Chapter 14 – Game Theory

1. Chapter 14 – Game Theory 14.1 Nash Equilibrium 14.2 Repeated Prisoners’ Dilemma 14.3 Sequential-Move Games and Strategic Moves

2. Game Theory and Life You are on a first date with the love of your dreams. You can propose 2 activities: • Safe activity (Coffee) • Exciting Activity (Waterpark) Your date could either want a safe activity or an exciting activity. There are different results if your ideas match up or clash:

3. You Mr/Miss Right First Date Game What is the outcome of this game? Payoff format is (Left, Top) Chapter Fourteen

4. Game Theory Components Players:agents participating in the game (You and Your Date Strategies: Actions that each player may take under any possible circumstance (Coffee, Waterpark) Outcomes: The various possible results of the game (four, each represented by one cell of the payoff matrix) Payoffs: The benefit that each player gets from each possible outcome of the game (the profits entered in each cell of the payoff matrix) Chapter Fourteen

5. Best Responses In all game theory games, players choose strategies without knowing with certainty what the opposing player will do. • Players construct BEST RESPONSES • -optimal actions given all possible actions of other players

6. You Mr/Miss Right First Date Game Best Responses If you know your date will pick coffee, you should pick coffee, since 10 > -5 If you know your date will pick waterpark, you should pick waterpark, since 20 > 0 Chapter Fourteen

7. You Mr/Miss Right First Date Game Best Responses If your date knows you will pick coffee, they should pick coffee, since 10 > -5 If your date knows you will pick waterpark, they should pick waterpark, since 20 > 0 Note that this game is SYMMETRICAL Chapter Fourteen

8. Nash Equilibrium Definition: A Nash Equilibrium occurs when each player chooses a strategy that gives him/her the highest payoff, given the strategy chosen by the other player(s) in the game. ("rational self-interest") Nash Equilibria occur when best responses line up The Date Game: Nash equilibria: Each proposes coffee or each proposes waterpark. Chapter Fourteen

9. Game Theory • A special kind of Best Response: DOMINANT STRATEGY • Strategy that is best no matter what the other player does.

10. Advertising A’s STRATEGY Don’t advertise Advertise B’sSTRATEGY A’s profit= $50 000 A’s profit= $75 000 Don’t advertise B’s profit = $50 000 B’s loss = $25 000 A’s loss = $25 000 A’s profit = $10 000 Advertise B’s profit = $10 000 B’s profit = $75 000

11. Dominant Strategy A’s dominant strategy is advertise B’s dominant strategy is advertise Don’t advertise Advertise A’s profit= $50 000 A’s profit= $75 000 Don’t advertise B’s profit = $50 000 B’s loss = $25 000 A’s loss = $25 000 A’s profit = $10 000 Advertise B’s profit = $10 000 B’s profit = $75 000

12. Prisoner’s Dilemma • This is an example of a prisoner’s dilemma type of game. • There is dominant strategy. • The dominant strategy does not result in the best outcome for either player. • It is hard to cooperate even when it would be beneficial for both players to do so • Cooperation between players is difficult to maintain because cooperation is individually irrational. • eg., The dominant strategy: advertise

13. 1 yearPrison Go free 1 year Prison 7 years Prison 5 years Prison 7 years Prison 5 years Prison Go free Classic Prisoners’ Dilemma Rocky’s strategies Deny Confess Dominant strategy: confess, even though they would both be better off if they both kept their mouths shut. Deny Ginger’s strategies Confess

14. Dominant Strategy Equilibrium Definition: A Dominant Strategy Equilibrium occurs when each player uses a dominant strategy. Toyota Honda

15. Dominated Strategy Definition: A player has a dominated strategywhen the player has another strategy that gives it a higher payoff no matter what the other player does. Toyota Honda Chapter Fourteen

16. Dominant or Dominated Strategy Why look for dominant or dominated strategies? A dominant strategy equilibrium is particularly compelling as a "likely" outcome Similarly, because dominated strategies are unlikely to be played, these strategies can be eliminated from consideration in more complex games. This can make solving the game easier. Chapter Fourteen

17. Dominated Strategy Toyota Honda "Build Large" is dominated for each player By eliminating the dominated strategies, we can reduce the game matrix.

18. Finding Nash Equilibrium Cases • Nash Equilibrium where Dominant Strategies overlap • Nash Equilibrium with one Dominant Strategy • Nash Equilibrium by eliminating Dominated Strategy • Nash Equilibrium through Best Responses Chapter Fourteen

19. Nash Equilibrium – Dominant Overlap Professor Student

20. Nash Equilibrium – One Dominant Professor Student

21. Nash Equilibrium – Eliminate Dominated Professor Student

22. Nash Equilibrium – Best Responses Professor Student

23. Nash Equilibrium • However it is found, a Nash Equilibrium ALWAYS occurs where Best Responses line up • If Multiple Nash Equilibria exist, we can’t conclude WHICH outcome will occur, only the possible outcomes that can occur • Also, it is often APPEARS that no Nash Equilibria exist:

24. No Nash Equilibrium Fred Barney

25. Mixed Strategies Pure Strategy – A specific choice of a strategy from the player’s possible strategies in a game. (ie: Rock) Mixed Strategy – A choice among two or more pure strategies according to pre-specified probabilities. (ie: Rock, Paper or Scissors each 1/3rd of the time) If Pure Strategies can’t produce a Nash Equilibrium, Mixed Strategies can: If both players randomize each choice 1/3rd of the time, nether have an incentive to deviate.