Chapter 12 Choices Involving Strategy McGraw-Hill/Irwin Copyright © 2008 by The McGraw-Hill Companies, Inc. All Rights Reserved.

Main Topics What is a game? Thinking strategically in one-stage games Nash equilibrium in one-stage games Games with multiple stages 12-2

What is a Game? A game is a situation in which each member of a group makes at least one decision, and cares both about his own choice and about others’ choices Includes any situation in which strategy plays a role Military planning, dating, auctions, negotiation, oligopoly Two types of games: One-stage game: each participant makes all choices before observing any choice by any other player Rock-Paper-Scissors, open-outcry auction Multiple-stage game: at least one participant observes a choice by another participant before making some decision of her own Poker, Tic-Tac-Toe, sealed-bid auction 12-3

Figure 12.1: How to Describe a Game Essential features of a one-stage game: Players Actions or strategies Payoffs Represented in a simple table 12-4

Thinking Strategically: Dominant Strategies Each player in the game knows that her payoff depends in part on what the other players do Needs to make a strategic decision, think about her own choice taking other players’ view into account A players’ best response is a strategy that yields her the highest payoff, assuming other players behave in a specified way A strategy is dominant if it is a player’s only best response, regardless of other players’ choices 12-5

The Prisoners’ Dilemma: Scenario Players: Oskar and Roger, both students The situation: they have been accused of cheating on an exam and are being questioned separately by a disciplinary committee Available strategies: Squeal, Deny Payoffs: If both deny, both suspended for 2 quarters If both squeal, both suspended for 5 quarters If one squeals while the other denies, the one who squeals is suspended for 1 quarter and the one who denies is suspended for 6 quarters 12-6

Figure 12.3: Best Responses to the Prisoners’ Dilemma Roger DenySqueal Oskar Deny Squeal (a) Oskar’s Best Response Roger DenySqueal Oskar Deny Squeal (b) Roger’s Best Response 12-7

Thinking Strategically: Iterative Deletion of Dominated Strategies Even if the strategy to choose is not obvious, can sometimes identify strategies a player will not choose A strategy is dominated if there is some other strategy that yields a strictly higher payoff regardless of others’ choices No sane player will select a dominated strategy Dominated strategies are irrelevant and can be removed from the game to form a simpler game Look again for dominated strategies, repeat until there are no dominated strategies left to remove Sometimes allows us to solve games even when no player has a dominant strategy 12-8

Nash Equilibrium in One-Stage Games Concept created by mathematician John Nash, published in 1950, awarded Nobel Prize Has become one of the most central and important concepts in microeconomics In a Nash equilibrium, the strategy played by each individual is a best response to the strategies played by everyone else Everyone correctly anticipates what everyone else will do and then chooses the best available alternative Combination of strategies in a Nash equilibrium is stable A Nash equilibrium is a self-enforcing agreement: every party to it has an incentive to abide by it, assuming that others do the same 12-9

Figure 12.8: Nash Equilibrium in the Prisoners’ Dilemma Roger DenySqueal Oskar Deny Squeal

Nash Equilibria in Games with Finely Divisible Choices Concept of Nash equilibrium also applies to strategic decisions that involve finely divisible quantities Determine each player’s best response function A best response function shows the relationship between one player’s choice and the other’s best response A pair of choices is a Nash equilibrium if it satisfies both response functions simultaneously 12-11

Figure 12.10: Free Riding in Groups 12-12

Mixed Strategies When a player chooses a strategy without randomizing he is playing a pure strategy Some games have no Nash equilibrium in pure strategies, in these cases look for equilibria in which players introduce randomness A player employs a mixed strategy when he uses a rule to randomize over the choice of a strategy Virtually all games have mixed strategy equilibria In a mixed strategy equilibrium, players choose mixed strategies and the strategy each chooses is a best response to the others players’ chosen strategies 12-13

Games with Multiple Stages In most strategic settings events unfold over time Actions can provoke responses These are games with multiple stages In a game with perfect information, players make their choices one at a time and nothing is hidden from any player Multi-stage games of perfect information are described using tree diagrams 12-14

Figure 12.13: Lopsided Battle of the Sexes 12-15

Thinking Strategically: Backward Induction To solve a game with perfect information Player should reason in reverse, start at the end of the tree diagram and work back to the beginning An early mover can figure out how a late mover will react, then identify his own best choice Backward induction is the process of solving a strategic problem by reasoning in reverse A strategy is one player’s plan for playing a game, for every situation that might come up during the course of play Can always find a Nash equilibrium in a multi-stage game of perfect information by using backward induction 12-16

Cooperation in Repeated Games Cooperation can be sustained by the threat of punishment for bad behavior or the promise of reward for good behavior Threats and promises have to be credible A repeated game is formed by playing a simpler game many times in succession May be repeated a fixed number of times or indefinitely Repeated games allow players to reward or punish each other for past choices Repeated games can foster cooperation 12-17

Figure 12.16: The Spouses’ Dilemma Marge and Homer simultaneously choose whether to clean the house or loaf Both prefer loafing to cleaning, regardless of what the other chooses They are better off if both clean than if both loaf 12-18

Repeated Games: Equilibrium Without Cooperation When a one-stage game is repeated, the equilibrium of the one-stage game is one Nash equilibrium of the repeated game Examples: both players loafing in the Spouses’ dilemma, both players squealing in the Prisoners’ dilemma If either game is finitely repeated, the only Nash equilibrium is the same as the one-stage Nash equilibrium Any definite stopping point causes cooperation to unravel 12-19

Repeated Games: Equilibria With Cooperation If the repeated game has no fixed stopping point, cooperation is possible One way to achieve this is through both players using grim strategies With grim strategies, the punishment for selfish behavior is permanent Credible threat of permanent punishment for non-cooperative behavior can be strong enough incentive to foster cooperation 12-20