Recap

1. Recap

2. The Late Assignment Game Partner 2 Partner 1 • What to do? • Each player should just hand it in and receive a 60% grade • Why? • Because handing it in is a strictly dominant strategy • Meaning that, no matter what the other partner does, it is always the better choice in terms of payoff

3. Nash Equilibrium • Developed by John Forbes Nash Jr. • Concept of a non-cooperative games • Set of strategies (one for each player) that are all a best response to the others • Each player is assumed to know the equilibrium strategies of the other players • No player has anything to gain by changing only their own strategy

4. Nash Equilibrium • Developed by John Forbes Nash Jr. • Concept of a non-cooperative games • Set of strategies (one for each player) that are all a best response to the others • Each player is assumed to know the equilibrium strategies of the other players • No player has anything to gain by changing only their own strategy

5. Nash Equilibrium: An Example • Consider two competing companies that each sell “widgets”. Each needs to set a price for their widgets, with either $1, $2 or $3 being the options. Each company must also keep in mind that higher prices will bring in more profit for sale, but could hurt total sales, particularly if the other company sells for less. After accounting for costs, the two companies’ payoffs for each set of strategies is as follows (in millions): • What is the Nash equilibrium? B A

6. Nash Equilibrium: An Example • Consider two competing companies that each sell “widgets”. Each needs to set a price for their widgets, with either $1, $2 or $3 being the options. Each company must also keep in mind that higher prices will bring in more profit for sale, but could hurt total sales, particularly if the other company sells for less. After accounting for costs, the two companies’ payoffs for each set of strategies is as follows (in millions): • What is the Nash equilibrium? B A

7. Example Coordination Game • Imagine a couple that agreed to meet this evening.The husband would most of all like to go to the football game. The wife would like to go to the opera. Both would prefer to go to the same place rather than different ones. If they cannot communicate, where should they go? • This is an example of a coordination game, often called “Battle of the Sexes” • It is also an example of multiple equilibria Wife Husband

8. Lecture 2

9. Another Example: The Hawk-Dove Game • Suppose two animals are engaged in a contest to decide how a piece of food will be divided between them. Each animal can choose to behave aggressively (the Hawk strategy) or passively (the Dove strategy). If the two animals both behave passively, they divide the food evenly, and each get a payoff of 3. If one behaves aggressively while the other behaves passively, then the aggressor gets most of the food, obtaining a payoff of 5, while the passive one only gets a payoff of 1. But if both animals behave aggressively, then they destroy the food. B A

10. Another Example: The Hawk-Dove Game • Suppose two animals are engaged in a contest to decide how a piece of food will be divided between them. Each animal can choose to behave aggressively (the Hawk strategy) or passively (the Dove strategy). If the two animals both behave passively, they divide the food evenly, and each get a payoff of 3. If one behaves aggressively while the other behaves passively, then the aggressor gets most of the food, obtaining a payoff of 5, while the passive one only gets a payoff of 1. But if both animals behave aggressively, then they destroy the food. • This is an example of an anti-coordination game, since Nash equilibria exist where no coordination takes place B A

11. Examples of Hawk-Dove • Chicken • Two people drive toward each other on a collision course • The first to swerve is declared a “chicken”, and thus deemed the loser • Nuclear brinkmanship • Two countries on the verge of nuclear war • Nash equilibrium for one to attack (hawk) and one not (dove) • Each carries their threat to act irrationally and retaliate to the “brink” in order to influence the opponent to the more favourable Nash equilibrium

12. Pareto Optimality • A choice of strategies is Pareto-optimal if there is no other choice of strategies in which all players receive payoffs at least as high, and at least one player receives a strictly higher payoff

13. Pareto Optimality • A choice of strategies is Pareto-optimal if there is no other choice of strategies in which all players receive payoffs at least as high, and at least one player receives a strictly higher payoff

14. Pareto Optimality • A choice of strategies is Pareto-optimal if there is no other choice of strategies in which all players receive payoffs at least as high, and at least one player receives a strictly higher payoff • The only non-Pareto-optimal outcome is the only Nash equilibrium! • Surely both parties would agree to change the outcome • Not enforceable in non-cooperative games. Exploitable!

15. Practice Problem 1 • Two individuals go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, he must have the cooperation of his partner in order to succeed (payoff of 4). An individual can get a hare by himself, but a hare is worth less than a stag (payoff of 3). • Draw the payoff table and find all Nash equilibria • Solution in text.

16. Practice Problem 2 • Create three different normal form games with Nash equilibria: • One with exactly one Nash equilibrium • One coordination game • One anti-coordination game • For each, don’t bother generating a narrative. Just generate the payoff table (2x2 is fine, bigger is OK too) and specify the Nash equilibria

17. Mixed Strategies • Matching Pennies game • Two roles: • Attacker • Defender • Each hold a penny and reveal heads or tails simultaneously: • If the two are different, the attacker wins • If the two are the same, the defender wins • Winner keeps the pennies

18. Matching Pennies • Payoff matrix: • Any dominant strategy? • Nash equilibria? Defender Attacker

19. Mixed Strategies • Don’t simply choose H or T directly i.e. the “pure” strategies • Instead, choose a “mix” of H or T according to some probability • E.g. 60% chance of playing H, 40% chance of playing T • The choice now is not H or T, but rather the probability p • Note that the set of mixed strategies includes the pure strategies • i.e. p(H) = 1 and p(H) = 0

20. Matching Pennies • Nash proved that if we allow mixed strategies, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium • What is the Nash equilibrium for the Matching Pennies game if mixed strategies are allowed? Defender Attacker

21. Matching Pennies • Nash proved that if we allow mixed strategies, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium • What is the Nash equilibrium for the Matching Pennies game if mixed strategies are allowed? • Attacker: p = ½; Defender: q = ½ Defender Attacker

22. Run-Pass Game • North American football. Each team needs to decide which play to employ next: the attacking team (offense) whether to pass the ball or run with it, the defending team (defense) to defend the pass or run. If the defender guesses correctly, the offense gains no yardage. However if they incorrectly try to defend against the run and the offense passes, a huge gain of 10 is made. Incorrectly defend against the pass when the offense runs results in a moderate gain of 5. • What is the mixed strategy Nash equilibrium?

23. Run-Pass Game • North American football. Each team needs to decide which play to employ next: the attacking team (offense) whether to pass the ball or run with it, the defending team (defense) to defend the pass or run. If the defender guesses correctly, the offense gains no yardage. However if they incorrectly try to defend against the run and the offense passes, a huge gain of 10 is made. Incorrectly defend against the pass when the offense runs results in a moderate gain of 5. • What is the mixed strategy Nash equilibrium? • Offense: p = 1/3; Defense: q = 2/3

24. Another Example: The Hawk-Dove Game • Suppose two animals are engaged in a contest to decide how a piece of food will be divided between them. Each animal can choose to behave aggressively (the Hawk strategy) or passively (the Dove strategy). If the two animals both behave passively, they divide the food evenly, and each get a payoff of 3. If one behaves aggressively while the other behaves passively, then the aggressor gets most of the food, obtaining a payoff of 5, while the passive one only gets a payoff of 1. But if both animals behave aggressively, then they destroy the food. • This is an example of an anti-coordination game, since Nash equilibria exist where no coordination takes place B A

25. Practice Problem 3 • Find the mixed-strategy Nash equilibria for the Hawk-Dove game

26. What Do Nash Equilibria Mean? • What do they tell us? How can they be used? • Descriptive uses • Study how people act in situations • Prediction: what agreements will be made, how partnerships will form, who will win and who will lose • Prescriptive uses • Study how people should act • Will a strategy from Nash equilibrium always be the best? No. But others can be exploited to the point that they’ll have worse outcomes • Mechanism uses • Ability to enforce agreement in non-cooperative games • Disincentivizes unwanted behaviour • All Nash equilibria give the space of such possible agreements