14 2 Repeated Prisoners Dilemma If the Prisoners

14. 2 Repeated Prisoner’s Dilemma If the Prisoner’s Dilemma is repeated, cooperation can come from strategies including: “Grim Trigger” Strategy – one episode of cheating by one player triggers the grim prospect of a permanent breakdown in cooperation for the remainder of the game. “Tit-for-Tat” Strategy – A strategy in which you do to your opponent in this period what your opponent did to you in the last period. 1

Prisoners’ Dilemma Review Rocky’s strategies Deny Ginger’s strategies Confess 1 year Prison 7 years Prison Go free Confess Go free 7 years Prison Can the Grim Trigger Strategy change the outcome? 5 years Prison 2

14. 2 “Grim Trigger” Strategy Example 1 “Grim Trigger” Strategy – “If you confess once, I’ll never trust you again and I’ll always confess” If player 1 confesses the first time, while the other denies, player 1 goes free. In each following term, they both confess for 5 years in prison. If the game is played N times, overall return is: 5(N-1) years in prison 3

“Grim Trigger” Strategy Example 1 If the players always co-operate and deny, each term they spend 1 year in prison. If the game is played N times, overall return is: N years in prison Since N < 5(N-1) for N>1, players will co-operate Note: It is important that players are UNCERTAIN about how many games will be played. 4

“Grim Trigger” Strategy Example 2 Consider the following Prisoner’s Dilemma: Can the Grim Trigger strategy (“If you cheat, I’ll never co-operate”) cause players to co-operate? 5

“Grim Trigger” Strategy Example 2 Returns can be summarized in the following graph: If N>1, it makes sense to co-operate. 6

Prisoners’ Dilemma Review Rocky’s strategies Deny Ginger’s strategies Confess 1 year Prison 7 years Prison Go free Confess Go free 7 years Prison Can a Tit-for. Tat Strategy change the outcome? 5 years Prison 7

“Tit-for-Tat” Strategy Example 1 “Tit-for-Tat” Strategy – “I will do whatever you did last term” If player 1 confesses the first time, while the other denies, player 1 goes free. In each following term, their partner will confess. Option 1: Confess Twice: 5 Years in Prison Option 2: Confess First time: 7 Years in Prison 8

“Tit-for-Tat” Strategy Example 1 If a player co-operates the first term, next term their partner will co-operate. In the second period: Option 1: Confess : 1 Year in Prison Option 2: Co-Operate: 2 Years in Prison In either case, Co-operating in turn 1 is a best response. 9

Repeated Prisoner’s Dilemma Likelihood of cooperation increases if: 1. The players are patient. 2. Interactions between the players are frequent. 3. Cheating is easy to detect. 4. The one-time gain from cheating is relatively small. 10 Chapter Fourteen

Repeated Prisoner’s Dilemma Likelihood of cooperation diminishes if: 1. The players are impatient. 2. Interactions between the players are infrequent. 3. Cheating is hard to detect. 4. The one-time gain from cheating is large in comparison to the eventual cost of cheating. 11 Chapter Fourteen

The Final Turn As long as the game continues, both strategies continue to be best responses It is important that players are UNCERTAIN about how many games will be played. If the game end’s is certain, it is a best response to cheat in that game. Since both players know the last turn will be cheating, it MAY be best to start cheating in other 12 turns.

14. 3 Game Trees A game tree shows the different strategies that each player can follow in the game and can show the order in which those strategies get chosen. Backward induction is a procedure for solving a game tree (finding Nash Equilibria) by starting at the end of the game tree and finding the optimal decision for the player at each decision point. 13 Chapter Fourteen

14. 3 Prisoner Dilemma Game Tree Confess B Deny (5 years, 5 years) (Go Free, 7 years) A Deny Confess B Deny (7 years, Go Free) (1 year, 1 year) 14 Chapter Fourteen

14. 3 Prisoner Dilemma Backwards Induction Confess A Deny Confess (5 years, 5 years) With a B definite end Deny (Go Free, 7 years) turn, Tit-for -Tat and Grim Confess Strategy (7 years, Go Free) are B dominated Deny by Confess. (1 year, 1 year) 15 Chapter Fourteen

14. 3 Prisoner Dilemma Backwards Induction Confess A Deny Confess (5 years, 5 years) Since B will B Confess, Deny (Go Free, 7 years) the only best response is Confess B (7 years, Go Free) Deny (1 year, 1 year) 16 Chapter Fourteen

Sequential Move Games In Sequential Move Games, one player (the first mover) takes an action before another player (the second mover). The second mover observes the action taken by the first mover before acting. A Subgame is all subsequent decisions players make given actions already made. 17 Chapter Fourteen

Sequential Move Games – Game Tree Each box represents a simultaneous decision. This game subgames: has 4 a) each of the 3 possible decisions for T, b) the entire game 18 Chapter Fourteen

Subgame Perfect Nash Equilibrium Backward induction can be used to find the Nash Equilibrium of each subgame. If each subgame is a Nash Equilibrium, the entire game is a Subgame Perfect Nash Equilibrium (SPNE) 19 Chapter Fourteen

Sequential Move Games – Game Tree 20 Chapter Fourteen

Simultaneous Game Comparison ØNote that the Simultaneous and Sequential Game outcomes differ ØGame trees may be more difficult to use in simultaneous games 21 Chapter Fourteen

Limiting Options By moving first and committing to “Build Large”, Honda limited its options, but was better off for it. Strategic moves are actions that a player takes in an early stage of a game that alter the player’s behavior and the other players’ behavior later in the game in a way that is favorable to the first player. Examples: i. Pod is incompatible with Windows Media Player, “We do not negotiate with terrorists. ” 22 Chapter Fourteen

Limiting Options It is important when limiting options that the actions be visible and difficult to reverse. Options include: 1) Investing capital that has no alternate uses 2) Signing contracts (ie: Most Favored Customer Clause) 3) Public statements 23 Chapter Fourteen

The Board Game When choosing board games simultaneously, Mr. Cool could play his least favourite game (Monopoly), or his favourite game (Small World): Mr. Boring Mr. Cool 24 Chapter Fourteen

The Board Game Mon Mr. B MR. C SW SM Mon Mr. B SW (-1, 40)) (-5, -5) By moving first (ie: deciding which games to bring), Mr. Cool limits his options and ensures highest utility. (-5, -5) (40, 15) 25 Chapter Fourteen

Chapter 14 Conclusion 1) A Nash Equilibrium occurs when best responses line up. 2) Dominant Strategies are always chosen, and Dominated Strategies have another strategy that is better. 3) If no Nash Equilibrium exists in pure strategies, a Nash Equilibrium can be found in mixed strategies. 4) If a game is repeated, the Grim Trigger Strategy and “Tit-for-tat” strategy can create a Nash 26 Equilibrium.

Chapter 14 Conclusion 5) Co-operation depends on many factors, often including no set last turn. 6) If each subgame has a Nash Equilibrium, the entire game has a Subgame Perfect Nash Equilibrium. 7) In sequential-move games, a game tree is useful to calculate a Subgame Perfect Nash Equilibrium. 8) Moving first and limiting one’s actions can be advantageous. 27 9) Small World is better than Monopoly.