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Game Theory II. Definition of Nash Equilibrium. A game has n players. Each player i has a strategy set S i This is his possible actions Each player has a payoff function p I : S ! R I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
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1. Game Theory II

2. Definition of Nash Equilibrium • A game has n players. • Each player i has a strategy set Si • This is his possible actions • Each player has a payoff function • pI: S ! R • A strategy ti2 Siis a best response if there is no other strategy in Si that produces a higher payoff, given the opponent’s strategies.

3. Definition of Nash Equilibrium • A strategy profile is a list (s1, s2, …, sn) of the strategies each player is using. • If each strategy is a best response given the other strategies in the profile, the profile is a Nash equilibrium. • Why is this important? • If we assume players are rational, they will play Nash strategies. • Even less-than-rational play will often converge to Nash in repeated settings.

4. An Example of a Nash Equilibrium Column a b a 1,2 0,1 Row b 1,0 2,1 (b,a) is a Nash equilibrium. To prove this: Given that column is playing a, row’s best response is b. Given that row is playing b, column’s best response is a.

5. Finding Nash Equilibria – Dominated Strategies • What to do when it’s not obvious what the equilibrium is? • In some cases, we can eliminate dominated strategies. • These are strategies that are inferior for every opponent action. • In the previous example, row = a is dominated.

6. Example • A 3x3 example: Column a b c a 73,25 57,42 66,32 Row b 80,26 35,12 32,54 c 28,27 63,31 54,29

7. Example Column • A 3x3 example: a b c a 73,25 57,42 66,32 Row b 80,26 35,12 32,54 c 28,27 63,31 54,29 c dominates a for the column player

8. Example Column • A 3x3 example: a b c a 73,25 57,42 66,32 Row b 80,26 35,12 32,54 c 28,27 63,31 54,29 b is then dominated by both a and c for the row player.

9. Example Column • A 3x3 example: a b c a 73,25 57,42 66,32 Row b 80,26 35,12 32,54 c 28,27 63,31 54,29 Given this, b dominates c for the column player – the column player will always play b.

10. Example Column • A 3x3 example: a b c a 73,25 57,42 66,32 Row b 80,26 35,12 32,54 c 28,27 63,31 54,29 Since column is playing b, row will prefer c.

11. Example Column a b c a 73,25 57,42 66,32 Row b 80,26 35,12 32,54 c 28,27 63,31 54,29 We verify that (c,b) is a Nash Equilibrium by observation: If row plays c, b is the best response for column. If column plays b, c is the best response by row.

12. Example #2 • You try this one: Column a b c a 2,2 1,1 4,0 Row b 1,2 4,1 3,5

13. Coordination Games • Consider the following problem: • A supplier and a buyer need to decide whether to adopt a new purchasing system. Buyer new old new 20,20 0,0 Supplier old 5,5 0,0 No dominated strategies!

14. Buyer new old new 20,20 0,0 Supplier old 5,5 0,0 Coordination Games • This game has two Nash equilibria (new,new) and (old,old) • Real-life examples: Beta vs VHS, Mac vs Windows vs Linux, others? • Each player wants to do what the other does • which may be different than what they say they’ll do • How to choose a strategy? Nothing is dominated.

15. Solving Coordination Games • Coordination games turn out to be an important real-life problem • Technology/policy/strategy adoption, delegation of authority, synchronization • Human agents tend to use “focal points” • Solutions that seem to make “natural sense” • e.g. pick a number between 1 and 10 • Social norms/rules are also used • Driving on the right/left side of the road • These strategies change the structure of the game

16. Finding Nash Equilibria – Simultaneous Equations • We can also express a game as a set of equations. • Demand for corn is governed by the following equation: • Quantity = 100000(10 – p) • Government price supports say that p must be at least 0.25 (and it can’t be more than 10) • Three farmers can each choose to sell 0-600000 lbs of corn. • What are the Nash equilibria?

17. Setup • Quantity (q) = q1 + q2 + q3 • Price(p) = a –bq (downward-sloping line) • Farmer 1 is trying to decide a quantity to sell. • Maximize profit = price * quantity • Maximize: pq1 =(a –bq) * q1 • Profit = (a – b(q1 + q2 + q3)) * q1 = = aq1 –bq12 –bq1q2 –bq1q3 Differentiate: Pr’ = a – 2bq1 –bq2 – bq3 To maximize: set this equal to zero.

18. Setup • So solutions must satisfy • a – b(q2 + q3) – 2bq1 = 0 • So what if q1 = q2 = q3 (everyone ships the same amount?) • Since the game is symmetric, this should be a solution. • a – 4bq1 = 0, a = 4bq1, q1 = a/4b. • q = 3a/4b, p = a/4. Each farmer gets a2 / 16b. • In this problem, a=10, b=1/100000. • Price = \$2.50, q1=250000, profit = 625,000 • q1=q2=q3=250000 is a solution. • Price supports not used in this solution.

19. Setup • What if farmers 2 & 3 send everything they have? • q2 + q3 = 1,200,000 • If farmer 1 then shipped nothing, price would be: • 10 - 1,200,000/100,000 = -2. • But prices can’t fall below \$0.25, so they’d be capped there. • Adding quantity would reduce the price, except for supports. • So, farmer 1 should sell all his corn at \$0.25, and earn \$125,000. • So everyone selling everything at the lowest price (q1 = q2 =q3 = 600,000) is also a Nash equilibrium. • These are the only pure strategy Nash equilibria.

20. Price-matching Example • Two sellers are offering the same book for sale. • This book costs each seller \$25. • The lowest price gets all the customers; if they match, profits are split. • What is the Nash Equilibrium strategy?

21. Price-matching Example • Suppose the monopoly price of the book is \$30. • (price that maximizes profit w/o competition) • Each seller offers a rebate: if you find the book cheaper somewhere else, we’ll sell it to you with double the difference subtracted. • E.g. \$30 at store 1, \$24 at store 2 – get it for \$18 from store 1. • Now what is each seller’s Nash strategy?

22. Price-matching example • Observation 1: sellers want to have the same price. • Each suffers from giving the rebate. • Profit = p1 – 2(p1 – p2) = -p1 –2p2 • Pr’ = -1. • There is no local maximum. So, to maximize profits, maximize price. • At that point, the rebate 2(p1 – p2) is 0, and p1 is as high as possible. • The 2 makes up for sharing the market.

23. Cooperative Games and Coalitions • When a group of agents decide to cooperate to improve their payments (for example, adopting a technology), we call them a coalition • Side payments, bribes, intimidation may be used to set up a coalition. • Example: A,B,C are running for class president. The president receives \$10, everyone else \$0 • Each player’s strategy is to vote for themselves. • A offers B \$5 to vote for her – now both A and B are happier and have formed a coalition.

24. Efficiency • We say that a coalition is efficient if there’s no choice of action that can improve one person’s profit without decreasing another. • Same reasoning as Nash equilibria, market equilibria. • If someone could change their strategy without hurting anyone and improve their payoff, it’s not efficient. • Money is left “on the table” • Example: cake-cutting.

25. Mixed strategies • Unfortunately, not every game has a pure strategy equilibrium. • Rock-paper-scissors • However, every game has a mixed strategy Nash equilibrium. • Each action is assigned a probability of play. • Player is indifferent between actions, given these probabilities.

26. Mixed Strategies • In many games (such as coordination games) a player might not have a pure strategy. • Instead, optimizing payoff might require a randomized strategy (also called a mixed strategy) Wife football shopping football 2,1 0,0 Husband shopping 1,2 0,0

27. Wife football shopping football 2,1 0,0 Husband shopping 1,2 0,0 Strategy Selection If we limit to pure strategies: Husband: U(football) = 0.5 * 2 + 0.5 * 0 = 1 U(shopping) = 0.5 * 0 + 0.5 * 1 = ½ Wife: U(shopping) = 1, U(football) = ½ Problem: this won’t lead to coordination!

28. Mixed strategy • Instead, each player selects a probability associated with each action • Goal: utility of each action is equal • Players are indifferent to choices at this probability • a=probability husband chooses football • b=probability wife chooses shopping • Since payoffs must be equal, for husband: • b*1=(1-b)*2 b=2/3 • For wife: • a*1=(1-a)*2 = 2/3 • In each case, expected payoff is 2/3 • 2/9 of time go to football, 2/9 shopping, 5/9 miscoordinate • If they could synchronize ahead of time they could do better.

29. Example: Rock paper scissors Column rock paper scissors 0,0 -1,1 1,-1 rock Row paper 1,-1 0,0 -1,1 scissors -1,1 1,-1 0,0

30. Setup • Player 1 plays rock with probability pr, scissors with probability ps, paper with probability 1-pr –ps • P2: Utility(rock) = 0*pr + 1*ps – 1(1-pr –ps) = 2 ps + pr -1 • P2: Utility(scissors) = 0*ps + 1*(1 – pr – ps) – 1pr = 1 – 2pr –ps • P2: Utility(paper) = 0*(1-pr –ps)+ 1*pr – 1ps = pr –ps Player 2 wants to choose a probability for each strategy so that the expected payoff for each strategy is the same.

31. Setup qr(2 ps + pr –1) = qs(1 – 2pr –ps) = (1-qr-qs) (pr –ps) • It turns out (after some algebra) that the optimal mixed strategy is to play each strategy ½ of the time. • Intuition: What if you played rock half the time? Your opponent would then play paper half the time, and you’d lose more often than you won. • So you’d decrease the fraction of times you played rock, until your opponent had no ‘edge’ in guessing what you’ll do.

32. Repeated games • Many games get played repeatedly • A common strategy for the husband-wife problem is to alternate • This leads to a payoff of 1, 2,1,2,… • 1.5 per week. • Requires initial synchronization, plus trust that partner will go along. • Difference in formulation: we are now thinking of the game as a repeated set of interactions, rather than as a one-shot exchange.

33. Repeated vs Stage Games • There are two types of multiple-action games: • Stage games: players take a number of actions and then receive a payoff. • Checkers, chess, bidding in an ascending auction • Repeated games: Players repeatedly play a shorter game, receiving payoffs along the way. • Poker, blackjack, rock-paper-scissors, etc

34. Analyzing Stage Games • Analyzing stage games requires backward induction • We start at the last action, determine what should happen there, and work backwards. • Just like a game tree with extensive form. • Strange things can happen here: • Centipede game • Players alternate – can either cooperate and get \$1 from nature or defect and steal \$2 from your opponent • Game ends when one player has \$100 or one player defects.

35. Analyzing Repeated Games • Analyzing repeated games requires us to examine the expected utility of different actions. • Assumption: game is played “infinitely often” • Weird endgame effects go away. • Prisoner’s Dilemma again: • In this case, tit-for-tat outperforms defection. • Collusion can also be explained this way. • Short-term cost of undercutting is less than long-run gains from avoiding competition.