Econ 419 Chapter 10

Finitely Repeated Games

(1) games in which players do not know when the game will end and (2) games in which players know when it will end.

Subgame Perfect Equilibrium

A condition describing a set of strategies that constitutes a Nash Equilibrium and allows no player to improve his own payoff at any stage of the game by changing strategies.

Advertising with Game Theory

A classic example of such a situation is the breakfast cereal industry, which is highly concentrated. By advertising its brand of cereal, a particular firm does not induce many consumers to eat cereal for lunch and dinner; instead, it induces customers to switch to its brand from another brand. This can lead to a situation where each firm advertises just to "cancel out" the effects of other firms' advertising, resulting in high levels of advertising, no change in industry or firm demand, and low profits.

Nash Equilibrium

A condition describing a set of strategies in which no player can improve her payoff by unilaterally changing her own strategy, given the other players' strategies

Normal-formal game

A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies. Example: In Table 10-1, the dominant strategy for player A is up. To see this, note that if player B chooses left, the best choice by player A is up since 10 units of profits are better than the −10 he would earn by choosing down. If B chose right, the best choice by A would be up since 15 units of profits are better than the 10 he would earn by choosing down. In short, regardless of whether player B's strategy is left or right, the best choice by player A is up. Up is a dominant strategy for player A.

Secure Strategy

A strategy that guarantees the highest payoff given the worst possible scenario. Example: In particular, player B in Table 10-1 should recognize that a dominant strategy for player A is to play up. Thus, player B should reason as follows: "Player A will surely choose up, since up is a dominant strategy. Therefore, I should not choose my secure strategy (right) but instead choose left." Assuming player A indeed chooses the dominant strategy (up), player B will earn 20 by choosing left, but only 8 by choosing the secure strategy (right).

Mixed (Randomized) Strategy

A strategy whereby a player randomizes over two or more available actions in order to keep rivals from being able to predict his or her action

Dominant Strategy

Check to see if you have a dominant strategy. If you have one, play it

Number of Firms vs. Collusion

Collusion is easier when there are few firms rather than many. If there are n firms in the industry, the total amount of monitoring that must go on to sustain the collusive arrangement is n × (n − 1). Economies of scale exist in monitoring. Monitoring and policing costs constitute a much greater share of total costs for small firms than for larger firms. Thus, it may be easier for a large firm to monitor a small firm than for a small firm to monitor a large firm. For example, a large firm (with, say, 20 outlets) can monitor the prices charged by a small competitor (with 1 outlet) by simply checking prices at the one store. But to check the prices of its rival, the smaller firm must hire individuals to monitor 20 outlets.

Strategy

In game theory, a decision rule that describes the actions a player will take at each decision point

Bertrand Duopoly Game

Example: Imagine that two gasoline stations are located side by side on the same block so that neither firm has a location advantage over the other. Consumers view the gasoline at each station as perfect substitutes and will purchase from the station that offers the lower price. The first thing in the morning, the manager of a gas station must phone the attendant to tell him what price to put on the sign. Since she must do so without knowledge of the rival's price, this "pricing game" is a simultaneous-move game.

Simultaneous-Move Game

Game in which each player makes decisions without knowledge of the other players' decisions

Sequential-Move Game

Game in which one player makes a move after observing the other player's move.

Put Yourself in Your Rival's Shoes

If you do not have a dominant strategy, look at the game from your rival's perspective. If your rival has a dominant strategy, anticipate that he or she will play it.

Factors Affecting Collusion in Pricing Games

It is easier to sustain collusive arrangements via the punishment strategies outlined earlier when firms know (1) who their rivals are, so they know whom to punish should the need arise; (2) who their rivals' customers are, so that if punishment is necessary they can take away those customers by charging lower prices; and (3) when their rivals deviate from the collusive arrangement, so they know when to begin the punishments. Furthermore, they must (4) be able to successfully punish rivals for deviating from the collusive agreement, for otherwise the threat of punishment would not work.

Finitely Repeated Games w/o knowledge of Ending

It turns out that when there is uncertainty regarding precisely when the game will end, the finitely repeated game in Table 10-9 exactly mirrors our analysis of infinitely repeated games

Multistage Games

Multistage games differ from the class of games examined earlier in that timing is very important. In particular, the multistage framework permits players to make sequential rather than simultaneous decisions.

Sequential-Move Bargaining Game

Specifically, suppose a firm and a labor union are engaged in negotiations over how much of a $100 surplus will go to the union and how much will go to management. Suppose management (M) moves first by offering an amount to the union (U). Given the offer, the union gets to decide to accept or reject the offer. If the offer is rejected, neither party receives anything. If the offer is accepted, the union gets the amount specified and management gets the residual. To simplify matters, suppose management can offer the union one of three amounts: $1, $50, or $99.

Sustaining Cooperative Outcomes with Trigger Strategies

Suppose a one-shot game is infinitely repeated and the interest rate is i. Further, suppose the "cooperative" one-shot payoff to a player is πCoop, the maximum one-shot payoff if the player cheats on the collusive outcome is πCheat, the one-shot Nash equilibrium payoff is πN, and Then the cooperative (collusive) outcome can be sustained in the infinitely repeated game with the following trigger strategy: "Cooperate provided no player has ever cheated in the past. If any player cheats, 'punish' the player by choosing the one-shot Nash equilibrium strategy forever after."

Coordination Game

The game in Table 10-4 has two Nash equilibria. One Nash equilibrium is for each firm to produce 120-volt appliances; the other is for each firm to produce 90-volt appliances. The question is how the firms will get to one of these equilibria. If the firms could "talk" to each other, they could agree to produce 120-volt systems. Alternatively, the government could set a standard that electrical outlets be required to operate on 120-volt, two-prong outlets. In effect, this would allow the firms to "coordinate" their decisions.

End of Period Problem

When players know precisely when a repeated game will end, what is known as the end-of-period problem arises. In the final period there is no tomorrow, and there is no way to "punish" a player for doing something "wrong" in the last period. Consequently, in the last period, players will behave just as they would in a one-shot game. In this section, we will examine some implications of the end-of-period problem for managerial decisions.

Infinitely Repeated Game

a game that is played over and over again forever. Players receive payoffs during each repetition of the game.

Trigger Strategy

a strategy that is contingent on the past plays of players in a game. A player who adopts a trigger strategy continues to choose the same action until some other player takes an action that "triggers" a different action by the first player. Example: To see how trigger strategies can be used to support collusive outcomes, suppose firm A and firm B secretly meet and agree to the following arrangement: "We will each charge the high price, provided neither of us has ever 'cheated' in the past (i.e., charged the low price in any previous period). If one of us cheats and charges the low price, the other player will 'punish' the deviator by charging the low price in every period thereafter." Thus, if firm A cheats, it pulls a "trigger" that leads firm B to charge the low price forever after, and vice versa. It turns out that if both firms adopt such a trigger strategy, there are conditions under which neither firm has an incentive to cheat on the collusive agreement.

Extensive-form Game

summarizes who the players are, the information available to the players at each stage of the game, the strategies available to the players, the order of the moves of the game, and the payoffs that result from the alternative strategies.

One-Shot Game

the underlying game is played only once

Repeated Game

the underlying game is played ore than once

Nash Bargaining Game

two players "bargain" over some object of value. In a simultaneous-move, one-shot bargaining game, the players have only one chance to reach an agreement, and the offers made in bargaining are made simultaneously. Example: Before concluding that you should ask for $100, think again. Suppose the union wrote down $50. Management's best response to this move would be to ask for $50. And given that management asked for $50, the union would have no incentive to change its amount. Thus, a 50-50 split of the $100 also would be a Nash equilibrium. Finally, suppose management asked for $0 and the union asked for the entire $100. This too would constitute a Nash equilibrium. Neither party could improve its payoff by changing its strategy given the strategy of the other. Thus, there are three Nash equilibrium outcomes to this bargaining game. One outcome splits the money evenly among the parties, while the other two outcomes give all the money to either the union or management.