cue values the person knew for the given cue. 5 This was done both in creating the weights and in using these weights to estimate populations. U n - like traditional procedures where weights are estimated from one half of the data, and inferences based on these weights are made for the other half, the regression algorithm had access to all the information in the Appendix (except, of course, the u n - known cue v a l u e s ) - - m o r e information than was given to any of the competitors. In the competition, multiple regression and, to a lesser degree, the weighted linear model approximate the ideal of the Laplacean Demon.

Results

Speed

The Take The Best algorithm is designed to enable quick de- cision making. Compared with the integration algorithms, how much faster does it draw inferences, measured by the a m o u n t of information searched in memory? For instance, in Figure 1, the Take The Best algorithm would look up four cue values (including the recognition cue values) to infer that a is larger than b. None of the integration algorithms use limited search; thus, they always look up all cue values.

Figure 5 shows the a m o u n t of cue values retrieved from memory by the Take The Best algorithm for various levels of limited knowledge. The Take The Best algorithm reduces search in memory considerably. Depending on the knowledge state, this algorithm needed to search for between 2 (the n u m - ber of recognition values) and 20 (the m a x i m u m possible cue values: Each city has n i n e cue values and one recognition value). For instance, when a person recognized half of the cities and knew 50% of their cue values, then, on average, only about

4 cue values (that is, one fifth of all possible) are searched for. The average across all simulated participants was 5.9, which was less than a third of all available cue values.

Accuracy

Given that it searches only for a limited a m o u n t of informa- tion, how accurate is the Take The Best algorithm, compared with the integration algorithms? We ran the competition for all states of limited knowledge shown in Figure 4. We first report the results of the competition in the case where each algorithm achieved its best performance: When 100% of the cue values were known. Figure 6 shows the results of the simulations, car- ried out in the same way as those in Figure 4.

To our surprise, the Take The Best algorithm drew as m a n y correct inferences as any of the other algorithms, and more than some. The curves for Take The Best, multiple regression, weighted tallying, and tallying are so similar that there are only slight differences among them. Weighted tallying performed about as well as tallying, and the unit-weight linear model per- formed about as well as the weighted linear m o d e l - - d e m o n - strating that the previous finding that weights may be chosen in a fairly arbitrary manner, as long as they have the correct sign ( Dawes, 1979), is generalizable to tallying. The two integration algorithms that make use of both positive and negative infor- mation, unit-weight and weighted linear models, made consid- erably fewer correct inferences. By looking at the lower-left and upper-right corners of Figure 6, one can see that all competitors do equally well with a complete lack of knowledge or with com- plete knowledge. They differ when knowledge is limited. Note that some algorithms can make more correct inferences when they do not have complete knowledge: a demonstration of the less-is-more effect mentioned earlier.

What was the result of the competition across all levels of limited knowledge? Table 2 shows the result for each level of limited knowledge of cue values, averaged across all levels of recognition knowledge. (Table 2 reports also the performance of two variants of the Take The Best algorithm, which we dis- cuss later: the Minimalist and the Take The Last algorithm.) The values in the 100% c o l u m n of Table 2 are the values in Figure 6 averaged across all levels of recognition. The Take The Best algorithm made as m a n y correct inferences as one of the competitors (weighted tallying) and more than the others. Be- cause it was also the fastest, we judged the competition goes to the Take The Best algorithm as the highest performing, overall.

To our knowledge, this is the first time that it has been dem- onstrated that a satisficing algorithm, that is, the Take The Best algorithm, can draw as m a n y correct inferences about a real-

4 We cannot claim that these integration algorithms are the best ones, nor can we know a priori which small variations will succeed in our bumpy real-world environment. An example: During the proof stage of this article we learned that regressing on the ranks of the cities does slightly better than regressing on the city populations. The key issue is what are the structures of environments in which particular algorithms and variants thrive.

5 If no single cue value was known for a given cue, the missing values were substituted by .5. This value was chosen because it is the midpoint of 0 and 1, which are the values used to stand for negative and positive cue values, respectively.

REASONING THE FAST AND FRUGAL WAY 659

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Results of the competition. The curve for the Take The Best algorithm is identical with the 100% curve in Figure 4. The results for proportion correct have been smoothed by a running median smoother, to lessen visual noise between the lines.

6 6 0 GIGERENZER AND GOLDSTE1N

Table 2 Results o f the Competition: Average Proportion o f Correct Inferences

Percentage of cue values known

Algorithm 10 20 50 75 100 Average

Take The Best .621 . 6 3 5 . 6 6 3 . 6 7 8 .691 .658 Weighted tallying .621 . 6 3 5 . 6 6 3 . 6 7 9 .693 .658 Regression .625 . 6 3 5 . 6 5 7 . 6 7 4 .694 .657 Tallying .620 . 6 3 3 . 6 5 9 . 6 7 6 .691 .656 Weightedlinear model .623 . 6 2 7 . 6 2 3 . 6 1 9 .625 .623 Unit-weight linear model .621 . 6 2 2 .621 . 6 2 0 .622 .621

Minimalist .619 .631 . 6 5 0 .661 .674 .647 Take The Last .619 . 6 3 0 . 6 4 6 . 6 5 8 .675 .645

Note. Values are rounded; averages are computed from the unrounded values. Bottom two algorithms are variants of the Take The Best algo- rithm.

world environment as integration algorithms, across all states of limited knowledge. The dictates of classical rationality would have led one to expect the integration algorithms to do substan- tially better than the satisficing algorithm.

Two results of the simulation can be derived analytically. First and most obvious is that if knowledge about objects is zero, then all algorithms perform at a chance level. Second, and less obvious, is that if all objects and cue values are known, then tallying produces as m a n y correct inferences as the unit-weight linear model. This is because, under complete knowledge, the score under the tallying algorithm is an increasing linear func- tion of the score arrived at in the unit-weight linear model. 6 The equivalence between tallying and unit-weight linear models under complete knowledge is an i m p o r t a n t result. It is known that unit-weight linear models can sometimes perform about as well as proper linear models (i.e., models with weights that are chosen in an optimal way, such as in multiple regression; see Dawes, 1979). The equivalence implies that under complete knowledge, merely counting pieces of positive evidence can work as well as proper linear models. This result clarifies one condition under which searching only for positive evidence, a strategy that has sometimes been labeled confirmation bias or positive test strategy, can be a reasonable and efficient inferen- tial strategy (Klayman & Ha, 1987; Tweney & Walker, 1990).

Why do the unit-weight and weighted linear models perform markedly worse under limited knowledge of objects? The rea- son is the simple and bold recognition principle. Algorithms that do not exploit the recognition principle in environments where recognition is strongly correlated with the target variable pay the price of a considerable n u m b e r of wrong inferences. The unit-weight and weighted linear models use recognition infor- mation and integrate it with all other information b u t do n o t follow the recognition principle, that is, they sometimes choose unrecognized cities over recognized ones. Why is this? In the environment, there are more negative cue values than positive ones (see the Appendix), and most cities have more negative cue values than positive ones. From this it follows that when a recognized object is compared with an unrecognized object, the (weighted) sum of cue values of the recognized object will often be smaller than that of the unrecognized object (which is - 1 for

the unit-weight model and - . 8 for the weighted linear model). Here the unit-weight and weighted linear models often make the inference that the unrecognized object is the larger one, due to the overwhelming negative evidence for the recognized ob- ject. Such inferences contradict the recognition principle. Tal- lying algorithms, in contrast, have the recognition principle built in implicitly. Because tallying algorithms ignore negative information, the tally for an unrecognized object is always 0 and, thus, is always smaller than the tally for a recognized ob- ject, which is at least 1 (for tallying, or .8 for weighted tallying, due to the positive value on the recognition cue). Thus, tallying algorithms always arrive at the inference that a recognized ob- ject is larger than an unrecognized one.

Note that this explanation of the different performances puts the full weight in a psychological principle (the recognition principle) explicit in the Take The Best algorithm, as opposed to the statistical issue of how to find optimal weights in a linear function. To test this explanation, we reran the simulations for the unit-weight and weighted linear models under the same con- ditions b u t replacing the recognition cue with the recognition principle. The simulation showed that the recognition principle accounts for all the difference.

C a n Satisficing A l g o r i t h m s G e t b y W i t h E v e n Less T i m e a n d K n o w l e d g e ?

The Take The Best algorithm produced a surprisingly high proportion of correct inferences, compared with more compu- tationally expensive integration algorithms. Making correct in- ferences despite limited knowledge is an i m p o r t a n t adaptive feature of an algorithm, b u t being right is not the only thing that counts. In m a n y situations, time is limited, and acting fast can be as i m p o r t a n t as being correct. For instance, if you are driving on an unfamiliar highway and you have to decide in an instant what to do when the road forks, your problem is not necessarily making the best choice, b u t simply making a quick choice. Pressure to be quick is also characteristic for certain types of verbal interactions, such as press conferences, in which a fast answer indicates competence, or commercial interactions, such as having telephone service installed, where the customer has to decide in a few minutes which of a dozen calling features to purchase. These situations entail the dual constraints of lim- ited knowledge and limited time. The Take The Best algorithm is already faster than any of the integration algorithms, because it performs only a limited search and does not need to compute weighted sums of cue values. Can it be made even faster? It can, if search is guided by the recency of cues in memory rather than by cue validity.

T h e T a k e T h e L a s t A l g o r i t h m

The Take The Last algorithm first tries the cue that discrimi- nated the last time. If this cue does n o t discriminate, the algo-

6 The proof for this is as follows. The tallying score t for a given object is the number n ÷ of positive cue values, as defined above. The score u for the unit weight linear model is n + - n - , where n - is the number of negative cue values. Under complete knowledge, n = n + + n - , where n is the number of cues. Thus, t = n + , and u = n + - n - . Because n - = n - n 4, by substitution into the formula for u, we find that u = n + - ( n - n +) = 2t - n.

REASONING THE FAST AND FRUGAL WAY 661

r i t h m then tries the cue that d i s c r i m i n a t e d the t i m e before last, and so on. The a l g o r i t h m differs from the Take The Best algo- r i t h m in Step 2, which is now r e f o r m u l a t e d as Step 2':

Step 2': Search for the Cue Values of the Most Recent Cue

For the two objects, retrieve the cue values o f the cue used m o s t recently. I f it is the first j u d g m e n t and there is n o d i s c r i m - ination record available, retrieve the cue values o f a r a n d o m l y chosen cue.

Thus, in Step 4, the a l g o r i t h m goes b a c k to Step 2'. Variants o f this search p r i n c i p l e have been studied as the "Einstellung effect" in the water j a r experiments ( L u c h i n s & Luchins, 1994), where the solution strategy o f the most recently solved p r o b l e m is tried first on the subsequent p r o b l e m . This effect has also been n o t e d in physicians' generation o f diagnoses for clinical cases (Weber, B6ckenholt, Hilton, & Wallace, 1993 ).

This a l g o r i t h m does n o t need a r a n k order o f cues according to their validities; all t h a t needs to be k n o w n is the direction in which a cue points. Knowledge a b o u t the r a n k order o f cue validities is replaced by a m e m o r y o f which cues were last used. Note t h a t such a r e c o r d can be built up independently o f any knowledge a b o u t the s t r u c t u r e o f an e n v i r o n m e n t and neither needs, nor uses, any feedback a b o u t whether inferences are right or wrong.

The Minimalist Algorithm

Can reasonably a c c u r a t e inferences be achieved with even less knowledge? W h a t we call the Minimalist a l g o r i t h m needs neither i n f o r m a t i o n a b o u t the r a n k ordering o f cue validities n o r the d i s c r i m i n a t i o n history o f the cues. In its ignorance, the a l g o r i t h m picks cues in a r a n d o m order. The a l g o r i t h m differs from the Take The Best a l g o r i t h m in Step 2, which is now re- f o r m u l a t e d as Step 2":

Step 2": Random Search

For the two objects, retrieve the cue values o f a r a n d o m l y chosen cue.

The M i n i m a l i s t a l g o r i t h m does not necessarily speed u p search, b u t it tries to get by with even less knowledge t h a n any other algorithm.

Results

Speed

How fast are the fast algorithms? The simulations showed t h a t for each o f the two variant algorithms, the relationship be- tween a m o u n t o f knowledge and the n u m b e r o f cue values looked up had the same f o r m as for the Take The Best a l g o r i t h m ( F i g u r e 5). T h a t is, unlike the integration algorithms, the curves are concave and the n u m b e r o f cues searched for is m a x - i m a l when knowledge o f cue values is lowest. The average n u m - ber o f cue values looked u p was lowest for the Take The Last a l g o r i t h m (5.29) followed by the M i n i m a l i s t a l g o r i t h m (5.64) and the Take The Best a l g o r i t h m (5.91). As knowledge be- comes m o r e and m o r e l i m i t e d ( o n b o t h dimensions: recogni- tion and cue values k n o w n ) , the difference in speed becomes

smaller and smaller. The reason why the M i n i m a l i s t a l g o r i t h m looks up fewer cue values than the Take The Best algorithm is t h a t cue validities and cue d i s c r i m i n a t i o n rates are negatively correlated (Table 1 ); therefore, r a n d o m l y chosen cues tend to have larger d i s c r i m i n a t i o n rates t h a n cues chosen by cue validity.

Accuracy

W h a t is the price to be p a i d for speeding u p search or reduc- ing the knowledge o f cue orderings and d i s c r i m i n a t i o n histories to nothing? We tested the p e r f o r m a n c e o f the two a l g o r i t h m s on the same e n v i r o n m e n t as all other algorithms. Figure 7 shows the p r o p o r t i o n o f correct inferences t h a t the M i n i m a l i s t algo- r i t h m achieved. For comparison, the p e r f o r m a n c e o f the Take The Best algorithm with 100% o f cue values known is indicated b y a d o t t e d line. N o t e that the M i n i m a l i s t algorithm p e r f o r m e d surprisingly well. The m a x i m u m difference a p p e a r e d when knowledge was c o m p l e t e and all cities were recognized. I n these circumstances, the M i n i m a l i s t a l g o r i t h m d i d a b o u t 4 percent- age points worse than the Take The Best algorithm. O n average, the p r o p o r t i o n o f correct inferences was only 1.1 percentage points less t h a n the best algorithms in the c o m p e t i t i o n (Ta- ble 2).

The p e r f o r m a n c e o f the Take The Last a l g o r i t h m is similar to Figure 7, and the average n u m b e r o f correct inferences is shown in Table 2. The Take The Last a l g o r i t h m was faster b u t scored slightly less than the M i n i m a l i s t algorithm. The Take The Last algorithm has an interesting ability, which fooled us in an earlier series o f tests, where we used a systematic (as opposed to a r a n - d o m ) m e t h o d for presenting the test pairs, starting with the largest city and pairing it with all others, and so on. A n integra- tion algorithm such as multiple regression c a n n o t "find o u t " t h a t it is being tested in this systematic way, and its inferences are accordingly independent o f the sequence o f presentation. However, the Take The Last a l g o r i t h m found out and won this first r o u n d o f the competition, o u t p e r f o r m i n g the other c o m - petitors b y some 10 percentage points. How d i d it exploit sys- tematic testing? Recall that it tries, first, the cue t h a t discrimi- n a t e d the last time. I f this cue does n o t d i s c r i m i n a t e , it proceeds with the cue t h a t d i s c r i m i n a t e d the t i m e before, and so on. In doing so, when testing is systematic in the way described, it tends to find, for each city t h a t is being p a i r e d with all smaller ones, the group o f cues for which the larger city has a positive value. Trying these cues first increases the chances o f finding a d i s c r i m i n a t i n g cue t h a t points in the right direction (toward the larger c i t y ) . We learned our lesson and reran the whole compe- tition with r a n d o m l y ordered o f pairs o f cities.

D i s c u s s i o n

The c o m p e t i t i o n showed a surprising result: The Take The Best a l g o r i t h m drew as m a n y correct inferences a b o u t un- known features o f a real-world e n v i r o n m e n t as any o f the inte- gration algorithms, and m o r e t h a n some o f them. Two further simplifications o f the a l g o r i t h m - - t h e Take The Last a l g o r i t h m (replacing knowledge a b o u t the r a n k orders o f cue validities by a m e m o r y o f the d i s c r i m i n a t i o n history o f cues) and the Mini- malist a l g o r i t h m (dispensing with b o t h ) showed a c o m p a r a -

662 GIGERENZER AND GOLDSTEIN

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Figure 7. Performance of the Minimalist algorithm. For comparison, the performance of the Take The Best algorithm (TTB) is shown as a dotted line, for the case in which 100% of cue values are known,

tively small loss in correct inferences, and only when knowledge about cue values was high.

To the best of our knowledge, this is the first inference com- petition between satisficing and "rational" algorithms in a real- world environment. The result is of importance for encouraging research that focuses on the power of simple psychological mechanisms, that is, on the design and testing of satisficing al- gorithms. The result is also of importance as an existence proof that cognitive algorithms capable of successful performance in a real-world e n v i r o n m e n t do not need to satisfy the classical n o r m s of rational inference. The classical n o r m s may be suffi- cient b u t are not necessary for good inference in real environments.

Cognitive Algorithms That Satisfice

In this section, we discuss the fundamental psychological mechanism postulated by the P M M family of algorithms: one- reason decision making. We discuss how this mechanism ex- ploits the structure of environments in making fast inferences that differ from those arising from standard models of rational reasoning.

One-Reason Decision M a k i n g

What we call one-reason decision making is a specific form of satisficing. The inference, or decision, is based on a single, good reason. There is no compensation between cues. One-reason decision making is probably the most challenging feature of the

PMM family of algorithms. As we mentioned before, it is a de- sign feature of an algorithm that is not present in those models that depict h u m a n inference as an optimal integration of all in- formation available (implying that all information has been looked up in the first place), including linear multiple regres- sion and nonlinear neural networks. One-reason decision mak- ing means that each choice is based exclusively on one reason (i.e., cue), b u t this reason may be different from decision to decision. This allows for highly context-sensitive modeling of choice. One-reason decision making is n o t compensatory. Com- pensation is, after all, the cornerstone of classical rationality, assuming that all commodities can be compared and everything has its price. Compensation assumes commensurability. How- ever, h u m a n minds do not trade everything, some things are supposed to be without a price (Elster, 1979). For instance, i f a person must choose between two actions that might help him or her get out of deep financial trouble, and one involves killing someone, then no a m o u n t of money or other benefits might compensate for the prospect of bloody hands. He or she takes the action that does not involve killing a person, whatever other differences exist between the two options. More generally, hier- archies of ethical and moral values are often noncompensatory: True friendship, military honors, and doctorates are supposed to be without a price.

Noncompensatory inference a l g o r i t h m s - - s u c h as lexico- graphic, conjunctive, and disjunctive r u l e s - - h a v e been dis- cussed in the literature, and some empirical evidence has been reported (e.g., Einhorn, 1970; Fishburn, 1988). The closest rel-

REASONING THE FAST AND FRUGAL WAY 663

ative to the P M M family o f satisficing a l g o r i t h m s is the lexico- graphic rule. The largest evidence for lexicographic processes seems to come from studies on decision under risk ( f o r a recent summary, see Lopes, 1995). However, despite e m p i r i c a l evi- dence, n o n c o m p e n s a t o r y lexicographic a l g o r i t h m s have often been dismissed at face value because they violate the tenets o f classical rationality ( K e e n e y & Raiffa, 1993; Lovie & Lovie, 1986). The P M M family is b o t h m o r e general and m o r e specific t h a n the lexicographic rule. It is m o r e general because only the Take The Best a l g o r i t h m uses a lexicographic p r o c e d u r e in which cues are ordered according to their validity, whereas the variant a l g o r i t h m s do not. It is m o r e specific, because several other psychological principles are integrated with the lexico- graphic rule in the Take The Best algorithm, such as the recog- nition p r i n c i p l e and the rules for confidence j u d g m e n t ( w h i c h a r e n o t dealt with in this article; see Gigerenzer et al., 1991 ).

Serious models t h a t c o m p r i s e n o n c o m p e n s a t o r y inferences are h a r d to find. O n e o f the few e x a m p l e s is in Breiman, Fried- man, Olshen, and Stone ( 1993 ), who r e p o r t e d a simple, non- c o m p e n s a t o r y a l g o r i t h m with only 3 binary, ordered cues, which classified h e a r t attack patients into high- and low-risk groups and was m o r e accurate t h a n s t a n d a r d statistical classi- fication methods t h a t used up to 19 variables. The practical rel- evance o f this n o n c o m p e n s a t o r y classification a l g o r i t h m is ob- vious: In the emergency r o o m , the physician can quickly o b t a i n the measures on one, two, or three variables and does n o t need to p e r f o r m any c o m p u t a t i o n s because there is no integration. T h i s group o f statisticians constructed satisficing algorithms t h a t a p p r o a c h the task o f classification ( a n d e s t i m a t i o n ) m u c h like the Take The Best a l g o r i t h m handles two-alternative choice. Relevance t h e o r y (Sperber, Cara, & G i r o t t o , 1995 ) pos- tulates t h a t people generate consequences from rules according to accessibility and stop this process when expectations o f rele- vance are met. Although relevance t h e o r y has n o t been as for- malized, we see its stopping rule as parallel to t h a t o f the Take The Best algorithm. Finally, o p t i m a l i t y t h e o r y (Legendre, Ray- mond, & Smolensky, 1993; Prince & Smolensky, 1991) p r o - poses t h a t hierarchical n o n c o m p e n s a t i o n explains how the g r a m m a r o f a language d e t e r m i n e s which structural description o f an i n p u t best satisfies well-formedness constraints. O p t i m a l - ity t h e o r y (which is actually a satisficing t h e o r y ) applies the same inferential p r i n c i p l e s as P M M t h e o r y to phonology and morphology.

Recognition Principle

The recognition p r i n c i p l e is a version o f one-reason decision m a k i n g that exploits a lack o f knowledge. The very fact t h a t one does n o t know is used to m a k e a c c u r a t e inferences. The recognition p r i n c i p l e is an intuitively plausible p r i n c i p l e t h a t seems n o t to have been used until now in models o f b o u n d e d rationality. However, it has long been used to g o o d advantage by h u m a n s and other animals. F o r instance, advertisement tech- niques as recently used by Benetton p u t all effort i n t o m a k i n g sure t h a t every c u s t o m e r recognizes the b r a n d name, with no effort m a d e to i n f o r m a b o u t the p r o d u c t itself. The idea behind this is t h a t recognition is a strong force in customers' choices. O n e o f o u r d e a r ( a n d well-read) colleagues, after seeing a draft o f this article, e x p l a i n e d to us how he makes inferences a b o u t which b o o k s are worth acquiring. I f he finds a b o o k a b o u t a

great topic b u t does n o t recognize the n a m e o f the author, he makes the inference that it is p r o b a b l y not worth buying. If, after an inspection o f the references, he does n o t recognize most o f the names, he concludes the b o o k is n o t even worth reading. The recognition p r i n c i p l e is also k n o w n as one o f the rules t h a t guide food preferences in animals. For instance, rats choose the food t h a t they recognize having eaten before ( o r having smelled on the b r e a t h o f fellow rats) and avoid novel foods (Gallistel, Brown, Carey, G e l m a n , & Keil, 1991 ).

The e m p i r i c a l validity o f the recognition p r i n c i p l e for infer- ences a b o u t u n k n o w n city populations, as used i n the present simulations, can be directly tested in several ways. First, partic- ipants are presented pairs o f cities, among t h e m critical pairs in which one city is recognized and the other unrecognized, and their task is to infer which one has m o r e inhabitants. The rec- ognition p r i n c i p l e predicts the recognized city. In o u r e m p i r i c a l tests, p a r t i c i p a n t s followed the recognition p r i n c i p l e in roughly 90% to 100% o f all cases ( G o l d s t e i n , 1994; G o l d s t e i n & Giger- enzer, 1996). Second, p a r t i c i p a n t s are taught a cue, its ecologi- cal validity, and the cue values for some o f the objects (such as whether a city has a soccer t e a m or n o t ) . Subsequently, they are tested on critical pairs o f cities, one recognized and one unrec- ognized, where the recognized city has a negative cue value (which indicates lower p o p u l a t i o n ) . The second test is a h a r d e r test for the recognition p r i n c i p l e t h a n the first one and can be m a d e even harder by using m o r e cues with negative cue values for the recognized object, and by other means. Tests o f the sec- ond k i n d hax;e been performed, and p a r t i c i p a n t s still followed the recognition principle m o r e t h a n 90% o f the time, providing evidence for its e m p i r i c a l validity (Goldstein, 1994; G o l d s t e i n & Gigerenzer, 1996).

The recognition principle is a useful heuristic in d o m a i n s where recognition is a p r e d i c t o r o f a target variable, such as whether a food contains a toxic substance. I n cases where rec- ognition does not predict the target, the P M M algorithms can still p e r f o r m the inference, b u t without the recognition princi- ple (i.e., Step 1 is canceled).

Limited Search

Both one-reason decision m a k i n g and the recognition princi- ple realize l i m i t e d search b y defining stopping points. Integra- tion algorithms, in contrast, do not provide any m o d e l o f stop- ping p o i n t s and implicitly assume exhaustive search (although they may provide rules for tossing out some o f the variables in a lengthy regression e q u a t i o n ) . Stopping rules are crucial for modeling inference under limited time, as in S i m o n ' s e x a m p l e s o f satisficing, where search among alternatives t e r m i n a t e s when a certain aspiration level is met.

Nonlinearity

L i n e a r i t y is a m a t h e m a t i c a l l y convenient tool t h a t has d o m i - n a t e d the t h e o r y o f rational choice since its inception in the mid-seventeenth c e n t u r y (Gigerenzer et al., 1989). The as- s u m p t i o n is t h a t the various c o m p o n e n t s o f an alternative a d d up independently to its overall estimate or utility. In contrast, n o n l i n e a r inference does n o t operate b y c o m p u t i n g linear s u m s o f (weighted) cue values. N o n l i n e a r inference has m a n y variet- ies, including simple principles such as in the conjunctive and

664 GIGERENZER AND GOLDSTEIN

disjunctive algorithms (Einhorn, 1970) and highly complex ones such as in n o n l i n e a r multiple regression and neural net- works. The Take The Best algorithm and its variants belong to the family of simple n o n l i n e a r models. One advantage of simple n o n l i n e a r models is transparency; every step in the P M M algo- rithms can be followed through, unlike fully connected neural networks with numerous hidden units and other free parameters.

O u r competition revealed that the unit-weight and weighted versions of the linear models lead to about equal performance, consistent with the finding that the choice of weights, provided the sign is correct, does often n o t matter much (Dawes, 1979). In real-world domains, such as in the prediction of sudden in- fant death from a linear combination of eight variables (Carpenter, Gardner, McWeeny & Emery, 1977), the weights can be varied across a broad range without decreasing predic- tive accuracy: a phenomenon known as the "fiat m a x i m u m effect" (Lovie & Lovie, 1986; von Winterfeldt & Edwards, 1982). The competition in addition, showed that the fiat maxi- m u m effect extends to tallying, with unit-weight and weighted tallying performing about equally well. The performance of the Take The Best algorithm showed that the fiat m a x i m u m can extend beyond linear models: Inferences based solely on the best cue can be as accurate as any weighted or unit-weight linear combination ofaU cues.

Most research in psychology and economics has preferred linear models for description, prediction, and prescription (Edwards, 1954, 1962; Lopes, 1994; von Winterfeldt & Ed- wards, 1982). Historically, linear models such as analysis of variance and multiple regression originated as tools for data analysis in psychological laboratories and were subsequently projected by means of the "tools-to-theories heuristic" into the- ories of m i n d (Gigerenzer, 1991 ). The sufficiently good fit of linear models in m a n y j u d g m e n t studies has been interpreted that h u m a n s in fact might combine cues in a linear fashion. However, whether this can be taken to mean that h u m a n s actu- ally use linear models is controversial ( H a m m o n d & Summers, 1965; H a m m o n d & Wascoe, 1980). For instance, within a cer- tain range, data generated from the ( n o n l i n e a r ) law of falling bodies can be fitted well by a linear regression. For the data in the Appendix, a multiple linear regression resulted in R 2 = .87, which means that a linear combination of the cues can predict the target variable quite well. But the simpler, nonlinear, Take The Best algorithm could match this performance. Thus, good fit of a linear model does not rule out simpler models of inference.

Shepard (1967) reviewed the empirical evidence for the claim that h u m a n s integrate information by linear models. He distinguished between the perceptual transformation of raw sensory inputs into conceptual objects and properties and the subsequent inference based on conceptual knowledge. He con- cluded that the perceptual analysis integrates the responses of the vast n u m b e r of receptive elements into concepts and prop- erties by complex n o n l i n e a r rules b u t once this is done, "there is little evidence that they can in t u r n be juggled and recom- bined with anything like this facility" (Shepard, 1967, p. 263 ). Although our minds can take account of a host of different fac- tors, and although we can remember and report doing so, "it is seldom more than one or two that we consider at any one time" (Shepard, 1967, p. 267). In Shepard's view, there is little evi-

C u e 1

C u e 2

C u e 3

a b c + - ?

Figure 8. Limited knowledge and a stricter discrimination rule can produce intransitive inferences.

dence for integration, linear or otherwise, in what we term in- ferences from memory--even without constraints of limited time and knowledge. A further kind of evidence does not sup- port linear integration as a model of memory-based inference. People often have great difiiculties in handling correlations be- tween cues (e.g., Armelius & Armelius, 1974), whereas inte- gration models such as multiple regression need to handle in- tercorrelations. To summarize, for memory-based inference, there seems to be little empirical evidence for the view of the m i n d as a Laplacean D e m o n equipped with the computational powers to perform multiple regressions. But this need n o t be taken as bad news. The beauty of the n o n l i n e a r satisficing algo- rithms is that they can match the D e m o n ' s performance with less searching, less knowledge, and less computational might.

Intransitivity

Transitivity is a cornerstone of classical rationality. It is one of the few tenets that the Anglo-American school of Ramsey and Savage shares with the competing Franco-European school of AUais (Fishburn, 1991 ). If we prefer a to b and b to c, then we should also prefer a to c. The linear algorithms in our com- petition always produce transitive inferences (except for ties, where the algorithm randomly guessed), and city populations are, in fact, transitive. The P M M family of algorithms includes algorithms that do not violate transitivity (such as the Take The Best algorithm), and others that do (e.g., the Minimalist algorithm). The Minimalist algorithm randomly selects a cue on which to base the inference, therefore intransitivities can re- suit. Table 2 shows that in spite of these intransitivities, overall performance of the algorithm is only about 1 percentage point lower than that of the best transitive algorithms and a few per- centage points better than some transitive algorithms.

An organism that used the Take The Best algorithm with a stricter discrimination rule (actually, the original version found in Gigerenzer et al., 1991 ) could also be forced into making intransitive inferences. The stricter discrimination rule is that search is only terminated when one positive and one negative cue value ( b u t n o t one positive and one u n k n o w n cue value) are encountered. Figure 8 illustrates a state of knowledge in which this stricter discrimination rule gives the result that a dominates b, b dominates c, and c dominates a.7

7 Note that missing knowledge is necessary for intransitivities to oc- cur. If all cue values are known, no intransitive inferences can possibly result. The algorithm with the stricter discrimination rule allows precise predictions about the occurrence of intransitivitics over the course of knowledge acquisition. For instance, imagine a person whose knowl- edge is described by Figure 8, except that she does not know the value of Cue 2 for object c. This person would make no intransitive judgments

REASONING THE FAST AND FRUGAL WAY 665

Biological systems, for instance, can exhibit systematic in- transitivities based on i n c o m m e n s u r a b i l i t y between two sys- t e m s on one d i m e n s i o n ( G i l p i n , 1975; Lewontin, 1968 ). Imag- ine three species: a , b, a n d c. Species a i n h a b i t s b o t h water and land; species b i n h a b i t s b o t h water and air. Therefor e, the two only c o m p e t e in water, where species a defeats species b. Species c i n h a b i t s land and air, so it only competes with b in the air, where it is defeated by b. Finally, when a and c meet, it is only on land, and here, c is in its element and defeats a . A linear m o d e l that estimates some value for the combative strength o f each species independently o f the species with which it is c o m - peting would fail to c a p t u r e this nontransitive cycle.

Inferences Without Estimation

Einhorn and H o g a r t h ( i 9 7 5 ) noted t h a t in the unit-weight m o d e l " t h e r e is essentially no estimation involved in its use" (p. 177), except for the sign o f the u n i t weight. A similar result holds for the algorithms r e p o r t e d here. The Take The Best algo- r i t h m does not need to estimate regression weights, it only needs to estimate a r a n k ordering o f ecological validities. The Take The Last and the M i n i m a l i s t a l g o r i t h m s involve essen- tially no estimation (except for the sign o f the cues). The fact t h a t there is no estimation p r o b l e m has an i m p o r t a n t conse- quence: A n organism can use as m a n y cues as it has experi- enced, without being concerned a b o u t whether the size o f the s a m p l e experienced is sufficiently large to generate reliable esti- mates o f weights.

Cue Redundancy and Performance

Einhorn and H o g a r t h ( 1 9 7 5 ) suggested t h a t unit-weight models can be expected to p e r f o r m a p p r o x i m a t e l y as well as p r o p e r linear models when ( a ) R 2 from the regression m o d e l is in the m o d e r a t e or low range ( a r o u n d .5 or smaller) and ( b ) predictors ( c u e s ) are correlated. Are these two criteria neces- sary, sufficient, or b o t h to explain the p e r f o r m a n c e o f the Take The Best algorithm? The Take The Best a l g o r i t h m and its vari- ants certainly can exploit cue redundancy: I f cues are highly correlated, one cue can do the j o b .

We have a l r e a d y seen t h a t in the present environment, R z = .87, which is in the high rather t h a n the m o d e r a t e nr lnw range. As m e n t i o n e d earlier, the pairwise correlations between the nine ecological cues ranged between - . 2 5 and .54, with an ab- solute average value o f .19. Thus, despite a high R 2 and only m o d e r a t e - t o - s m a l l correlation between cues, the satisficing al- g o r i t h m s p e r f o r m e d quite successfully. T h e i r excellent perfor- m a n c e in the c o m p e t i t i o n can be e x p l a i n e d only p a r t i a l l y by cue redundancy, because the cues were only m o d e r a t e l y correlated. High cue redundancy, thus, does seem sufficient b u t is not nec-

comparing objects a, b, and c. If she were to learn that object c had a negative cue value for Cue 2, she would produce an intransitive judg- ment. If she learned one piece more, namely, the value of Cue 1 for object c, then she would no longer produce an intransitive judgment. The prediction is that transitive judgments should turn into intransitive ones and back, during learning. Thus, intransitivities do not simply de- pend on the amount of limited knowledge but also on what knowledge is missing.

essary for the successful p e r f o r m a n c e o f the satisficing algorithms.

A New Perspective on the Lens Model

Ecological theorists such as Brunswik (1955) emphasized that the cognitive system is designed to find m a n y pathways to the world, substituting missing cues by whatever cues h a p p e n to be available. Brunswik labeled this ability vicarious functioning, in which he saw the m o s t f u n d a m e n t a l p r i n c i p l e o f a science o f perception and cognition. His proposal to model this adaptive process by linear multiple regression has inspired a long t r a d i - tion o f neo-Brunswikian research (B. Brehmer, 1994; H a m - mond, 1990), although the e m p i r i c a l evidence for mental multiple regression is still controversial (e.g., A. B r e h m e r & B. Brehmer, 1988). However, vicarious functioning need n o t be equated with linear regression. The P M M family o f a l g o r i t h m s provides an alternative, nonadditive m o d e l o f vicarious func- tioning, in which cue substitution operates without integration. This gives a new perspective o f B r u n s w i k ' s lens model. In a one- reason decision making lens, the first d i s c r i m i n a t i n g cue t h a t passes through inhibits any other rays passing through and de- t e r m i n e s j u d g m e n t . N o n c o m p e n s a t o r y vicarious functioning is consistent with some o f Brunswik's original examples, such as the substitution o f behaviors in Hull's h a b i t - f a m i l y hierarchy, and the alternative manifestation o f s y m p t o m s according to the psychoanalytic writings o f Frenkel-Brunswik (see Gigerenzer & Murray, 1987, chap. 3).

It has been r e p o r t e d sometimes that teachers, physicians, and other professionals claim that they use seven or so criteria to make j u d g m e n t s (e.g., when grading papers or making a differ- ential diagnosis) b u t t h a t experimental tests showed t h a t they in fact often used only one criterion (Shepard, 1967). A t first glance, this seems to indicate t h a t those professionals m a k e out- rageous claims. But it need n o t be. I f experts' vicarious func- tioning works according to the P M M algorithms, then they are correct in saying t h a t they use m a n y predictors, b u t the decision is m a d e by only one at any time.

What Counts as Good Reasoning?

M u c h o f the research on reasoning in the last decades has assumed that sound reasoning can be r e d u c e d to principles o f internal consistency, such as additivity o f probabilities, confor- m i t y to t r u t h - t a b l e logic, and transitivity. For instance, research on the Wason selection task, the " L i n d a " p r o b l e m , and the " c a b " p r o b l e m has evaluated reasoning a l m o s t exclusively b y some measure o f internal consistency (Gigerenzer, 1995, 1996a). Cognitive algorithms, however, need to meet m o r e im- p o r t a n t constraints t h a n internal consistency: ( a ) They need to be psychologically plausible, ( b ) they need to be fast, and ( c ) they need to m a k e a c c u r a t e inferences in real-world environ- ments. In real t i m e and real environments, the possibility t h a t an a l g o r i t h m (e.g., the M i n i m a l i s t a l g o r i t h m ) can make intran- sitive inferences does not m e a n t h a t it will m a k e t h e m all the t i m e or that this feature o f the a l g o r i t h m will significantly h u r t its accuracy. W h a t we have not addressed in this article are con- straints on h u m a n reasoning t h a t emerge from the fact t h a t Homo sapiens is a social animal (Gigerenzer, 1996b). For in- stance, some choices (e.g., who to treat first in an emergency

6 6 6 GIGERENZER AND GOLDSTEIN

room) need to be justified (Tetlock, 1992 ). Going with the sin- gle best reason, the strategy of the Take The Best algorithm, has an immediate appeal for justification and can be more convinc- ing and certainly easier to communicate than some complicated weighting of cues.

F u r t h e r R e s e a r c h

Among the questions that need to be addressed in future re- search are the following. First, how can we generalize the pres- ent satisficing algorithm from two-alternative-choice tasks to other inferential tasks, such as classification and estimation? The reported success of the classification and regression tree models (Breiman et al., 1993 ), which use a form of one-reason decision making, is an encouraging sign that what we have shown here for two-alternative-choice tasks might be generaliz- able. Second, what is the structure of real-world environments that allows simple algorithms to perform so well? We need to develop a conceptual language that can capture i m p o r t a n t as- pects of the structure of environments that simple cognitive al- gorithms can exploit. The traditional proposal for understand- ing the structure of environments in terms of ecological validi- ties defined as linear correlations (Brunswik, 1955 ) may not be adequate, as the power of the n o n l i n e a r satisficing algorithms suggests.

C a n R e a s o n i n g B e R a t i o n a l a n d P s y c h o l o g i c a l ?

At the beginning of this article, we pointed out the c o m m o n opposition between the rational and the psychological, which emerged in the nineteenth century after the breakdown of the classical interpretation of probability (Gigerenzer et al., 1989). Since then, rational inference is commonly reduced to logic and probability theory, and psychological explanations are called on when things go wrong. This division of labor is, in a nutshell, the basis on which much of the current research on j u d g m e n t under uncertainty is built. As one economist from the Massa- chusetts Institute of Technology put it, "either reasoning is ra- tional or it's psychological" (Gigerenzer, 1994). Can not rea- soning be both rational and psychological?

We believe that after 40 years of toying with the notion of bounded rationality, it is time to overcome the opposition be- tween the rational and the psychological and to reunite the two. The P M M family of cognitive algorithms provides precise models that attempt to do so. They differ from the Enlighten- ment's unified view of the rational and psychological, in that they focus on simple psychological mechanisms that operate under constraints of limited time and knowledge and are sup- ported by empirical evidence. The single most i m p o r t a n t result in this article is that simple psychological mechanisms can yield about as many (or more) correct inferences in less time than standard statistical linear models that embody classical proper- ties of rational inference. The demonstration that a fast and fru- gal satisficing algorithm won the competition defeats the wide- spread view that only "rational" algorithms can be accurate. Models of inference do not have to forsake accuracy for sim- plicity. The m i n d can have it both ways.

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Appendix

The Environment

S o c c e r S t a t e F o r m e r E a s t I n d u s t r i a l L i c e n c e I n t e r c i t y E x p o s i t i o n N a t i o n a l

C i t y P o p u l a t i o n t e a m c a p i t a l G e r m a n y b e l t p l a t e t r a i n l i n e s i t e c a p i t a l U n i v e r s i t y

B e r l i n 3 , 4 3 3 , 6 9 5 - + - - + + + + +

H a m b u r g 1 , 6 5 2 , 3 6 3 + + - - - + + - +

M u n i c h 1 , 2 2 9 , 0 2 6 + + - - + + + - +

C o l o g n e 9 5 3 , 5 5 1 + - - - + + + - +

F r a n k f u r t 6 4 4 , 8 6 5 + - - - + + + - +

E s s e n 6 2 6 , 9 7 3 - - - + + + + - +

D o r t m u n d 5 9 9 , 0 5 5 + - - + - + + - +

S t u t t g a r t 5 7 9 , 9 8 8 + + - - + + + - +

D i J s s e l d o r f 5 7 5 , 7 9 4 - + - - + + + - +

B r e m e n 5 5 1 , 2 1 9 + + - - - + - - +

D u i s b u r g 5 3 5 , 4 4 7 - - - + - + - - +

H a n n o v e r 5 1 3 , 0 1 0 - + - - + + + - +

L e i p z i g 5 1 1 , 0 7 9 - - + - + + + - +

N u r e m b e r g 4 9 3 , 6 9 2 + - - - + + + - +

D r e s d e n 4 9 0 , 5 7 1 + - * + - - + - - +

B o c h u m 3 9 6 , 4 8 6 + - - + - + - - +

W u p p e r t a l 3 8 3 , 6 6 0 - - - + + + - - +

B i e l e f e l d 3 1 9 , 0 3 7 . . . . . + - - +

M a n n h e i m 3 1 0 , 4 1 1 . . . . . + - - +

H a l l e 3 1 0 , 2 3 4 - - + - - + - - -

C h e m n i t z 2 9 4 , 2 4 4 - - + - + . . . .

G e l s e n k i r c h e n 2 9 3 , 7 1 4 + - - + - + - - -

B o n n 2 9 2 , 2 3 4 . . . . + - - +

M a g d e b u r g 2 7 8 , 8 0 7 - + + - - + - - -

K a r l s r u h e 2 7 5 , 0 6 1 + . . . . + - - -

W i e s b a d e n 2 6 0 , 3 0 1 - + - - - + - - -

M f i n s t e r 2 5 9 , 4 3 8 . . . . . + - - +

M 6 n c h e n g l a d b a c h 2 5 9 , 4 3 6 + . . . . . . . .

B r a u n s c h w e i g 2 5 8 , 8 3 3 . . . . . + - - +

A u g s b u r g 2 5 6 , 8 7 7 . . . . + + - - +

R o s t o c k 2 4 8 , 0 8 8 - - + - - + - - -

K i e l 2 4 5 , 5 6 7 - + - - - + - - +

K r e f e l d 2 4 4 , 0 2 0 - * . . . . . . . .

A a c h e n 2 4 1 , 9 6 1 . . . . . + - - +

O b e r h a u s e n 2 2 3 , 8 4 0 - - - + - + - - -

L / i b e c k 2 1 4 , 7 5 8 . . . . . + - - -

H a g e n 2 1 4 , 4 4 9 - - - + - + - - -

E r f u r t 2 0 8 , 9 8 9 - + + - - + - - -

K a s s e l 1 9 4 , 2 6 8 . . . . . + - - +

S a a r b r i J c k e n 1 9 1 , 6 9 4 + + - - - + + - +

R E A S O N I N G T H E F A S T A N D F R U G A L W A Y 669

S o c c e r S t a t e F o r m e r E a s t I n d u s t r i a l L i c e n c e I n t e r c i t y E x p o s i t i o n N a t i o n a l

C i t y P o p u l a t i o n t e a m c a p i t a l G e r m a n y b e l t p l a t e t r a i n l i n e s i t e c a p i t a l U n i v e r s i t y

F r e i b u r g 1 9 1 , 0 2 9 . . . . . . + - - +

H a m m 1 7 9 , 6 3 9 - - - + - + - - -

M a i n z 1 7 9 , 4 8 6 - + - - - + - - +

H e r n e 1 7 8 , 1 3 2 - - - + . . . . .

M / i l h e i m 1 7 7 , 6 8 1 - - - + . . . . .

S o l i n g e n 1 6 5 , 4 0 1 . . . . . . + - - -

O s n a b r i i c k 1 6 3 , 1 6 8 . . . . . + - - +

L u d w i g s h a f e n 1 6 2 , 1 7 3 . . . . . + - - -

L e v e r k u s e n 1 6 0 , 9 1 9 + . . . . . . . .

N e u s s 1 4 7 , 0 1 9 . . . . . . . . .

O l d e n b u r g 1 4 3 , 1 3 1 . . . . . + - - +

P o t s d a m 1 3 9 , 7 9 4 - + + - + + - - -

D a r m s t a d t 1 3 8 , 9 2 0 . . . . . + - - +

H e i d e l b e r g 1 3 6 , 7 9 6 . . . . . + - - +

B r e m e r h a v e n 1 3 0 , 4 4 6 . . . . . + - - -

G e r a 1 2 9 , 0 3 7 - - + - + + - - -

W o l f s b u r g 1 2 8 , 5 1 0 . . . . . . . . .

W i i r z b u r g 1 2 7 , 7 7 7 . . . . . + - - +

S c h w e r i n 1 2 7 , 4 4 7 - + + - - + - - -

C o t t b u s 1 2 5 , 8 9 1 - - + . . . . . .

R e c k l i n g h a u s e n 1 2 5 , 0 6 0 - - - + - + - - -

R e m s c h e i d 1 2 3 , 1 5 5 . . . . . . . . .

G S t t i n g e n 1 2 1 , 8 3 1 . . . . . + - - +

R e g e n s b u r g 1 2 1 , 6 9 1 . . . . + + - - +

P a d e r b o r n 1 2 0 , 6 8 0 . . . . . . . . . +

B o t t r o p 1 1 8 , 9 3 6 - - - + . . . . .

H e i l b r o n n 1 1 5 , 8 4 3 . . . . . . . . .

O f f e n b a c h 1 1 4 , 9 9 2 . . . . . . + - -

Z w i c k a u 1 1 4 , 6 3 6 - - + - + . . . .

S a l z g i t t e r 1 1 4 , 3 5 5 . . . . . . . . .

P f o r z h e i m 1 1 2 , 9 4 4 . . . . . + - - -

U l m 1 1 0 , 5 2 9 . . . . . + - - +

S i e g e n 1 0 9 , 1 7 4 . . . . . . . . +

K o b l e n z 1 0 8 , 7 3 3 . . . . . + - - +

J e n a 1 0 5 , 5 1 8 - - + - + + - - +

l n g o l s t a d t 1 0 5 , 4 8 9 . . . . . + - - -

W i t t e n 1 0 5 , 4 0 3 - - - + . . . . .

H i l d e s h e i m 1 0 5 , 2 9 1 . . . . . + - - +

M o e r s 1 0 4 , 5 9 5 - - - + . . . . .

B e r g i s c h G l a d b a c h 1 0 4 , 0 3 7 . . . . . . . . .

R e u t l i n g e n 1 0 3 , 6 8 7 . . . . . . . . .

F i i r t h 1 0 3 , 3 6 2 . . . . . + - - -

E r l a n g e n 1 0 2 , 4 4 0 . . . . . + - - +

Note. C i t y p o p u l a t i o n s w e r e t a k e n f r o m Fischer Welt Almanach ( 1 9 9 3 ) . * T h e t w o s t a r r e d m i n u s v a l u e s a r e , i n r e a l i t y , p l u s v a l u e s . B e c a u s e o f t r a n s c r i p t i o n e r r o r s , w e r a n a l l s i m u l a t i o n s w i t h t h e s e t w o m i n u s v a l u e s . T h e s e

d o n o t a f f e c t t h e r a n k o r d e r o f c u e v a l i d i t i e s , s h o u l d n o t h a v e a n y n o t i c e a b l e e f f e c t o n t h e r e s u l t s , a n d a r e i r r e l e v a n t f o r o u r t h e o r e t i c a l a r g u m e n t .

R e c e i v e d M a y 2 0 , 1 9 9 5

R e v i s i o n r e c e i v e d D e c e m b e r 4 , 1 9 9 5

A c c e p t e d D e c e m b e r 8 , 1 9 9 5 •