STRATEGIC DECISION MAKING GAME THEORY STRATEGIC DECISION MAKING

What is Game Theory? GT is an analytical tool for social sciences that is used to model strategic interactions or conflict situations. Strategic interaction: When actions of a player influence payoffs to other players

GT: science or art? GT is the science of rational behavior in interactive situations. Good strategists mix the science of GT with their own experience.

Why is GT important? Facilitates strategic thinking. Provides a standard taxonomy that is needed for a scientific approach in analyzing strategic interactions. Helps confirm long held beliefs. Provides new insights. “To be literate in the modern age, you need to have a general understanding of GT.” P.Samuelson

GT in the News GT, long an intellectual passtime, came into its own as a business tool. Forbes, July 3, 1995 FCC hired game theorists to construct rules of an auction for new wireless phone systems’ licenses. In response, communications companies hired game theorists first to negotiate with FCC and then help prepare optimal bids given the rules of the auction. Business Week, Mar. 14, 1994. GT is hot. The Wall Street Journal, Feb. 13, 1995.

GT in the News (cont.) …lately game theorists have focused on real-world issues—how to raise auction proceeds by revealing the bids and how Wal-Mart can coexist with local retailers. Consultants are jumping in. McKinsey & Co. has set up a game theory unit. Forbes, Nov. 7, 1994.

Where can we use GT? Any situation that requires us to anticipate our rival’s response to our action is a potential context for GT. Games: Checkers, poker, chess, tennis, soccer etc. Economics: Industrial Organization, Micro/Macro/ International/Labor/Natural resource Economics, and Public Finance Political science: war/peace (Cuban missile crisis) Law: Designing laws that work Biology: animal behavior, evolution Information systems: System competition/evolution

Where can we use GT? (cont.) Business: Games against rival firms: Pricing, advertising, marketing, auctions, R&D, joint ventures, investment, location, quality, take over etc. Games against other players Employee/employer, managers/stockholders Supplier/buyer, producer/distributor, firm/government

Some Terminology Strategy Payoffs Rationality Common knowledge of rules Equilibrium

Strategic (Normal) Form Games Static Games of Complete and Imperfect Information

What is a Normal Form Game? A normal (strategic) form game consists of: Players: list of players Strategies: all actions available to all players Payoffs: a payoff assigned to every contingency (every possible strategy profile as the outcome of the game)

Prisoners’ Dilemma Two suspects are caught and put in different rooms (no communication). They are offered the following deal: If both of you confess, you will both get 5 years in prison (-5 payoff) If one of you confesses whereas the other does not confess, you will get 0 (0 payoff) and 10 (-10 payoff) years in prison respectively. If neither of you confess, you both will get 2 years in prison (-2 payoff)

Easy to Read Format of Prisoner’s Dilemma Confess Don’t Confess -5, -5 0, -10 -10, 0 -2, -2 Prisoner 1

Assumptions in Static Normal Form Games All players are rational. Rationality is common knowledge. Players move simultaneously. (They do not know what the other player has chosen). Players have complete but imperfect information.

Solution of a Static Normal Form Game Equilibrium in strictly dominant strategies A strictly dominant strategy is the one that yields the highest payoff compared to the payoffs associated with all other strategies. Rational players will always play their strictly dominant strategies.

Solution of a Static Normal Form Game Iterated elimination of strictly dominated strategies Rational players will never play their dominated strategies. Eliminating dominated strategies may solve the game.

Solution of a Static Normal Form Game (cont.) Nash Equilibrium (NE): In equilibrium neither player has an incentive to deviate from his/her strategy, given the equilibrium strategies of rival players. Neither player can unilaterally change his/her strategy and increase his/her payoff, given the strategies of other players.

Definition of Nash Equilibrium A strategy profile is a list (s1, s2, …, sn) of the strategies each player is using. If each strategy is a best response given the other strategies in the profile, the profile is a Nash equilibrium. Why is this important? If we assume players are rational, they will play Nash strategies. Even less-than-rational play will often converge to Nash in repeated settings.

An Example of a Nash Equilibrium Column a b a 1,2 0,1 Row b 1,0 2,1 (b,a) is a Nash equilibrium. To prove this: Given that column is playing a, row’s best response is b. Given that row is playing b, column’s best response is a.

Finding Nash Equilibria – Dominated Strategies What to do when it’s not obvious what the equilibrium is? In some cases, we can eliminate dominated strategies. These are strategies that are inferior for every opponent action. In the previous example, row = a is dominated.

Example A 3x3 example: Column a b c a 73,25 57,42 66,32 Row b 80,26 35,12 32,54 c 28,27 63,31 54,29

Example A 3x3 example: Column a b c a 73,25 57,42 66,32 Row b 80,26 35,12 32,54 c 28,27 63,31 54,29 c dominates a for the column player

Example A 3x3 example: Column a b c a 73,25 57,42 66,32 Row b 80,26 35,12 32,54 c 28,27 63,31 54,29 b is then dominated by both a and c for the row player.

Example A 3x3 example: Column a b c a 73,25 57,42 66,32 Row b 80,26 35,12 32,54 c 28,27 63,31 54,29 Given this, b dominates c for the column player – the column player will always play b.

Solution of Prisoners’ Dilemma Dominant Strategy Equilibrium Confess Don’t Confess -5, -5 0, -10 -10, 0 -2, -2 Prisoner 1

Solution of Prisoners’ Dilemma Iterated Elimination Procedure Confess Don’t Confess -5, -5 0, -10 -10, 0 -2, -2 Prisoner 1

Solution of Prisoners’ Dilemma Cell-by-cell Inspection Confess Don’t Confess -5, -5 0, -10 -10, 0 -2, -2 Prisoner 1

NE of Prisoners’ Dilemma The strategy profile {confess, confess} is the unique pure strategy NE of the game. In equilibrium both players get a payoff of –5. Inefficient equilibrium; (don’t confess, don’t confess) yields higher payoffs for both.

A Pricing Example 100, 100 -10, 140 140, -10 0, 0 High Price Low Price Firm 2 High Price Low Price 100, 100 -10, 140 140, -10 0, 0 Firm 1

3x3 Game Using Iterated Elimination Player 2 Left Center Right Top 1, 0 1, 3 3, 0 Middle 0, 2 0, 1 Bottom 2, 4 5, 3 Player 1

A Coordination Game Battle of the Sexes Husband Opera Movie 2, 1 0, 0 1, 2 Wife

Battle of the Sexes: After 30 Years of Marriage Husband Opera Movie 3, 2 0, 0 1, 2 Wife

Mixed strategies Unfortunately, not every game has a pure strategy equilibrium. Rock-paper-scissors However, every game has a mixed strategy Nash equilibrium. Each action is assigned a probability of play. Player is indifferent between actions, given these probabilities.

Mixed Strategies In many games (such as coordination games) a player might not have a pure strategy. Instead, optimizing payoff might require a randomized strategy (also called a mixed strategy) Wife football shopping football 2,1 0,0 Husband shopping 1,2 0,0

A Strictly Competitive Game Matching Pennies Player 2 Heads Tails 1, -1 -1, 1 Player 1 No NE in pure strategies

Dynamic Games of Complete and Perfect Information Extensive Form Games Dynamic Games of Complete and Perfect Information

What is a Game Tree? Right Left A B C D P11 P21 P12 P22 P13 P23 P14 Player 1 Right Left Player 2 Player 2 A B C D P11 P21 P12 P22 P13 P23 P14 P24

An Advertising Example Migros Normal Aggressive Wal-Mart Wal-Mart Enter Enter Stay out Stay out 680 -50 730 700 400 800

Assumptions in Dynamic Extensive Form Games All players are rational. Rationality is common knowledge Players move sequentially. (Therefore, also called sequential games) Players have complete and perfect information Players can see the full game tree including the payoffs Players can observe and recall all previous moves

Solution of an Extensive Form Game Subgame Perfect Equilibrium: For an equilibrium to be subgame perfect, it has to be a NE for all the subgames as well as for the entire game. A subgame is a decision node from the original game along with the decision nodes and end nodes. Backward induction is used to find SPE

Advertising Example: 3 proper subgames Migros Wal-Mart Wal-Mart 680 -50 730 700 400 800

Solution of the Advertising Game Subgame 1 Subgame 2 Wal-Mart Wal-Mart Enter Enter Stay out Stay out 680 -50 730 700 400 800

Solution of the Advertising Game (cont.) Migros Normal Aggressive 730 700 400 SPE of the game is the strategy profile: {aggressive, (stay out, enter)}

Properties of SPE The outcome that is selected by the backward induction procedure is always a NE of the game with perfect information. SPE is a stronger equilibrium concept than NE SPE eliminates NE that involve incredible threats.

Suppose WM threatens to enter no matter what Migros does Suppose WM threatens to enter no matter what Migros does. Is this a credible threat? Migros Normal Aggressive Wal-Mart Wal-Mart Enter Enter Stay out Stay out 680 -50 730 700 400 800

JUST PLAYING! Repeated Games

Game 1 Firm 1 Firm 2 Low Output High Output 40, 40 60, 30 30, 60 50, 50 Firm 1

Game 2 Greece War Peace 1, 1 3, 0 0, 3 2, 2 Turkiye

Repeated Normal Form Games REPEATED GAMES Repeated Normal Form Games

Prisoners’ Dilemma Revisited Suppose that the two suspects play the same game every time they get caught. Can they coordinate their choices in order to get the best outcome for both of them? Finitely repeated game Infinitely repeated game

Twice-repeated PD: First stage payoff matrix after adding NE payoffs of the second stage Prisoner 2 Confess Don’t Confess -10, -10 -5, -15 -15, -5 -7, -7 Prisoner 1

N-times repeated PD In a finitely repeated (n times repeated game where n  2) PD game, the cooperative outcome (don’t confess, don’t confess) cannot be enforced. Since in the last stage (nth stage) the NE is (confess, confess) and all players know this, in all previous stages the same NE will prevail.

Infinitely Repeated PD When the game is played infinitely or players do not know when the game is going to end, the backward induction breaks down. Following trigger strategies can enforce the cooperative outcome. Trigger strategy: A player cooperates as long as the other players cooperate, but any defection from cooperation on the part of the rivals triggers the player to behave noncooperatively for a specified period of time (period of punishment).

Trigger Strategies Grim strategy: A trigger strategy in which the punishment period lasts till the end of the game. Grim strategy for PD game: Play “don’t confess” in the first period. In period t, play “don’t confess” if the outcome was (don’t confess, don’t confess) in all preceding t-1 periods, and play “confess” otherwise.

Trigger Strategies (cont.) Tit-for-tat (TFT): A trigger strategy in which the punishment period lasts as long as the rival keeps on cheating (returning back to cooperative periods of game play is possible). TFT strategy for PD game: Play “don’t confess” in the first period. In period t, play “don’t confess” if the rival’s most recent play was (don’t confess, don’t confess), and play “confess” otherwise.