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

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

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

Prisoner’s dilemma Yes No –1, –1 +3, –6 –6, +3 +1, +1 Player 2 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 Yes No –1, –1 +3, –6 –6, +3 +1, +1 Player 1

Prisoner’s dilemma What is the NE? Yes No –1, –1 +3, –6 –6, +3 +1, +1 Player 2 What is the NE? Let’s underline Yes No –1, –1 +3, –6 –6, +3 +1, +1 Player 1

Prisoner’s dilemma Yes No –1, –1 +3, –6 –6, +3 +1, +1 What is the NE? Player 2 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 Yes No –1, –1 +3, –6 –6, +3 +1, +1 Player 1

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

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

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

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

Battle of the Sexes Bar Play +3, +1 +0, +0 +1, +3 Player 2 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) Bar Play +3, +1 +0, +0 +1, +3 Player 1 Two people plan a date, and each knows that the date is either at the bar or a play

Battle of the Sexes: Other things to note Player 2 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 Bar Play +3, +1 +0, +0 +1, +3 Player 1

Battle of the Sexes Two NE Player 2 Two NE (Bar, Bar) (Play, Play) As in cases before when there are multiple NE, we cannot determine which outcome will actually occur Bar Play +3, +1 +0, +0 +1, +3 Player 1

Battle of the Sexes Player 2 Battle of the Sexes is known as a coordination game Both get a positive payout if they show up to the same place Bar Play +3, +1 +0, +0 +1, +3 Player 1

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”

Chicken Swerve Straight +0, +0 –1, +1 +1, –1 –10, –10 Player 2 Swerve Straight +0, +0 –1, +1 +1, –1 –10, –10 Player 1 Next step: Underline as before

Chicken Swerve Straight +0, +0 –1, +1 +1, –1 –10, –10 Player 2 Swerve Straight +0, +0 –1, +1 +1, –1 –10, –10 Player 1 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

Matching pennies 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 Heads Tails +1, –1 –1, +1 Player 1 Two players each choose Heads or Tails

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

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

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

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

Subordinate pig/Dominant pig The numbers on the previous slide translate to the payoff matrix seen dominant pig Yes No 3, 7 –1, 12 10, 1 0, 0 subordinate pig Next: Underline test

Subordinate pig/Dominant pig 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 dominant pig Yes No 3, 7 –1, 12 10, 1 0, 0 subordinate pig Exactly 1 NE The dominant pig presses lever

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

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