Static Games of Complete Information

1. Static Games of Complete Information .

2. Games and best responses • Bob is a florist • One bunch of flowers sells for $10, but to sell 100 bunches market price should be zero • Thus, each extra bunch reduces market price by a dime • Cost is $2 a bunch

3. Games and best responses . $ 10 Price $ 2 80 100 Quantity

4. Games and best responses • What is Bob’s optimal quantity as a monopolist? • It turns out to be 40 units • Suppose now Ann is contemplating entry • How will the total quantity be allocated between the two?

5. Games and best responses • It depends on how they expect each other to act • Suppose Ann is a sophisticated thinker and responds optimally to Bob • Ann’s best response to Bob is (80-QB)/2 • What should Bob do? • Note: Bob wants the total quantity to be the midpoint between Ann’s quantity (QA) and the break-even point (80)

6. Games and best responses • Demand curve facing Bob is • To maximize profits, Bob wants the total quantity to be the midpoint between Ann’s quantity (QA) and the break-even point (80) $ 10 Price $ 2 QA (80+QA)/2 80 100 Quantity Bob

7. Games and best responses • Four scenarios for Bob’s reasoning

8. What if Bob’s quantity is a given? • How many bunches should Bob bring, knowing how Ann would react to him? • Now, total quantity is (80-QB)/2+QB= (80+QB)/2 • Thus, mkt price is $10-$0.10{(80+QB)/2} • Bob’s optimal quantity is 40; and so Ann’s quantity is 20 • A significant first-mover advantage! • Why does Bob produce monopoly quantity?

9. Strategic-Form Games • Elements of strategic-form games: - finite set of players, i I ={1,2,…,I} - pure-strategy space Si for each player i - payoff functions ui(s) for each player i and for each strategy profile s=(s1, s2,…, sI) • Each player goal is to maximize his own payoff, and NOT to ‘beat’ other players • A Zero-Sum game is where =0 • Most applications in the Social Sciences are non zero-sum games

10. Mixed-Strategies • A mixed-strategy σiis a probability distribution over pure-strategies • σi(si) is probability that σi assigns to si • Space of player i’s mixed-strategies is ∑i • Space of mixed-strategy profiles is ∑= with elements σ • Each player’s randomization is independent of those of other players • Player i’s payoff to profile σ is

11. Computing payoff from a mixed-strategy • Player 1 plays along rows; 2 along columns • Let σ1={⅓, ⅓, ⅓} andσ2={0, ½, ½ }. • What is player 1’s payoff?

12. Dominated Strategies • Is there an obvious prediction about how the previous game will be played? • Iterated Strict Dominance predicts (U, L) • Let “–i” denote all players other than i • Then strategies of other players s-i S-i and a strategy profile is (si , s-i ) • Defn: Pure strategy si is strictly dominated for player i if there exists ∑i such that, for all s-i S-i

13. Some implications of dominance • A mixed-strategy that assigns positive probability to a dominated pure strategy, is dominated • A mixed-strategy my be dominated even though it assigns positive probability only to undominated pure strategies. • Examples: -Prisoner’s Dilemma -Second-Price Auction

14. Nash Equilibrium • Defn: A mixed-strategy profile σ* is a Nash Equilibrium if, for all players i, for all for all s-i S-i • Common examples in economics: - Cournot quantity-setting game - Bertrand price-setting game

15. Assumptions • Rationality • Players aim to maximize their payoffs • Players are perfect calculators • Common Knowledge • Each player knows the rules of the game • Each player knows that each player knows the rules • Each player knows that each player knows that each player knows the rules • Each player knows that each player knows that each player knows that each player knows the rules • Each player knows that each player knows that each player knows that each player knows that each player knows the rules • Etc. Etc. Etc.

16. Properties and implications • Property 1: A player must be indifferent between all pure strategies in the support of a mixed-strategy • Property 2: One only needs to check for pure-strategy deviations • Property 3: If a single strategy survives iterated deletion of strictly dominated strategies it is a Nash Equilibrium • Question: Why play mixed strategies when all pure strategies in the support have same payoff?

17. Further comments on N.E. • In many games pure strategy NE do not exist -example is game of “Matching Pennies” • Sometimes there are multiple Nash Equilibria - “Battle of the Sexes”; “Chicken” • Schelling’s theory of “focal points” • Pre-play communication

18. Baseball anyone? • 1986 baseball National League championship series • The New York Mets won a crucial game against he Houston Astros • Len Dykstra hit Dave Smith’s second pitch for a two-run home run • Later the two players talked about this critical play

19. Analysis of a home run • Dykstra said, “He threw me a fastball on the first pitch and I fouled it off. I had a gut feeling then that he’d throw me a forkball next, and he did. I got a pitch I saw real well, and I hit it real well”. • Smith said, “What it boils down to is that, it was a bad pitch selection…if I had to do it over again, it would be [another] fastball”. • Would Dykstra not have been prepared for a fastball? • Again, randomization is the only way to go

20. But how do you randomize? • In a game of tennis, suppose receiver’s forehand is stronger than backhand • Consider following probabilities of successfully returning serve Server’s Aim Receiver’s Move

21. Reducing receiver’s effectiveness by randomizing • Suppose server tosses a coin before each serve • Aims to forehand or backhand according to coin turning heads/tails • When receiver moves to forehand, his successful return rate is 55% • When receiver moves to backhand, his successful return rate is 45% • Given server’s randomization, receiver should move to forehand • The server has already an improved outcome compared to serving the same way all the time!!

22. What is server’s best mix? • Consider following graph 90 Anticipate Forehand 60 Anticipate Backhand 48 Percentage successful returns 30 20 0 40 100 Percentage of times server aims serve to forehand

23. The mixing probabilities • The 40:60 mixture of forehands to backhands is the equilibrium • This mixture is the only one that cannot be exploited by the receiver to his own advantage • With this mixture the receiver does equally well with either of his choices • Both ensure the receiver a success rate of 48%

24. Nash bargaining • Two players, N=1, 2, bargain over the partition of a cake • The set of Agreements, A, has elements A={(a1, a2)єR2: ai≥0 for i=1, 2} • The event of Disagreement is D • Players have utilities ui, i=1, 2, s.t. ui: A {D}→R • Let, S={(u1(a), u2(a)) for a є A}, and d=(u1(D), u2(D)) • A Bargaining ProblemB is the set of pairs <S, d>, and the Bargaining Solution is a function f :B → R2 that assigns to each <S, d> єB an element of S

25. Nash’s axioms 1. INV (invariance to equiv utility representations): If <S/, d/> is obtained from <S, d> by si , then, fi(S/, d/)= 2. SYM (symmetry) If <S, d> is symmetric, then f1(S, d)= f2(S, d) 3. PAR (Pareto efficiency) Suppose <S, d> is a bargaining problem, with si , ti єS , and ti >si , for i=1,2, then f(S, d)≠s 4. IIA (independence of irrelevant alternatives) If <S, d> and <T, d> are bargaining problems with S T and f(T, d) єS, then f(S, d) = f(T, d).

26. Nash bargaining solution Theorem (Nash 1950) There is a unique bargaining solution fN:B → R2 satisfying the above axioms given by fN(S, d)=

27. Application: Splitting a dollar • Set of agreements is A={(a1, a2)єR2: a1+ a2≤1 and ai≥0 for i=1, 2} • Players are risk-averse, i.e. uiare concave • Disagreement point is d=(u1(0), u2(0))=(0, 0) • So <S, d> is a bargaining problem Result 1:With equal risk-aversion, players are symmetric & SYM, PAR give (u(1/2), u(1/2))

28. Role of risk-aversion • Let player 2 be more risk-averse than player 1 • Let player 2’s utility be v2 =hu2,where h is concave, and v1 = u1 • Let <S/, d/> be bargaining problem with utilities vi • The optimizing program for <S, d> gives solution zu where zu solves: • The optimizing program for <S/, d/> gives solution zv where zv solves:

29. Role of risk-aversion • The first program gives, • The second program gives, • Result 2:If player 2 becomes more risk-averse, then Player 1’s share of the dollar in the Nash solution increases.

30. Existence of NE Theorem (Nash 1950) Every finite strategic-form game has a mixed-strategy equilibrium. Sketch of Proof: 1. Use players’ Reaction Correspondences r(σ) 2. Realize that a NE is a Fixed Point of r(σ) 3. Show that conditions for Kakutani’s Fixed Point Theorem are satisfied in this case

31. Existence of NE (cont) Theorem (Shizuo Kakutani 1941) If : • ∑ is compact and convex • A correspondence r(σ) :∑→∑ is non-empty and convex • r(σ) has a closed graph Then r(σ) has a fixed point, that is, there exists σ* such that, r(σ*)= σ*

32. Nash equilibrium is a fixed point • For any strategy profile σ, player i’s reaction correspondence ri maps σ to the mixed strategy that maximizes his payoff, given his opponents play σ-i . Thus, ri (σ)= ri (σi ,σ-i ) = ui(σ/i , σ-i ) • Each player i=1,2,..n, has a reaction function • Let us form the n-tuple, r(σ)=(r1 (σ), r2 (σ),…, rn(σ)), where r(σ)є∑ • Suppose there exists σ* є∑such that,for all players, σ*i =ri(σ*)= ri(σ*i , σ*-i), then σ* is a Nash equilibrium • But σ*i =ri(σ*) for all i,means that σ*=r(σ*), i.e. σ* is a fixed point of the reaction correspondence r(.)

33. Applying Kakutani • ∑ is compact and convex: 1. Mixed strategies convexify the strategy space, and so each ∑i is a convex space of dimension (#Si-1) 2. Theorem (Heine-Borel): Closed & bounded subsets in Rn are compact • r(σ) is non-empty: 1. Theorem (Weierstrass): A continuous real-valued function defined on a compact space achieves its maximum values 2. Player’s i’sutility functions are continuous in σi

34. Applying Kakutani • r(σ) is convex: Suppose σ/,σ//є r(σ) . uiis linear, therefore for λє(0, 1) ui(λσi/+ (1- λ)σi//, σ-i)= λ ui(σi/, σ-i) + (1- λ)ui(σi//, σ-i). Thus, if σi/,σi//are best responsestoσ-i, then so is their weighted average. Thus, λσi/+ (1- λ)σi// є r(σ) • r(σ) has a closed graph: If with , then