More game theory

1. More game theory Today: Some classic games in game theory

2. Last time… • Introduction to game theory • Games have players, strategies, and payoffs • Based on a payoff matrix with simultaneous decisions, we can find Nash equilibria (NE) • In sequential games, some NE can be ruled out if people are rational

3. Today, some classic game theory games • Games with inefficient equilibria • Prisoner’s Dilemma • Public Goods game • Coordination games • Battle of the Sexes • Chicken • Zero-sum game • Matching pennies • Animal behavior • Subordinate pig/Dominant pig

4. Prisoner’s dilemma • Why is this game called prisoner’s dilemma? • Think about a pair of criminals that have a choice of whether or not to confess to a crime Player 2 Player 1

5. Prisoner’s dilemma • What is the NE? • Let’s underline Player 2 Player 1

6. Prisoner’s dilemma • What is the NE? • Let’s underline • Each player has a dominant strategy of choosing Yes • However, both players get a better payout if each chooses No Player 2 Player 1

7. Prisoner’s dilemma and cartels • Cartels are usually unstable since each firm has a dominant strategy to charge a lower price and sell more • See Table 11.4 (p. 327) for an example

8. Public goods game • You can decide whether or not you want to contribute to a new flower garden at a local park • If you decide Yes, you will lose $200, but every other person in the city you live in will gain $10 in benefits from the park • If you decide No, you will cause no change to the outcome of you or other people

9. Public goods game • What is each person’s best response, given the decision of others? • We need to look at each person’s marginal gain and loss (if any) • Choose yes  Gain $10, lose $200 • Choose no  Gain $0, lose $0

10. Public goods game • Which is the better choice? • Choose no (Gain nothing vs. net loss of $190) • NE has everybody choosing no • Efficient outcome has everybody choosing yes • Why the difference? • Each person does not account for others’ benefits when making their own decision

11. Two people plan a date, and each knows that the date is either at the bar or a play Neither person knows where the other is going until each person shows up If both people show up at the same place, they enjoy each other’s company (+1 for each) Battle of the Sexes Player 2 Player 1

12. Player 1 gets additional enjoyment from the bar if Player 2 is there too, since Player 1 likes the bar more Player 2 enjoys the play more than Player 1 if both show up there As before, we underline the best strategy, given the strategy of the other player Battle of the Sexes:Other things to note Player 2 Player 1

13. Two NE (Bar, Bar) (Play, Play) As in cases before when there are multiple NE, we cannot determine which outcome will actually occur Battle of the Sexes Player 2 Player 1

14. Battle of the Sexes is known as a coordination game Both get a positive payout if they show up to the same place Battle of the Sexes Player 2 Player 1

15. Chicken • Two cars drive toward each other • If neither car swerves, both drivers sustain damage to themselves and their cars • If only one person swerves, this person is known forever more as “Chicken”

16. Chicken Player 2 • Next step: Underline as before Player 1

17. Chicken Player 2 • Notice there are 2 NE • One player swerves and the other goes straight • This game is sometimes referred to as an “anti-coordination” game • NE results from each player making a different decision Player 1

18. Two players each choose Heads or Tails If both choices match, Player 1 wins If both choices differ, Player 2 wins This is an example of a zero-sum game, since the sum of each box is zero Matching pennies Player 2 Player 1

19. Underlining shows no NE A characteristic of zero-sum games Whenever I win, the other player must lose Matching pennies Player 2 Player 1

20. Subordinate pig/Dominant pig • Two pigs are placed in a cage • Left end of cage: Lever to release food • 12 units of food released when lever is pressed • Right end of cage: Food is dispensed here

21. Subordinate pig/Dominant pig • If both press lever at the same time, the subordinate pig can run faster and eat 4 units of food before the dominant pig “hogs” the rest • If only the dominant pig presses the lever, the subordinate pig eats 10 of the 12 units of food • If only the subordinate pig presses the lever, the dominant pig eats all 12 units • Pressing the lever exerts a unit of food

22. Subordinate pig/Dominant pig • Who do you think will get more food in equilibrium? • Who thinks ? • Who thinks ?

23. Next: Underline test The numbers on the previous slide translate to the payoff matrix seen Subordinate pig/Dominant pig dominant pig subordinate pig

24. Exactly 1 NE The dominant pig presses lever In Nash equilibrium, the dominant pig always gets the lower payout Why? The subordinate pig has a dominant strategy: No The dominant pig, knowing that the subordinate pig will not press the lever, will want to press the lever Subordinate pig/Dominant pig dominant pig subordinate pig

25. Do people always play as Nash equilibrium predicts? • No • Many papers have shown that people often are not selfish, and donate into public goods • Norms are often established to make sure that people are encouraged to act in the best interest of society

26. Summary • Today, we looked at some well-known games • Some games have NE; others do not • However, people do not always behave as NE would predict