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lecture_123_week_5.docxA Nash equilibrium is a pair of strategies, one for each player, in which each strategy is a best response against the other.
When players act rationally, optimally, and in their own self-interest, it’s possible to compute the likely outcomes of games. By studying games, we learn where the pitfalls are and how to avoid them.
Sequential games include a potential first-mover advantage, or disadvantage, and players can change the outcome by committing to a future course of action.. Credible commitments are difficult to make because they require that players threaten to act in an unprofitable way—against their self-interest.
In simultaneous-move games, players move at the same time.
In the prisoners’ dilemma, conflict and cooperation are in tension—self-interest leads to outcomes that no one likes. Studying the games can help you figure a way to avoid these bad outcomes.
In repeated games, it is much easier to get out of bad situations. Here are some general rules of thumb:
Be nice: No first strikes.
Be easily provoked: Respond immediately to rivals.
Be forgiving: Don’t try to punish competitors too much.
Don’t be envious: Focus on your own slice of the profit pie, not on your competitor’s.
Be clear: Make sure your competitors can easily interpret your actions.
A significant portion of our studies so far have revolved around situations in which the firm making the decision does not need to consider the effects of the decisions of other competitors in the market. In some markets, the firm we were examining was the only firm. In other markets, the firm was one of many firms and any individual firm’s decision would not affect the market significantly. In this chapter, we begin our study of situations in which there are a relatively small (but more than one!) number of actors in the market, and one actor’s decision will affect the outcomes of the other actors. A new tool is developed, the Nash Equilibrium, which will be used to study such situations.
Definition of a Game
A game (in the Game Theory sense) is any situation that has participants, a set of rules, and payoffs that depend on the actions chosen by all participants. The rules include such things as what actions are available to the participants, what information is available to each participant when they make their decisions, and the order in which participants make their decisions. The definition of a game is quite broad, and intentionally so. Chess, poker, checkers, baseball, football, Jeopardy!, Go, auctions, negotiations, competition in business, behavior in cartels, how to monitor employees, what to do when serving or receiving in tennis, which way to shoot when taking a penalty shot in soccer, and what to do when approaching a stoplight either are games or can be modeled using Game Theory.
Nash Equilibrium
The idea behind Nash Equilibrium is that if and when all the participant’s actions are revealed, no participant wishes to change their action. In other words, every participant’s choices are a best response to the strategies of all other participants; given the other participants’ strategies no participant can change their action and receive a better outcome. Note the use of the word strategy; sometimes mixed strategies are used in which the participant chooses randomly between several actions. In the tennis example, if the server always serves down the middle, then the receiver’s best response is to defend that area. That is not an equilibrium, since the server’s choice must be a best response to what the receiver does. In this case, if the receiver defends the middle then the server can do better by serving to the outside. We will see later in the lecture that there is a Nash equilibrium in which the server chooses to serve down the middle with some probability and serves to the outside the remaining portion of the time. In that equilibrium the receiver will defend the middle with some (usually different) probability and defend the outside the remainder of the time. Since neither the server or receiver can predict the action taken by the other, the Nash equilibrium in this case is based on the probabilities rather than the particular choice of either participant.
Example
Consider a hypothetical example of political negotiation. In this example, there are Democrats and Republicans negotiating over how to balance the federal budget. Democrats have their preferred way of balancing the budget (say, raising taxes) and Republicans have a different preference (reducing spending). Both sides decide whether to take a strong stance at the negotiating table or to seek compromise. If both sides take a strong stance, no agreement is reached and the government defaults on its debt. This is the worst result for everybody. Assign a payoff of −100 to both sides for this outcome. If one side takes a strong stance and the other seeks compromise, the side taking the strong stance dictates how the budget is balanced. That is the best outcome for the side taking the strong stance (assign a payoff of 1) and the worst outcome except for a government shutdown for the side seeking compromise (assign a payoff of −1). If both sides seek compromise, then the budget is balanced through both tax increases and spending cuts. This is worse for each participant than getting their way, but it is better outcome than letting the other side get their way. We will assign a payoff of 0 – or something between 1 and −1 – to this outcome.
To determine the Nash equilibria, we can use the underlining method discussed in the book. We draw a grid with the actions of each player labeling the columns or rows (one player is the row player, one player is the column player) and fill in each box in the grid with the payoffs to each participant based on the outcomes determined by the strategies of the row and column associated with the box. The game can be represented as follows, where S stands for “Strong negotiating stance” and C stands for “Compromise”:
Republicans
S
C
Democrats
S
-100, -100
1, -1
C
-1, 1
0, 0
The first payoff is for the Democrats, the second in the pair is for the Republicans. To find the Nash equilibria, examine the Democrats’ best action given what the Republicans do and underline those payoffs. Following this procedure would underline the −1 in the lower left cell, as Democrats would rather compromise (-1) than negotiate strongly (-100) if the Republicans negotiate strongly (this is the first column). Similarly, if the Republicans choose compromise, then the Democrats would prefer to negotiate strongly (1) instead of compromising (0) so the 1 in the upper right box would be underlined. These underlines represent the Democrats’ best response to what the Republicans choose. The Republicans’ best responses can be found in a similar fashion. The 1 in the lower left box and the −1 in the upper right box would receive underlines. If both payoffs are underlined, that means the action of each party is a best response to the actions of the other party and therefore is a Nash Equilibrium. In this case, the two equilibria are where one party negotiates strongly and the other side gives in. Obviously the Democrats prefer one Nash equilibrium over the other, and the Republicans have the opposite preference. How is the particular equilibrium chosen? Game theory does not give an answer to that question. It only predicts that one of the equilibria is chosen. However, we can use game theory to determine ways of influencing which equilibrium is chosen.
Changing the Rules
In the budget negotiation example above, both sides would like the other to believe that they will negotiate strongly and will not deviate from that strategy. If one side believes the other will negotiate strongly, then their best response will be to compromise. However, absent any other considerations, the threat to negotiate strongly and maintain the strong stance (if the game is repeated over several days or weeks in the case where both sides chose “strong”) is not credible because it is not in the best interest of either party to maintain a strong negotiating position if the other side remains strong. One side simply saying they will take a strong negotiating stance is meaningless.
However, if one could somehow commit to negotiating strongly then the threat may become credible. For instance, Republicans could sign Grover Norquist’s pledge to not raise taxes. One could view this pledge as removing the compromise option from the Republicans’ possible choices if the politicians are committed to keeping their word. Alternatively, this could adjust the payoffs that the Republicans receive if they compromise and the Democrats negotiate strongly. If those strategies were chosen, with or without the pledge the Republicans would not get their way. However, the pledge results in additional negative repercussions for Republicans, including the possibility of much stronger competition in their next primary which significantly reduces the payoffs associated with compromising. If the payoffs are reduced enough, it may be that the Republicans’ best response to Democrats negotiating strongly is to adopt a strong negotiation stance as well. In that case, a Republican threat to negotiate strongly is credible; the strong stance is in the Republicans’ best interests.
The game of “Chicken” follows the payoffs in the matrix above. If the strategy “S” stands for “go Straight” and “C” stands for “Chicken out”, then the payoffs from Chicken follow the same payoff structure. One way to demonstrate that you have restricted your actions is to detach your steering wheel and throw it out your window to show your opponent that your options are now limited. The clip of “Dr. Strangelove” regarding the Doomsday device (https://www.youtube.com/watch?v=2yfXgu37iyI) represents the Russians attempting such a strategy, but as Dr. Strangelove points out, not letting the Americans know about the Doomsday decide defeats the purpose.
An example of payoffs changing is demonstrated in the negotiations between the NFL referee association and the league. The league and referees were engaged in a labor dispute and the NFL was using replacement referees to start the season. Entering the Monday night game of week three, there had been a few questionable calls by the replacement referees but there were no major missed calls. The league and the referees remained in a stalemate. Then on the last play of the Green Bay-Seattle game on Monday night – a nationally televised game – Seattle was behind and threw a desperation pass into the end zone as time was running out. The ball came down in someone’s hands. Replays would clearly show that the Green Bay defender caught the ball, but after looking at each other for a couple seconds, one referee signaled a touchdown while another referee signaled interception. After some discussion the referees awarded Seattle a touchdown and the win. The play and call were ubiquitous – it was even discussed on the morning news programs. The payoff for the NFL choosing a “strong negotiating stance” was greatly reduced because of the disgust that fans of the NFL felt towards the skill level of the replacement referees. Two days later the league and referees reached an agreement, most likely with the referees receiving most of their requests. Click on the following link for some commentary after the call but before the agreement:
Video: Green Bay Packers, Seattle Seahawks Blown Call
While the referee’s union did not change the payoff themselves, perhaps they had the foresight to predict an event similar to the one that occurred.
Randomization
Sometimes no pure strategy equilibria exist. Pure strategy equilibria are equilibria in which one person chooses some action every time and the other person takes some action all the time. In the negotiation example, there were two pure strategy Nash equilibria – one side cooperates and the other side negotiates strongly. There are be cases where a mixed strategy in which each participant randomly choses from several actions is one equilibrium of a game, and sometimes the only equilibrium of a game. To solve such equilibria, we will consider the initial concept of an equilibrium that was discussed earlier: both sides are content with their strategy given what the other person is doing.
If we assume both participants are using mixed strategies, then for one person to be happy choosing one strategy some of the time and the other strategy the rest of the time, the payoffs from the two strategies must be equal (in expectation – this is addressed a little more in the uncertainty chapter this week. You may want to skip this section for now and come back to it after you read the chapter on uncertainty.). If the payoffs are unequal, then the participant would not be willing to play both strategies. The expected payoff of one player’s strategy will depend on how often the other player chooses each available option. So we can use the indifference condition for one player to determine the other player’s strategy in order to be an equilibrium. As an example, consider a soccer penalty shot. The shooter can choose to kick left or right. The goalie can also choose to dive left or right. Suppose the payoffs are as given in the payoff matrix below (the payoff to the shooter is the probability of a goal, the payoff to the goalie is the probability that the goalie saves the shot).
Goalie
L
R
Shooter
p
(1-p)
q L
10, 90
95, 5
(1-q) R
80, 20
15, 85
The goalie will dive left p percent of the time, and right (1-p) percent of the time. The shooter will shoot left q percent of the time and right (1-q) percent of the time. For the shooter to be willing to randomly choose between shooting left or right, the payoffs of each action must be equal. The payoff for shooting left is the payoff associated with the goalie diving left times the probability the goalie dove left plus the payoff associated with the goalie diving right times the probability the goalie goes right. That is, 10p + 95 (1 - p) = 95 − 85 p. The payoff for shooting right is 80p + 15 (1 - p) = 65 p + 15. Setting these equal to each other and solving for p results in p = 80/150. This means if the goalie dives left with a probability of 53.33% the shooter is indifferent between shooting left or right. Similarly, the payoff for the goalie diving left is 90q + 20 (1 - q) and the payoff for the goalie diving right is 5q + 85 (1 - q). Setting those payoffs equal and solving for q results in q = 65/150. Thus if the shooter kicks left 43.33% of the time, the goalie is just as well off diving left as diving right. Thus if the goalie dives left 53.33% of the time and the shooter kicks left 43.33% of the time neither the kicker or goalie would want to change their actions.
Lecture 2
· Bargaining can be modeled as either a simultaneous-move or sequential-move game.
· A player can gain an advantage by 1) turning a simultaneous-move game into a sequential move game with a first-mover advantage; or by 2) committing to a position.
· Credible commitments are difficult to make because they require players to commit to a course of action against their self-interest. Thus, the best threat is one you never have to use.
· The strategic view of bargaining focuses on how the outcome of bargaining games depends on who moves first and who can commit to a bargaining position, as well as whether the other player can make a counteroffer.
· The non-strategic view of bargaining does not focus on the explicit rules of the game to understand the likely outcome of the bargaining. This view focuses on the gains from bargaining relative to alternatives.
· The gains from agreement relative to the alternatives to agreement determine the terms of any agreement.
· Anything you can do to increase your opponent’s gains from reaching agreement or to decrease your own will improve your bargaining position.
Game Theoretical View of Negotiations In the previous chapter, we discussed some examples of negotiations to illustrate how and where games can be used to analyze economic situations. We can add a little more realism to the game by considering a sequential move game – a game in which one side moves first and the other side observes and may base its actions on what the first side chose. To these games we apply a concept called Subgame Perfect Equilibrium (SPE), a modification of Nash equilibrium. This modification of a Nash equilibrium accounts for the order of decisions or actions. To analyze a game using SPE, find the last decisions that are made and identify the choice that maximizes the decision maker’s payoff. Replace the decision with the payoffs – since we know what the decision-maker will choose, we may make the decision for him. With the last decisions replaced with the payoffs, find the decisions that are now the last ones to be made and repeat the process until every decision has been replaced by the payoffs.
In the toy game illustrated in the text, the Union gets to present Management with an offer, and Management may accept or reject the offer. Management has the last decision, so to apply SPE to this game we analyze Management’s two decisions – one when the Union decided to extend a low offer and the other one when the Union decided to extend a generous offer. In either case, accepting the offer results in a positive payoff for management and rejecting the offer results in a payoff of zero, so for both decisions Management will accept the offer. Now the Union’s decision can be analyzed; if they make a low offer, they will receive a payoff of 75 while a generous offer will result in a payoff of 25. Seventy-five is better than 25, so they should make the low offer.
In this game, the Union acted first and the result was the equilibrium most beneficial for the Union. In a sense, the Union was able to commit to a hard bargaining stance. This is an example of the first-mover advantage. Note that the Management was not able to commit to bargain hard and reject a low offer. If it could somehow do so before the Union made their offer, then Management would then have the upper hand. Perhaps that may be accomplished through a rainy-day fund so that the firm’s unavoidable expenses can be covered. It may be through statements accepting the inevitability of a strike or lock out and vowing to remain firm in their stance so that eventually the Union will capitulate. It may be through statements to the public to engender support. Of course, the Union may take similar actions. As I am writing this, I am reminded of the dispute between Time Warner Cable and CBS. The agreement between Time Warner Cable and CBS has expired and they have been negotiating a new agreement for a while. Currently, no agreement has been reached and Time Warner customers in New York, Los Angeles and Dallas can not watch CBS on their cable system (Reuters). Both sides have been issuing press releases trying to position themselves as the “good guys” who are negotiating in good faith and trying to be fair. The particular article I cite below discusses how the standoff may end – when the football season begins. Note the payoffs from bargaining change at that time as well – Time Warner customers may endure the inconvenience during the last part of summer re-runs, but once the football and the fall TV season begin Time Warner may face many customers leaving if no deal has been reached.
Click the following link:
· Article: Time Warner-CBS Blackout May Last Until Sept: Analysts
Practical Uses The textbook presents a stylized look at bargaining using something called the BATNA, or Best Alternative to a Negotiated Agreement. The textbook postulates that the result from bargaining equally splits the gains from bargaining. In other words, if side A’s BATNA is worth $50 and side B’s BATNA is worth $30, and an agreement will be worth a combined total of $100, then the gains from a successful negotiation are $100 - ($50 + $30) = $20, so the agreement should increase both side’s payoffs by $10. Other economic models modify this result slightly – the split may depend on how patient the sides are relative to each other or other similar factors. During a strike or lockout, if one side is more patient than the other, the final agreement will be more favorable to that side.
Thus, there are several options that immediately present themselves in order to obtain a more favorable outcome. Being more patient, or at least appearing more patient and out-waiting the other side will increase the benefits of a successful negotiation. Increasing one’s BATNA will also increase the payoff of a successful negotiation. Suppose side B increases their BATNA to $40. In that case, only $10 of value is added by a successful negotiation and thus according to the book’s model, $5 of value would be earned by either side. This results in side B obtaining a payoff of $45, which is $5 more than when their BATNA was $30 (they would receive $40—their BATNA of $30 plus $10).
In a real-world example of this, Gneezy, List, and Price discuss an experiment in which handicapped and non-handicapped people go to an automobile repair shop and ask for estimates to repair a car. Their results show that “disabled agents receive roughly a $213 (or 30 percent) higher quote price than the non-disabled” which was statistically significant (not likely to have occurred from random factors). This could be because the customers in wheelchairs face a higher cost to obtain other estimates and would therefore be expected to have fewer competing estimated. In fact, a survey reported in Gneezy et. al. show that disabled consumers visit on average 1.67 mechanics while non-disabled consumers visit on average more than twice that number. Mechanics are cognizant of these patterns, and so can infer that the BATNA for a disabled person is worse than that of a non-disabled person and make their offers based on this inference. In a follow-up experiment, Gneezy et. al. repeated the experiment except had “both the disabled and abled agents explicitly stated to the mechanic that they are ‘getting a few price quotes.’” This should indicate to the mechanic that the disabled customer’s BATNA has now increased to be commensurate with the non-disabled consumer’s BATNA. If this is the case, the estimates should be the same between the two groups and that is exactly what was found.
Gneezy, List, and Price. “Toward an Understanding of Why People Discriminate: Evidence from a Series of Natural Field Experiments.” NBER working paper 17855.
Main Points
· When you’re uncertain about the costs or benefits of a decision, assign a simple probability distribution to the variable and compute expected costs and benefits.
· When customers have unknown values, you face a familiar trade-off: Price high and sell only to high-value customers, or price low and sell to all customers.
· If you can identify high-value and low-value customers, you can price discriminate and avoid the trade-off. To avoid being discriminated against, high-value customers will try to mimic the behavior and appearance of low-value customers.
· Difference-in-difference estimators are a good way to gather information about the benefits and costs of a decision. The first difference is before versus after the decision or event. The second difference is the difference between a control and an experimental group.
· If you are facing a decision in which one of your alternatives would work well in one state of the world, and you are uncertain about which state of the world you are in, think about how to minimize expected error costs.
Another wrinkle that has not been discussed much in this course is uncertainty and how it can affect the decision-making process. Typically, many quantities in a forward looking projection will not be exact – there will be a range over which the parameter may vary. For an example, let us slightly modify the hotel booking example from last week. If the price for a room is set at $30, then assume the hotel will receive 98 people arriving at the hotel with reservations 15% of the time, 99 people 25% of the time, 100 people 30% of the time, 101 people 20% of the time and 102 people the remaining 10% of the time. In this case, there is uncertainty regarding how many visitors will arrive at the hotel.
Expected Value The expected value of guests is the average number of guests that would arrive at the hotel if we repeat the situation many times. If we observed the hotel for 100 days, then 15% of the time we would see 98 people arrive. Since 15% of 100 days is 15 days, on 15 of the 100 days we would observe 98 guests arriving. On 25 days we would see 99 people, and so on. We could then add the total number of guests that arrive during those 100 days and then divide by the number of days to calculate the average number of guests per day which I will call X. This calculation would look like:
X = (98 * 15 + 99 * 25 + 100 * 30 + 101 * 20 + 102 * 10) / 100
We could calculate this to determine the numerical value. However, we can also distribute the 100 in the denominator and obtain
X = (98 * 15/100) + (99 * 25/100) + (100 * 30/100) + (101 * 20/100) + (102 * 10/100)
= (98 * 0.15) + (99 * 0.25) + (100 * 0.30) + (101 * 0.20) + (102 * 0.10)
Or
X = (98 * 15%) + (99 * 25%) + (100 * 30%) + (101 * 20%) + (102 * 10%).
This is a simpler formula to apply generally; instead of needing to find a particular number of days such that each case occurs a whole number of days we can apply a formula similar to above. For each potential outcome, multiply the outcome by the probability of the outcome. Then add together those intermediate calculations to determine the average, or expected, value of the outcome.
Using Uncertainty in Calculations In our hotel example the quantity of interest to the hotel is not the number of guests who arrive, but rather the profit. Profit can be calculated by examining the revenues and costs. Let us assume that the additional costs incurred from a single overbooking is $50. This number includes costs associated with finding the guest other accommodations, transportation costs to sent the guest to the other accommodations, an estimate of the loss of revenue from bad word-of-mouth advertising, and whatever other costs may be created by an overbooking. With this information we can determine expected revenues and expected costs. The expected revenue will be the sum of the room rate times the number of rooms rented times the probability of that many rooms being rented, or:
ER = $30 * 98 * 15% + $30 * 99 * 25% + $30 * 100 * 60% = $2983.50
Note that I combined the cases where 100, 101 and 102 guests arrive, since in all three of those cases exactly 100 rooms will be rented and the revenues will be identical. We could calculate the marginal costs associated with cleaning, guest services, and other such costs in a similar way. If the variable cost per occupied room is $5, then the expected variable costs will be $497.25. The expected losses from overbooking can be estimated as
EC = $0 * 70% + $50 * 20% + $100 * 10% = $20.
Here there is a 70% chance the hotel is not overbooked, or in other words only 98, 99 or 100 guests arrive and thus there is room for all. A 20% chance exists that the hotel will be overbooked by one room and a 10% chance two people will not find lodgings available.
The profit maximizing price is calculated by determining the marginal revenue and marginal cost associated with a small change in the price and finding the price at which the marginal cost is equal to the marginal revenue. When uncertainty is present, this rule is changed to finding the expected values of the marginal cost and marginal price. If the price is adjusted slightly, the probability that a certain number of guests arrive is likely to change. Therefore, if the price of the room is changed, the calculations outlined above can be recomputed with the new outcomes and probabilities and the change in expected revenue and expected costs can be determined.
For example, suppose lowering the price to $29.90 results in 99 guests arriving 20% of the time, 100 guests 35% of the time, 101 guests 20% of the time, 102 guests 15% of the time, and 103 guests 10% of the time. In that case, the expected revenue would be
ER = $29.90 * 99 * 20% + $29.90 * 100 * 80% = $2984.02.
So the expected marginal revenue is positive since the expected revenue in this case is larger. The variable costs per occupied room would be $499. Immediately we can see that the expected marginal cost ($499 - $497.25 = $1.75) is larger than the expected marginal revenue ($0.52). The change in expected costs from overbooking simply adds more to the marginal cost of decreasing the room rate. To be complete, the expected costs from overbooking are now:
EC = $0 * 55% + $50 * 20% + $100 * 15% + $150 * 10% = $40.
Thus the motel should not lower its price to increase revenue. This analysis suggests that the room rate should be decreased in order to increase profit but the probability of particular numbers of guests arriving would need to be known before definitively making that statement.
Managing Uncertainty The text discusses several ways to manage uncertainty or risk. I’ll comment or expand on a couple. The “Uncle Joe” example on page 203 of your text is reminiscent of the “Best/Worst/Average Case” analysis. As the name implies, in a Best/Worst/Average case analysis, the best case, worst case, and expected case outcomes are examined in order to determine the potential range of profits. For instance, if a firm was considering a project to build jet engines, several factors would determine the profitability of the project. Among them would be the costs associated with materials, the number of competitors and the size of the market. The best case scenario would use the most favorable projections for the costs, number of competitors, demand for the product, and any other factor affecting the project’s profitability that are likely. The worst case would use the least favorable projections – high costs, many competitors, few demanders – and the expected case would use the expected or average projections. Note that this analysis is only used to give a range of potential outcomes; once the best, worst and average case are determined the analysis ends; they are not combined into a single projection.
A second way to manage uncertainty is to perform regional experiments to obtain better estimates of the unknown variables. For instance, suppose Taco Bell is considering a new product, the “Doritos Locos taco.” Based on existing products, Taco Bell should have a very good idea of the marginal costs associated with the new product. However, while Taco Bell likely has some estimates of the demand for the product, there is significant uncertainty surrounding the demand for the new taco. Taco Bell could decide to begin a national distribution of the taco, but a more safe route may be to introduce the taco to a small number of regional markets like Toledo. It can then observe the demand for the product in these representative markets and make a more informed decision whether to roll out the product nationally, potentially avoiding a costly mistake if it has overestimated the product’s demand. Note that Taco Bell should select several regional markets that represent a cross-section of potential consumers; testing the product in Toledo, Detroit, Cleveland, Cincinnati and Pittsburgh would not be as good as testing in Toledo, Dallas, Sacramento, Atlanta and Boston since the latter combination better represents the national market. That is, if there is something about the product that appeals to consumers in Ohio and the immediately surrounding areas, the first combination would give an overly optimistic prediction of the demand.
