Module 4
Information, Theory, Conflict
a. Perfect and Imperfect Information
Such games have perfect information: Each player, when deciding his move, must
have complete information about the current position of the board (I include in "position"
qualities that may be physically undetectable, such as whether a player may castle), or
equivalently, about the original position of the board and all moves made so far.
Examples of perfect information games would include Chess and Backgammon; games
like Stratego, Kriegspiel, or the recent Stealth Chess are not perfect information games.—
J. Mark Thompson, "Defining the Abstract"
Parlett describes card games as games of "imperfect information," due to the fact
that the two-sided nature of cards permits them to hide their informational value. In the
quote above, mathematician and game aficionado J. Mark Thompson points out the
opposite kind of system: games of "perfect information." Perfect and imperfect
information refer to the relationship a player has to the information contained in the
formal system of the game.
Perfect information exists in a game when all players have complete knowledge
about every element in the game at all times. Thompson includes Chess and
Backgammon in this category. In games of imperfect information, some of the
information may be hidden from players during the game. Thompson names Stratego, a
game in which the value of each piece on the gameboard is kept hidden from each
player's opponent, as a game of imperfect information. Card games are also good
examples of games of imperfect information. Although it would be possible to design a
card game in which all of the cards are visible for the entire game, almost every card
game does feature imperfect information in some form.
Games of perfect and imperfect information can both provide meaningful play,
but they do so in different ways. Games of perfect information tend to be more
analytically competitive games, in which players are pitted directly against each other,
each player's moves and strategies available for the other to see. Games of imperfect
information add an element of mystery and uncertainty to a game. Imperfect information
invites treachery, trickery, and deception, and can be used as a design element in games
meant to inspire mistrust among players.
Although imperfect information can heighten a feeling of uncertainty in a game,
actual randomness is not intrinsically tied to either perfect or imperfect information. A
game can have chance mechanisms as part of the game and still possess either perfect or
imperfect information. As Thompson points out, both Chess and Backgammon are games
of perfect information, even though Backgammon makes use of die rolls every turn. An
imperfect information game such as Stratego does not use any kind of chance
mechanism, whereas Poker does, in the form of a shuffled deck.
Chance is not related to our definitions of perfect and imperfect information. But
whether or not a game has a random element does affect the kind of information that
exists in a game. In The Interactive Book, designer and scholar Celia Pearce presents a
different typology for understanding the ways games manifest information.
Pearce's categories offer another way of describing the informational component
of a game. She differentiates between two kinds of hidden information, one in which each
player possesses private information and another in which the game system itself hides
information from all of the players. As she illustrates through her example of Gin, any
card game with a shuffled deck and private hands for each player contains both kinds of
hidden information. Pearce also designates randomly generated information within its
own informational category.
One disagreement we have with Pearce's typology regards the inclusion of the
rules of a game in her information model. Pearce states that the rules of Chess should be
considered information known to all players. Although this is arguably the case, we don't
feel that including rules as part of the perfect information of a game is particularly useful,
since in all games, some or all of the game rules are known to all players. There may be
hidden strategic relationships in a game that are gradually uncovered (like the strengths
and weaknesses of particular unit combinations) but these are not rules. Rules are the
formal foundation of a game that allows players to manipulate information. Rules
generally do not constitute the information being manipulated during play.
There are other slightly ambiguous aspects of her model as well. For example,
although a deck of cards in Gin might contain information "known to the game only," one
could imagine a deck of cards that is reshuffled every turn, functioning in exactly the
same way as a random die roll—Pearce's "randomly generated information".Perhaps the
subtle difference between these two types of information is whether or not the exact
makeup of the remaining deck is completely known to all players. If it is, then the deck
would function like a random die roll, in which the precise chance to roll a particular
number (or draw a particular card) is known to all players. On the other hand, when
players draw cards in Gin, the face-down deck contains information known only to the
game.
Despite our critique of Pearce's model, the four categories she proposes are in fact
quite useful. For example, information is not always either public or private, and often
moves between categories. As a deck of face-down cards are exposed to players, there is
movement from once-private information to informa-tion that is public and shared.
Similarly, in Battleship, both play- ers try to uncover information that is hidden from
them, but known to their opponent. As Battleship proceeds, the positions of the players'
ships gradually become public knowledge.
b. Enchanted Information
Whether you prefer a model of perfect and imperfect informa-tion or Pearce's four
categories, it is important to consider the ways that games manipulate information to
generate mean-ingful play. Consider the game Enchanted Forest, a children's board game
that uses information in a number of ways. In Enchanted Forest, players are seeking a
number of treasures that are hidden on the board. When a game begins, a set of tree
objects are shuffled and placed in set locations on the board. On the bottom of each tree is
an image of one of the treasures in the game.There is also a deck of cards, each card
containing the image of one of the treasures. At the start of the game, the deck is shuf fled
and the top card is turned over. This is the treasure that all of the players are initially
seeking.
By rolling two dice and moving on a network of paths, playersmaneuver their
pieces to specific trees. When a player lands on atree, she can privately look at the
treasure under the tree. If a player thinks that she knows the location of the treasure
pictured on the face-up card, she can in subsequent turns travel to the castle on one end of
the board and pick up the tree where she thinks the pictured treasure is hidden. If she is
correct, she shows the tree to the other players, puts it back on the board, keeps the card,
and turns over a new card, which becomes the next treasure for all of the players to seek.
If she is not correct, she puts the tree back without revealing it to the other players and
moves her piece to the village, which is located on the far end of the board from the
castle. The first player to collect three cards and get to the castle wins.
Other special rules in the game allow a player to bump another player back to the
village by landing on his piece. Rolling doublesallows a player to optionally take a
special action instead of moving. Special actions include shuffling the card deck and
picking a new card, looking under any tree, and moving directly to the castleEnchanted
Forest manipulates game information in a variety of ways. . Let's use Pearce's four
categories to analyze the use of information in the game. Enchanted Forest contains all
four kinds of information:
Information known to all players: There are many elements of Enchanted Forest
not hidden from players. For example, the networks of paths on the boards, the locations
of the players' pieces, the cards each player has collected, and the treasure that players are
currently seeking represent information known to everyone.
Information known only to one player: The one kind of information that is known
only to one player is the loca-tions of the treasures that players secretly uncover as they
move about the forest. In most games that make use of private information, that
information is kept in a hand of cards or an otherwise concealed collection of game
components. In Enchanted Forest, however, the privately held information is something
that each player keeps in memory, adding to the challenge of using the information.
Information known to the game only: Initially, the locations of the treasures under
each tree are kept hidden from all of the players. However, as the game proceeds, the
hidden information about the treasure locations shifts subtly from this category to the
others. As they move about the board, each player begins to piece together a larger
picture of the information hidden under the trees. Sometimes, when a player correctly
guesses the location of a treasure, the information is temporarily made public.
Randomly generated information: Enchanted Forest creates random information
through a two-die roll. Players use each die separately when deciding where to move, so
that rolling a 4 and a 5 can be used to move one space (moving five in one direction and
then doubling back four spaces). These flexible rules of movement offer players many
possibilities for navigating the network of paths on the game board. Moving onto a
particular space can let a player look at a tree, enter the castle, or bump another player
back to the village. Rolling doubles also allows special actions. Even the raw, randomly
generated information of the die roll is used in a number of ways in the game.
As a game of Enchanted Forest proceeds, players gradually discover the hidden
information stored under the trees, allowing them to build their own personal system of
information and accompanying strategies. How will you navigate the board? Do you
want to seek out trees that no other player has looked at, so that you have an
informational advantage? Or perhaps you want to shadow other players and make sure
that no one else has an advantage over you. Maybe you should just look at the trees near
the castle, so that you can more quickly move there if a treasure you have seen turns up
on a card.
If each player were allowed to take notes and store the data they gather, then a
gradual accumulation of the game's hidden information would be inevitable. However,
because players are storing all of this information in their memories, it is easy to forget
exactly where a treasure is, even after they have seen it. The fact that players must
memorize the locations of the treasure under each tree forces them to focus their attention
on the game, making each guess in the castle a little bit risky. Furthermore, because
strategic use of information dominates the game, there are many ways to acquire
information, beyond looking under trees. If another player has just looked under a tree
and is now heading for the castle, is it because she has just seen the treasure that is on the
current face-up card? If you are closer to the castle, can you beat her there and make an
educated guess based on her behavior? Or is it all a bluff to get you to waste your time
making a false guess so that you will be sent back to the village?
The same kind of dilemma confronts you when you roll doubles and can take a
special action instead of moving. If another player is about to reach the castle and make a
guess, you can shuffle the deck and draw a new card. But that might only reveal a card
that the player already knows. If you think you know the tree where the treasure is, you
can teleport next to the castle. Or perhaps you just want to ignore the other player's
movement toward the castle and use your doubles to look under a tree on the far side of
the forest.
c. Each turn, Enchanted
Forest players must make tough decisions about what action to take. This is a
good sign that meaningful play is taking place. Each choice a player makes in Enchanted
Forest is bound up in the fluid flow of information, as direct knowledge and educated
hunches are balanced with strategic navigation, challenging memorization, and risky
guesswork.
Economies of Information The other informational component of Enchanted
Forest is the deck of cards that is uncovered as the game progresses. Because players
know that each card in the deck corresponds to one of the treasures, the predictability of
the identity of the next card being revealed changes as the game proceeds. At the
beginning, there is a low chance that a particular card might turn up. But near the end of
the game, when there are only two or three cards left in the deck, there is much less
uncertainty about the possible outcome. That means that using a special action to
reshuffle the information in the deck, as a way of making a particular card appear, is a
more effective strategy later in the game.
This strategic phenomenon occurs because the information contained on each card
does not exist in isolation from the others. Instead, the information is part of a larger
system: an economy of information that grants each card its relative value. Meaning in a
game, or in any system, for that matter, emerges out of relational identity. To use a non-
game example, the meaning of the word "bird" depends on the larger sentence and
performative context of use. Its relationship to other words and meanings will determine
whether "bird" references a feathered animal, the jazz musician Charlie Parker, a rudely
flipped-up middle finger, or a slang term for "woman."
In a game, the same holds true: the value of the information known to a player
gains meaning within the larger system of the game. Furthermore, the systemic nature of
information in a game and the way that it helps generate meaningful play has two facets:
the actual make-up of the information structures in a system, and the apprehension of that
information by players. We call a game's information structures objective information
and the player's understanding of these structures perceived information. The interaction
of these two aspects of game information determines the way that information operates
within the system of a game.
In a trick-taking game such as Euchre, for example, if all of the trumps have
already been played during a round, the once-private but now-public information
regarding the location of the trumps becomes extremely important for making game
decisions. The fact that any given card can be a trump or non-trump is a function of the
objective information of the game. But the systemic fact that the trumps have all been
played is only meaningful because of the player's ability to turn the objective information
into perceived information and make decisions accordingly. Good Euchre players track
not only the trump cards, but every card that is played in the game. This information
determines the potential value of the cards you are holding in your own hand, as well as
the cards you think other players will play next, directly informing your decision-making
process. Euchre contains a tightly woven system of information, in which meaningful
play emerges from the elegantly architected value of the card deck, as well as from
players' shifting certainties, speculations, and ignorance about the cards left to be played.
The game of Scrabble also contains objective informational structures that are
gradually revealed to players as perceived information over the course of play. For
example, knowledge about the ratio of the letters in the overall mix is part of how you
play the game. If you desperately need to pick a Z tile to make a 7-letter word and get a
big bonus, you need to decide whether to play a single letter, make a small score this turn,
and hold out for that Z, or else abandon the large-word strategy altogether. Your
judgment will involve not only the probability of drawing a Z (dependent on the objective
informational economy of the pool of letters), but also on perceived informa-tion:
whether or not you know that the single Z has already been played on the board. The fact
that the ratios of letters are printed on the Scrabble game board, next to the grid where
letters are placed, points to the way that the structure of Scrabble's informational system
is central to the play of the game.
d. Hiding and Revealing Systems
We conclude our discussion of games as information systems by looking at some
examples of digital games that successfully integrate information into the overall game
design. These games take advantage of the ability of digital games to manipulate
information in complex ways to engender meaningful play: he Fog of War: A wonderful
method by which digital games manipulate information is the "fog of war" feature in real-
time strategy games such as Starcraft. Although a real-time strategy game level might
take place on a large map of terrain, the game only reveals information about structures
that are gradually revealed to players as perceived the map near a player's own units.
Initially, you only know about your own local area of the playfield. Sending out scouts
and spies to learn more about the terrain and the location of your enemies becomes an
important part of the game. The strategic unveiling of hidden information adds suspense
and tension to the game experience.
Secret Locations and Hidden Moves: Many digital games possess locations and
features which are hidden to players at the beginning of a game and are slowly revealed
through play. In Super Mario 64, not only does the player slowly gain enough coins to
unlock new game worlds, but some of these worlds contain items that give Mario new
powers and abilities. Although any level-based game could be considered a system of
information revealed to players over the course of a game, Super Mario 64 masterfully
integrates the uncovering of this information into the geography of the overall game
space. Such an approach creates a rhythm of discovery that keeps players engaged
throughout the game.
Item Economies: Many digital games feature complex. economies of items. In
LEGO Drome Racing Challenge, a multiplayer online game in which players customize
their cars by purchasing parts, the game limits a player's ability to purchase car parts
based on their License Class: as a player progresses, they gain the ability to purchase
more and more powerful parts. But the game doesn't limit the informational access to all
of the parts: players can "browse" the more powerful parts, but not acquire them.
Although this is the opposite design strategy that Super Mario 64 takes, it works well in
the context of this particular design. By granting access to the entire set of part
information, this structure gives players a better understanding of the relative value of the
items available for purchase, while also allowing them to make meaningful decisions
about what to buy. Rather than purchasing a currently accessible item, a player can
choose to "save up" for a more expensive item that can only be purchased once he or she
has advanced to another License class.
Rules as Information: The automated nature of game systems, combined with
their ability to manipulate information, forms a powerful design tool: the ability to reveal
not just static information, but dynamic behaviors and relationships as well. In the game
FLUID,the player has little information at the beginning of the game experience. The
interaction consists of poking, stroking, and prodding a touchscreen, interacting with the
elements of the game. Each game element is part of a miniature ecosystem; as the player
plays with FLUID,interaction and observation reveals the underlying principles of the
system. In this case, the hidden information gradually revealed through play is the rules
of the simulation itself. Part of the play of FLUID is the discovery of the game rules as
information.
Many digital games rely on vast sets of information rewards for player
interaction. Huge worlds to explore, complex economies of items, and hidden fighting
moves are the "stuff" with which digital game designers fill their systems. One challenge
for digital game designers is not to rely so much on hidden content in order to generate
player interest. Making use of information systems within your games requires that the
information be made meaningful through player interaction. If there is too much
information, or if the information is neither discernable nor integrated, the design has
failed to support meaningful play. In contrast, non-digital games typically consist of more
limited, replayable systems: imperfect information might be a core component of Poker,
but you are not going to uncover a previously hidden card in the deck after 30 hours of
play.
The kind of information system you design, therefore, depends on the kind of
game you want to create. A simple game system that players fully know from the start
can create the obsessive play of Tetris or the opportunities for mastery of Go. On the
other hand, a game that contains large amounts of information —strategically unveiling
informational complexity, rewarding progress via new levels, powers, or narrative
segments—can let players slowly come to know the game without feeling overwhelmed.
Your challenge as a game designer is to identify the kind of experience you want to
create and design a system that finds the proper balance of meaningful informational
play.
e. Strategies in Game Theory
Decision trees help us understand how players move through the space of
possibility of a game. To see how this works, think back to the Tic-Tac-Toe decision tree.
The tree contains every conceivable move, in every possible iteration of the game. This is
actually more information than we need. Most players will not randomly pick their next
square, but will actively try and score three in a row while keeping an opponent from
doing the same. With this in mind, we can start to trim all of the "stupid move" branches
from our tree. Poundstone describes what this process of "trimming" would be like: Go
through the diagram and carefully backtrack from every leaf. Each leaf is someone's last
move, a move that creates a victory or a tie. For instance, at Point A, it is X's move, and
there is only one empty cell. X has no choice but to fill it in and create a tie.
From this pruned-down version of Tic-Tac-Toe, it is possible to create what game
theory calls a strategy. A strategy in game theory parlance offers a more precise meaning
than what is commonly meant by "strategy." A common understanding of a strategy in
Starcraft might be: "If you're playing the Zergs, create a lot of Zerglings at the beginning
of the game and rush your opponent's central structures before they have time to build
power." A strategy in this casual sense is a set of general heuristics or rules of thumb that
will help guide you as you play. However, a strategy in game theory means a complete
description of how you should act at every moment of the game. Once you select a
strategy in the game theory sense of the word, you do not make any other choices,
because the strategy already dictates how you should act for the rest of the game,
regardless of what the other player does. This can make game theory strategies quite
intricate. Poundstone lists a sample strategy for Tic-Tac-Toe for the first player X.
As you can see, the strategy for even a simple game such as Tic-Tac-Toe is
somewhat complex. A complete strategy is ultimately a methodology for navigating the
branches of a decision tree. A strategy proscribes exact actions for the player utilizing the
strategy, but it also has to take into account all of the possible branches that an opposing
player could select. In Poundstone's example, the strategy dictates the way that the first
player would move from the root of the tree to the first of nine possible points. From
there the opposing player could move to any of the other eight points, a move that the
strategy has to take into account.
f. Game Theory Games
Now that we have outlined decision trees and strategies, we are ready to take a
look at what it is that game theory calls a game. As the mathematician Richard Epstein
points out, game theory games are not about real-world situations or about all kinds of
games. Game theorists look at very particular kinds of situations in a very narrow way.
What kind of situations? We can summarize a game theory game in the following way: a
game theory game consists of rational players who simultaneously reveal a strategy to
arrive at an outcome that can be defined in a strict measure of utility. Usually, game
theory limits itself to games with only two players.
Rational play, simultaneity, strategy, outcome, utility, and two players. Let us
look at each of these elements separately. First, game theory focuses its attention on
rational players. Rational players are perfectly logical players that know everything there
is to know about a game situation. Furthermore, rational players play to win. As
Poundstone puts it, "Perfectly rational players would never miss a jump in checkers or
'fall into a trap'in Chess. All legal sequences of moves are implicit in the rules of these
games, and a perfectly logical player gives due consideration to every possibility." As we
know from our detailed investigation of Tic-Tac-Toe, if two rational players played the
game, the outcome will always end in a draw, because both players would select
strategies that would stalemate the other player. Rational players are a fiction, of course,
as Epstein makes clear. Real-world players are not like game theory players, as rational
as Mr. Spock, completely immune from "emotional, ethical, and social" liabilities. But
rational players are still a useful theoretical construct, for they allow us to look at games
in a very isolated and controlled way.
The fact that rational players follow a strategy is an important aspect of a game
theory game as well. As we mentioned previously, a strategy is comprehensive. It is a
complete plan for playing an entire game, from start to finish. A strategy includes explicit
instructions for playing against any other strategy an opponent selects. In a game theory
game, both rational players simultaneously choose and reveal their strategies to each
other. In other words, instead of the "I take my turn, you take your turn" pattern of many
games, in a game theory game, players only make one decision, at the same time, without
knowing what the other player will do. In making a simultaneous decision, a player has to
take into account not just the current state of the game, but also what the opponent is
thinking at that very moment. A classic example of a simultaneous decision game is
Rock-Paper-Scissors, in which both players have to decide what they are going to do
based on the anticipated action of the other player.
So although game theory does not study psychology directly, there is a
psychological element in game theory games, where players might consider "bluffing" or
using other indirect strategies against each other. Though they might take these kinds of
actions, rational players are still psychologically predictable. Players in a game theory
scenario are never going to be vindictive, forgetful, self-destructive, or lazy, as this would
change their status as rational players. In game theory games one can always assume that
both rational players are acting in their own best interest and are developing strategies
accordingly.
Why would game theory choose blind, simultaneous decision making as the game
play process that it studies? Remember that game theory is not a form of game design: it
is a school of economic theory. Within an economic situation, decisions have to be made
without knowledge of how the other "players" are going to act. Should you sell your
stock in Disney, or buy more shares? Should you purchase two gallons of milk this week,
or buy one and wait to see if the price goes down? Should a nation increase or decrease
import taxes? All of these micro-and macro-economic scenarios involve making
decisions. But the outcome of the decision is based on factors outside the decision
maker's direct control. Simultaneous, blind decisions offer a way of simulating this
decision making context, a context that lies at the intersection of mathematics and
psychology.
Another important component of a game theory game is utility, which is a
mathematical measure of player satisfaction. In order to make a formal theory of decision
making, it was necessary that Von Neumann and Morganstern numerically quantify the
desire of a player to achieve a certain outcome. In a game theory game, for every kind of
outcome that a decision might have, a utility is assigned to that decision.
Utility can become more complex when multiple factors come into play. For
example, if you were building ahouse for yourself on beachfront property, thinking in
game theory terms, you could measure different locations of your house in terms of
utility. You might be able to get the highest utility, say +10, if you built right on the
beach. There might be a lower utility, such as +5 or +2, if you had to build it several
meters away from the shoreline.
On the other hand, if you had to build the house so far away from the beach that
the ocean was no longer in view, your utility might go into the negative numbers,
indicating an outcome that you would find unpleasant. Of course, you might not have the
money to afford the situation with the highest utility. For example, you might require a
house of a certain size and if it were directly on the beach it couldn't have a basement and
would have to be smaller. Or the cost of the house might be higher on the beach because
of the extra architectural complexity required to build in the sand. Cost, size, and location
would all be assigned different values. In making your decision, you would try and
maximize the total utility given your available options.
These examples touch on the ways that game theory employs the concept of
utility. It might seem silly to turn something like human satisfaction into a numerical
value, given the innumerable complexities that go into our feelings of pleasure,
butMorganstern and Von Neumann felt very strongly that a scientific theory of
economics necessitated such an approach. In their book, they use an analogy to physical
properties such as heat. Before scientists developed a way of conceptualizing and
measuring heat, it was an unknown, fuzzy property that seemed impossible to measure: a
sensation that occurred as one approached a flame. But the precise measurement of heat
is now an important part of contemporary physics. The aim of Von Neumann and
Morganstern was to begin a similar revolution in economics, by quantifying pleasure as a
measure of utility.
Utility may well be an oversimplification of human desire, but it does make a
good fit with the formal qualities of games. As we know from our definition of games, all
games have a quantifiable outcome: someone wins, or loses, everyone wins or loses, or
player performance is measured in points, time, or some other numerical value. The
concept of assigning a numerical utility to decision outcomes is really just another way of
creating a quantifiable outcome. When looking at games through a formal frame, we do
not have the luxury of being non-numerical. The formal systems of both digital and non-
digital games require an exactness that does in fact come down to numbers. How many
kills did you earn that round? What qualifying time do you need on the next heat in order
to continue the race? Which team won the game? These very simple game results are all
quantifiable outcomes, and are all examples of utility as well.
The last component of most game theory games is that they are usually played by
only two players. This was not part of the original formulation of game theory as
proposed in Theory of Games and Economic Behavior. The original idea was that the
theory could cover n-player games, where n was a number of any size that indicated the
number of players. But Von Neumann and Morganstern found that, as with the problem
of three planetary bodies discussed in Games as Emergent Systems, their theory became
vastly more complex when it took three or more players into account. As a result, most
game theory work has focused on two player games. We follow suit in the material to
follow.
g. Cake Division
It is finally time to take a look at a real game theory game. The following
description is taken from Prisoner's Dilemma and is the classic "cake division" game
theory problem: Most people have heard of the reputed best way to let two bratty children
split a piece of cake. No matter how carefully a parent divides it, one child (or both!)
feels he has been slighted with the smaller piece. The solution is to let one child divide
the cake and let the other choose which piece he wants. Greed ensures fair division. The
first child can't object that the cake was divided unevenly because he did it himself.The
second child can't complain since he has his choice of pieces.
The cake division problem contains all of the elements of a game theory "game"
listed earlier. There are two rational players (the children motivated by self interest).
These two players choose a strategy about how to behave (how to cut or select the
pieces).These strategies result in some kind of utility for the two players, measured in
how much cake they get. Note that even though the "play" of this very simple game
consists of a two-part action (first slice the cake and then choose a slice), the two players
can still reveal and enact their strategies simultaneously. For example, the strategy of the
player that chooses from the two pieces is always going to be "take the bigger piece." A
rational piece-choosing player is going to choose this strategy regardless of the strategy
that the cake-cutting player takes. (Note that although these "strategies" may seem like
forgone conclusions rather than choices, this is because this game theory game has a
saddle point, a concept explained in detail later on.)
A powerful analytical tool provided by game theory is to map this decision
making process into a grid. One axis of the grid represents one player's decision. The
other axis represents the other player's decision. The cells in the grid represent the
outcomes reached depending on which decisions were made. A game theory table of this
sort is called a payoff matrix (payoff being another term for utility). Note that William
Poundstone makes the assumption that the cake slicing is going to happen in an imperfect
world, so that even if the child that cuts the cake tries to slice it evenly, the two resulting
slices will still differ a tiny bit, say by one crumb.
Along the left side of the matrix are strategies that the cutter can take: either cut
the cake evenly or cut it unevenly. Although there are any number of ways to cut the
cake, these are the two essential strategies from which the cutter can choose. Across the
top of the matrix are strategies the chooser can take: choose the bigger piece or choose
the smaller piece. The cells show the utility or payoff for only one of the players (the
cutter), but it can be assumed that the inverse payoff would happen for the chooser. If the
payoff matrix indicates that the cutter receives the "small piece," the chooser would
therefore receive the "big piece." This is also true for half of the cake plus or minus a
crumb.
The cake division problem illustrates two important game theory concepts. The
first is the concept of a zero-sum game. In a zero-sum game, the utilities of the two
players for each game outcome are the inverse of each other. In other words, for every
gain by one player, the other player suffers an equal loss. For example, playing a version
of Poker in which everyone puts money into a pot is a zero-sum game. At the end of the
game, every dollar won by one player is a dollar lost by another player. A group of
gamblers playing Roulette is not a zero-sum game between the players, because they are
not playing directly against each other. On the other hand, if we frame Roulette so that
one player is playing against the casino, then it is a zero-sum situation: if a player wins a
dollar, it is taken from the house, and vice-versa.
Many games are zero-sum games, even those that do not involve money. When
one player wins a game of Checkers and the other player loses, the loss by one player
equals a gain for the other player. In this case, game theory would assign a utility of –1
for the loss and +1 for the gain. The utilities add up to zero, which is exactly why it is
called a "zero-sum" game. Some games, such as the cooperative board game Lord of the
Rings, are not zero sum games. In the basic version of Lord of the Rings, players
cooperate against the game system itself. Players other. If one player receives half of the
cake minus a crumb (-1) the other player will receive half of the cake plus a crumb (+1).
The total is zero. Cake division is a zero-sum game.
Why is this important? Because, according to game theory, every finite, zero-sum,
two-player game has a solution (a proper way to play the game), the strategy that any
rational player would take. What is the solution to the cake division problem? The game
will always end in the upper left corner. The cutter will get half of the cake minus a
crumb and the chooser will get half of the cake plus a crumb. Why is this so? Look at the
cut-ter's strategies. The cutter would love to end up with the lower right cell, where he
gets the big piece. So perhaps he should choose the strategy of cutting the pieces
unequally. But the cutter also knows that if the chooser is given the chance to choose, the
chooser will always choose the bigger piece. As a result, the cutter has to minimize the
bigger piece that the chooser will select by cutting the cake as evenly as possible. The
game resolves to the upper left corner.
This situation clearly illustrates another key game theory concept: the saddle point
property of payoff grids. In cake division, each player is trying to maximize his own
gains while minimizing the gains of the other player. When the choices of both players
lead to the same cell, the result is what Von Neumann and Morganstern call a saddle
point. A saddle point refers to a saddle-shaped mountain pass, the intersection of a valley
that goes between two adjacent mountains. The height of the pass is both the minimum
elevation that a traveler going across the two mountains will reach, as well as the
maximum elevation that a valley traveler crossing the mountain pass will achieve. The
mathematical proof of saddle points in games is called the minimax theorem, which Von
Neumann first published in 1928, many years before the 1944 publication of Theory of
Games and Economic Behavior.
The concept of saddle points is extremely important in game design. In general,
you want to avoid them like the plague. Remember, a saddle point is an optimal solution
to a game. Once a player finds it, there is no other reason to do anything else. Think
about the cake division saddle point: if either player deviates, that player will lose even
more cake. If you think of the space of possibility that you are crafting as a large 3D
structure carefully crafted to give a certain shape to the experience of your players, saddle
points are short-circuits in the structure that allow players to make the same decision over
and over. That kind of play experience does not usually provide very meaningful play.
Why? Because if there is always a knowable saddle point solution to a game, a best
action regardless of what other players do or what state the system is in, the game loses
the uncertainty of possible action. Meaningful play then goes out the window.
Saddle points do not just occur in game theory games. Many fighting games are
ruined, for example, because despite all of the special moves and combinations that are
designed into the game, the best strategy to use against opponents is simply to use the
same powerful attack again and again and again. Saddle point! Another common
occurrence of saddle points involves the programming of computer opponents. In many
real-time strategy games there are "holes" or weaknesses in the AI that allow for saddle
points. If a player discovers that the computer opponent does not know how to defend
well against a certain type of unit, he is likely to abandon all other game strategies and
simply hammer on the AI's weakness over and over, regardless of how much care went
into carefully designing missions that require different kinds of problem-solving. Saddle
point!
This style of play, based on exploiting a strategic saddle point, is called an exploit
or degenerate strategy. A degenerate strategy is a way of playing a game that ensures
victory every time. The negative connotation of the terms "exploit" and "degenerate"
imply that players are consciously eschewing the designed experience in favor of the
shortest route to victory. There are some players that will refuse to make use of
degenerate strategies, even after they find out about them, because they wish to play the
game in a "proper" manner. On the other hand, many players will not hesitate to employ a
degenerate strategy, especially if their winnings are displayed in a larger social space
outside the game, such as an online high score list.
Degenerate strategies can be painful for game designers, as players shortcut all of
the attention lavished on a game's rich set of possibilities. Try to find degenerate
strategies and get rid of them! We learned in the previous schema that positive and
negative feedback systems can emerge unexpectedly from within a game's structure and
can ruin a game experience for players. The same is true of degenerate strategies. A close
analysis of your game design can sometimes reveal them but the only real way to root
them out is through rigorous playtesting. If you see players drawn to a particular set of
strategies again and again, they may be exploiting a weakness in your design.
h. Conflict case Studies
Our first example is the arcade game Centipede, in which the player uses a
trackball controller and fire button to move a character at the bottom of the screen and
shoot at objects coming down from the top. Centipede might seem at first glance to have
a simple and straightforward structure of conflict. But in fact, the formal system provides
many ways for players to struggle and pursue goals. As a single-player experience, you
compete against the program.The game compiles an ongoing "score" based on your
performance, and the presumed goal of the game is to achieve the highest score.
There are many ways that you might pursue goals related to the high score goal of
the game. You might have a general idea of what constitutes a "good score," which you
try to achieve. Or you might try to surpass your previous game's score, or attain a new
personal best score. You might set other goals besides those involving your score. For
example, you might try to play for a certain · amount of time, get to a certain level in the
game, or destroy every enemy of a particular type that appears. Several of these goals
might co-exist with each other and with the score-oriented goals.
Who knew so many different forms of conflict were lurking under the surface of a
simple arcade game? Our next example adds even more. In Joust, two players maneuver
bird-mounted knights, attacking enemies controlled by the program. Both players can
play the game simultaneously, instead of alternat-· ing turns. This structure opens up
whole new forms of competition. Joust can be a single-player game. Individual players
receive a score and there is a list of player high scores, including separate rankings for
daily high scores and "all time" high scores. Most of the forms of competition in ·
Centipede also occur in Joust.
In Joust, the two-player simultaneous structure adds new layers to the possibilities
of game conflict. In Gauntlet, our third arcade game example, up to four players can play
at once. The players take fixed roles (Warrior, Valkerie, Thief, or Wizard) as members of
a team. Together the team explores the game spaces, fights computer-generated enemies,
and gathers resources that boost their abilities to let them explore further. Like Joust and
Centipede, Gauntlet can be played by a single player. Gauntlet players also receive a
score; if the score is high enough, players record high scores and player · initials. All of
the single-player and high score list forms of competition apply to Gauntlet as well.
i. Competition and Cooperation
So far, we have spoken somewhat loosely about competition and cooperation as
they relate to the conflict in a game. But what do these terms really mean? Competition
occurs when players struggle against each other within the artificial conflict of a game.
Perhaps our clearest model of competition comes from game theory: the zero-sum game.
In a zero-sum game, one player's winnings equal another player's losses. If one player is
the victor in a two-player zero-sum game, the other player will necessarily lose. Winning
is always equally balanced by losing, making the end sum zero. A common criticism
leveled against games is that they are all competitive, and that competition is somehow
undesirable. Framed in this way, competition is something to avoid in order to ensure a
positive play experience.
DeKoven's point is that when the winning and losing of competition enters into
the conflict of a game, it becomes the paramount concern of the game's participants,
eclipsing everything else the game has to offer. With all due respect, we disagree. It
seems quite clear to us that competitive games can offer genuinely meaningful
experiences. Sometimes that meaning can stem from the joy of play itself (DeKoven's
"playing well together"), but certainly much meaning derives from the competitive
struggle of a game, from trying to become a winner while avoiding a loss.
The competitive striving toward a goal is fundamental in giving shape to the
structure of a game and the way that the game creates meaning. The idea, for example,
that in meaningful play a player's actions are integrated into the larger context of a game
is dependent on the competitive nature of games. Without a goal toward which players
strive, it is very difficult for a player to measure his or her progress through the system of
a game. Without a measure of progress to give a player feedback on the meaning of his or
her decisions, meaningful play is not possible. Remember the "horrible" game The Grid
in Games as Emergent Systems? That game had no goal, and no way for players to
compete with each other. There was nothing to motivate players to move their pieces this
way instead of that way. Meaningful play was impossible.
Our opinion is that all games are competitive. All games involve a conflict,
whether that conflict occurs directly between players or whether players work together
against the challenging activity presented by the game system. Without a clearly defined
goal, games generally become less formalized play activities. However, just because all
games are competitive does not mean that they are not cooperative as well. Although we
can assert with confidence that all games are competitive, it is equally true that all games
are cooperative. Are these two statements contradictory? Can all games be both
competitive and cooperative? Theidea that games are both competitive and cooperative is
only contradictory if the two terms are mutually exclusive, which they are not. The root
of the word "compete"is the Latin con petire, which means "to seek together."
In what ways are all games cooperative? Recall the magic circle and the lusory
attitude, and the way that these aspects of a game create meaning.To play a game is to
submit your behavior to the rules of the game, to enter into the time and space that the
game demarcates, to traffic in the special meanings that the game offers up. To play a
game is to participate in the discourse of the game with the other players. Players can
play Basketball together because they both speak the "language" of Basketball. When two
players hit the courts for a game of one-on-one, that is exactly what they are doing.
j. New Games
In the earlier critique of Bernard DeKoven's ideas about the negative aspects of
competition, we were not quite playing fair. It is true that DeKoven questions traditional
forms of competitive play. It is also true that we do not agree with all of his ideas on the
subject. But DeKoven's concepts have to be understood within the larger context of his
important work on games. In his book The Well-Played Game, DeKoven argues for a
new understanding of play, governed by a shift in emphasis away from competition.
Instead, DeKoven is an advocate for more improvisational games in which players take
on the role of game designers.
DeKoven was not alone in his ideas. He was one of the early members of the New
Games Movement, a group of game designers and play advocates that had a tremendous
impact on the culture of games. Founded by Stewart Brand (the same man who started
The Whole Earth Catalog) in the late 1960s, the New Games Movement was an
organization dedicated to the promotion of play and its positive impact on society. During
the late 1960s and 1970s, the New Games Movement organized a number of large-scale
public game "tournaments" in the San Francisco Bay Area and other parts of the world.
Part art happening, part community action, and part playground carnival, New Games
Movement Tournaments embodied a uniquely game-centric, community-based politics of
a scale that has not been seen since.
The New Games Movement had a large impact on physical education and the
integration of games and play into schools. If you grew up playing with a parachute or
huge rubber "Earth Ball" in your elementary school gym class, it is probably due to the
direct or indirect influence of the New Games Movement. The New Games Movement
published two books (The New Games Book and More New Games) that cataloged their
playful game designs. How does the New Games Movement fit into an understanding of
games as systems of conflict? The New Games Movement confronted the idea of
competition and cooperation head on, creating games and ways of thinking about game
design that challenged conventional notions of games as conflict.
Many people think of New Games as non-competitive. Of course this isn't the
case. Most of the games in this book involve competi-tion—it's what gives New Games
its vitality. …The effort each player makes to overcome the resistance and achieve the
goal is the heart of the game and what makes it enjoyable and gratifying. In most games,
the resistance is supplied by your opponent trying to achieve her goal. Your opponent is
therefore your partner in the game. The best games are those in which you can play your
hardest and still count on our opponent to meet your effort—to compete with you.
Although DeKoven may rail against competition in some of his writings, he also
helped instill in New Games the more balanced notions of competition embodied in the
quote above, taken from an essay he wrote for the New Games Book. DeKoven's main
point is that in the context of a game, the struggle of players against each other is also a
struggle with each other, as players meet the challenges that they provide for one another.
In this way, New Games affirms the interdependent relationship between competition and
cooperation, the systemic cooperation that is part of all games.
Catch the Dragon's Tail purposefully blurs the lines between competition and
cooperation. On the one hand, all of the players are cooperating to hold on to each other
to become a single dragon. But at the same time, the front part of the dragon is chasing
the rear part, with the people in the middle not given a clear role to play in the conflict.
Catch the Dragon's Tail makes playfully explicit the ways that players must work
together even as they compete within the limited space of a game. Catch the Dragon's
Tail also embodies an important lesson for game design: all of our preconceptions about
games can be questioned. Normally we might think that all players of a game must have a
clearly defined goal, or that lines of competition must be sharply defined, or that a game
with player cooperation cannot also have vigorous competition—but Catch the Dragon's
Tail debunks all of these assumptions. If nothing else, game design is about playing with
ideas, and even seemingly fundamental ideas about competition in games are subject to
playful intervention.
k. The Goal of a Game
In addition to competition and cooperation, another essential component of a
game as a system of conflict is a goal. Goals are fundamental to games. In the explication
of Centipede, Joust, and Gauntlet, goals into each form of conflict. At the outcome of a
game, the goals are either reached or not reached, and this quantifiable outcome is part of
our definition of games. Very often, it is a clear and quantifiable goal and outcome that
distinguishes games from other play activities. Add a goal to informal play and usually
you will have a game. Casual skiing for fun is a leisure play activity. But race your friend
to the bottom of the mountain and suddenly you're taking part in a game.
A game's goal is defined by its rules and is tightly interwoven into the formal
structure of the game as a whole. A game's goal is a central feature of its formal structure.
When players come together to play a game, the goal is at the center of the magic circle,
the pole that holds aloft the circular tent of the game while the players are inside the
structure, at play with one other. The goal sustains their interest, their engagement, and
their desire. Without a clear goal, meaningful game play is not possible; if players cannot
judge how their actions are bringing them closer to or farther away from winning the
game, they cannot properly understand the significance of their actions, and the game
collapses into a jumbled heap of ambiguity.
A game's goal defines its endpoint; once it is reached, the game is over. In this
sense, a game's goal is the death of play, the mark of the end, foretelling the moment the
magic circle will disappear. There is a curious poetic quality to the struggle of game
players as they make their way through the system of a game, playing to no end but the
one provided by the game itself, even as their joyful pursuit of that end means the death
of their pleasure. Until, of course, the next game begins.
Most games have an end in which one or more players achieve victory. However,
in games such as Space Invaders, in which the game structure repeats itself with
increasing challenge to the player, there is no single victorious endpoint. In this form of
game, the goal is to play as long as possible or achieve the highest score. This formal
structure heightens the sense of inevitable death. The player is living on borrowed time,
stavingoff the inevitable end of a game that occurs when conditions of failure are met.
The space of possibility of a game is a plane stretched between two anchorage
points: the beginning and the end of the game. The players journey from one end to
another, making their way from the start to the finish. In a well-designed game that
supports meaningful play, this journey between points should be taut and efficient, with
every element contributing directly or indirectly to the larger experience. In case this all
sounds too goal-oriented, we must acknowledge that goals are not the only reason people
play games. Play can be an end in itself, or a way to achieve social interaction, or affect
cultural change. But seen as a formal system, the goal of a game needs to be recognized
as a primary structure that shapes the game as a whole.
l. The Level Playing Field of Conflict
Competition and cooperation, goals and struggle, victory and loss: how does it all
add up? What are the general conditions of a game conflict? One core principle of
conflict in games is that it is fair. Game conflict is impartial conflict: it is premised on the
idea that all players have an equal chance at winning, that the game system is intrinsically
equitable, that the game's contest takes place on a level playing field, which does not
favor one side over the other. Anthropologist Roger Caillois points this out in speaking
about competitive forms of play: "A whole group of games would seem to be
competitive, that is to say, like a combat in which equality of chances is artificially
created, in order that the adversaries should confront each other under ideal conditions,
susceptible of giving precise and incontestable value to the winner's triumph."
Why would a game strive so forcefully to create equality in this way? As our
definition states, a game is an artificial conflict. The game structure creates an artificial
arena, in which everything is removed except for the factors involved in the conflict.
Chess is a context for intellectual strategic competition. In a gymnastics competition,
only gymnastics skills matter.
In real life, the conflicts and struggles faced are never so clearly articulated and
understood as in a game. The idea that players are entering into a fair conflict, where they
won't be fooled or tricked by the game itself, is a key component of the lusory attitude.
Even though games may have elements of uncertainty, the structure within which that
uncertainty plays out is known in advance. The qualities of rules themselves make this so.
As we know from Defining Rules, rules of a game limit player action, are explicit,
unambiguous, binding, and shared by all players. Within the magic circle, players
experience a kind of equality and fairness that is not present outside games.
Is it really true that games strive to create spaces of equality, where only the play
of the game can determine the winner of a game? Are games so pure and separate from
the real world that no other factors possibly enter into the play? Generally speaking, no.
And in a formal sense, yes, games are spaces of pure conflict, separate from the outside
world.
There is indeed a contradiction at work in the idea of equality within games. As
Caillois points out, equality is something that is sought after in games, but somehow
never quite achieved- at least, in the non-digital game examples he cites. What about
digital games? In some ways, they have an advantage when it comes to creating a level
playing field. The constrained context of a computer system allows for greater control
over the exact conditions of play. On the other hand, the complex automated nature of
digital games can place players at some distance from the rules. Players can easily grow
suspicious of an unfair network lag, a "cheating" AI, or a low processing speed "stutter."
This kind of player distrust, whether or not it is based in reality, can ruin a game.
The magic circle is fragile, easily dispelled when players fail to invest faith in the
game. If your players feel that your game is unfair, that it lacks a level playing field, it is
unlikely that they will want to play. Within the magic circle, a game is suspended
between the ideal notion of a level playing field and the reality of inevitable unfairness, a
reality that creeps into every game, even while the magic circle's border holds it at bay.
Perhaps games do not take place on an absolutely level playing field. But they are
premised on the very real idea of fairness and equality. This struggle is part of what gives
games their vitality.
