In the Name of God. Sharif University of Technology. Microeconomics 2. Graduate School of Management and Economics. Dr. S.

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1 In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics ( nd term) - Group 2 Dr. S. Farshad Fatemi Chapter 8: Simultaneous-Move Games

2 Simultaneous-Move Game: A situation in which players play a oneshot game (play only in one round) and move simultaneously (or sequentially when the second player is not able to observe the action of the first player prior to making her decision). We use the normal form representation Γ N = [I, {S i }, {u i (. )}] to study this class of games. Microeconomics 2 Dr. F. Fatemi Page 28

3 Dominant and Dominated Strategies: Definition (MWG 8.B.1): A strategy s i S i is a strictly dominant strategy for player i in game Γ N = [I, {S i }, {u i (. )}] if for all s i s i, we have u i (s i, s i ) > u i (s i, s i ) for all s i S i. (we show a strategy profile for player i s opponents by s i S i ). Note: In this definition for the player a pure strategy is strictly better than all other pure strategies regardless of what her opponents do. Microeconomics 2 Dr. F. Fatemi Page 29

4 Example: Prisoners Dilemma Prisoner 2 Confess Don t Confess Prisoner 1 Confess -4, -4-1, -10 Don t Confess -10, -1-2, -2 Confess is a strictly dominant strategy for both players. Microeconomics 2 Dr. F. Fatemi Page 30

5 Definition (MWG 8.B.1): A strategy s i S i is a strictly dominated strategy for player i in game Γ N = [I, {S i }, {u i (. )}] if there exists another strategy s i s i, such that u i (s i, s i ) < u i (s i, s i ) for all s i S i. In this case we say s i strictly dominates s i. Note: In this definition there should be a strategy that the player would be worsen off playing that strategy in comparison with just another strategy. Usually when we just use term dominated strategy we mean strictly dominated. Microeconomics 2 Dr. F. Fatemi Page 31

6 Definition (MWG 8.B.1): A strategy s i S i is a weakly dominated strategy for player i in game Γ N = [I, {S i }, {u i (. )}] if there exists another strategy s i s i, such that s i S i then u i (s i, s i ) u i (s i, s i ), and s i S i such that u i (s i, s i ) < u i (s i, s i ). In this case we say s i weakly dominates s i. A strategy is a weakly dominant strategy if it weakly dominates every other strategy. Microeconomics 2 Dr. F. Fatemi Page 32

7 Iterated Elimination of Strictly Dominated Strategies: When we assume rationality is common knowledge, players are able to anticipate the strategies that their opponent is not going to play for sure. 1) Eliminate strictly dominated strategies for all players. Left with reduced game, with each player having fewer strategies 2) In the reduced game, eliminate strictly dominated strategies for all player... further reduced game. 3) Repeat, till no further elimination is possible. Microeconomics 2 Dr. F. Fatemi Page 33

8 Example: A different version of Prisoners Dilemma Prisoner 2 Confess Don t Confess Prisoner 1 Confess -4, -4-1, -10 Don t Confess -10, -1 0, -2 Microeconomics 2 Dr. F. Fatemi Page 34

9 Example: L C R T -1, -2 1, -3 0, -2 M -2, 1 0, 0 10, 0 B 2, 2-4, 1 3, -1 Microeconomics 2 Dr. F. Fatemi Page 35

10 Domination in Mixed Strategies: We now extend the definition of domination to the mixed strategies. Assume (S i ) denotes all possible mixed strategies for player i. Definition (MWG 8.B.1): A strategy σ i (S i ) is strictly dominated for player i in game Γ N = [I, { (S i )}, {u i (. )}] if there exists another strategy σ i (S i ), such that σ i j i S j then u i (σ i, σ i ) < u i (σ i, σ i ). In this case we say σ i strictly dominates σ i. Microeconomics 2 Dr. F. Fatemi Page 36

11 Proposition (MWG 8.B.1): Player i s pure strategy s i S i is strictly dominated in game Γ N = [I, { (S i )}, {u i (. )}] if and only if there exists another strategy σ i (S i ), such that s i S i then u i (s i, s i ) < u i (σ i, s i ). Example: L R T 10, 1 0, 4 M 4, 2 4, 3 B 0, 5 10, 2 Microeconomics 2 Dr. F. Fatemi Page 37

12 Nash Equilibrium: Definition (MWG 8.D.1): A strategy profile s = (s 1,, s I ) is a Nash equilibrium of game Γ N = [I, {S i }, {u i (. )}] if for every i = 1,, I u i (s i, s i ) u i (s i, s i ) for all s i S i. In other words, none of the players has incentive to deviate from a Nash equilibrium. Microeconomics 2 Dr. F. Fatemi Page 38

13 Example: Stag Hunt: Two hunters should help each other to be able to catch a stag. Alternatively each can hunt a hare on their own (the initial idea of this game is from Jean-Jacques Rousseau). The payoffs are: Player 2 Stag Hare Player 1 Stag 2, 2 0, 1 Hare 1, 0 1, 1 (Stag, Stag) and (Hare, Hare) are the two NEs of this game. Microeconomics 2 Dr. F. Fatemi Page 39

14 Definition: Player i s best-response correspondence (function) in the game Γ N = [I, {S i }, {u i (. )}] is the correspondence that assigns to each s i S i the set: b i (s i ) = {s i S i u i (s i, s i ) u i (s i, s i ); s i S i }. Then we can restate that s is the Nash equilibrium of the game Γ N = [I, {S i }, {u i (. )}] if and only if s i = b i (s i ) for i = 1,, I All individual strategies in s are best responses to each other. Microeconomics 2 Dr. F. Fatemi Page 40

15 Discussion of the Concept of Nash Equilibrium: Why NE makes sense as an equilibrium concept for a given game: 1) NE as a consequence of rational inference. 2) NE as a necessary condition if there is a unique predicted outcome to a game. 3) If a game has a focal point, then it is necessary a NE. (Schelling (The strategy of conflict; 1960) introduced the concept of focal points). 4) NE as a self-enforcing agreement. 5) NE as a stable social convention. Microeconomics 2 Dr. F. Fatemi Page 41

16 Mixed Strategy Nash Equilibrium: Definition (MWG 8.D.2): A mixed strategy profile σ = (σ 1,, σ I ) is a Nash equilibrium of game Γ N = [I, { (S i )}, {u i (. )}] if for every i = 1,, I u i (σ i, σ i ) u i (σ i, σ i ) for all σ i (S i ). Example: NE of Matching Pennies H T H 1, -1-1, 1 T -1, 1 1, -1 Claim: A mixed strategy NE of the game is when players play H and T with equal probabilities (investigate?). Microeconomics 2 Dr. F. Fatemi Page 42

17 Proposition (MWG 8.D.1): Let S i + S i shows the set of pure strategies that player i plays with positive probability in mixed strategy profile σ = (σ 1,, σ I ). Strategy profile σ is a Nash equilibrium of game Γ N = [I, { (S i )}, {u i (. )}] if and only if for every i = 1,, I i) u i (s i, σ i ) = u i (s i, σ i ) for all s i, s i S + i. ii) u i (s i, σ i ) u i (s i, σ i ) for all s i S + i and for all s + i S i Therefore, a necessary and sufficient condition for a mixed strategy profile to be a NE is to make players indifferent over the pure strategies appearing in the mixed NE with positive probabilities. These pure strategies also should be at least as good as any other pure strategy (which are not part of the mixed NE). Microeconomics 2 Dr. F. Fatemi Page 43

18 Example: NE of Meeting in New York Grand Central Empire State p Grand Central 500, 500 0, 0 1 p Empire State 0, 0 100, 100 Row players chooses p in order to make the column player indifferent between her options: u C (GC s R = p. GC (1 p). ES) = 500 p + 0 u C (ES s R = p. GC (1 p). ES) = (1 p) Then u C (GC) = u C (ES) iff = 1. 6 In Eq every player plays GC with probability 1 6. (Note: the game is symmetric) Microeconomics 2 Dr. F. Fatemi Page 44

19 In a mixed strategy NE, a player chooses her probabilities just in order to make other players indifferent over their strategies. The probabilities have no real effect on player s own payoffs. Microeconomics 2 Dr. F. Fatemi Page 45

20 Existence of Nash Equilibria: There are few equilibrium existence propositions; here we just include two which are more important in economics applications. Proposition (MWG 8.D.2): Every game Γ N = [I, { (S i )}, {u i (. )}] in which the sets S 1,, S I have a finite number of elements has a mixed strategy NE. This NE might be in pure strategies. There might exist more than one NE. Microeconomics 2 Dr. F. Fatemi Page 46

21 Proposition (MWG 8.D.3): A NE exist in game Γ N = [I, {S i }, {u i (. )}] if for all = 1,, I : a) S i is a nonempty, convex, and compact subset of some Euclidean space R M, and b) u i (s 1,, s I ) is continuous in (s 1,, s I ) and quasiconcave in s i. These conditions assure us that there is a NE in pure strategies. It does not mean that in situation when these conditions do not hold there is no NE; we just cannot prove the existence. Microeconomics 2 Dr. F. Fatemi Page 47

22 Games of Incomplete Information: Bayesian - Nash Equilibrium In many cases in economics we can find the games when a player is not sure about the payoff of the other players. A simple example is two firms competing in a market, but firm 1 is not sure about the cost function of firm 2 (With some probability firm 2 can reduce his cost by acquiring a new technology). However, Firm 2 is informed about the state. Microeconomics 2 Dr. F. Fatemi Page 48

23 Example: A different version of Prisoners Dilemma Strategic form: State 1 (prob. μ) State 2 (prob. 1 μ) Prisoner 2 Prisoner 2 DC C DC C Prisoner 1 DC 0, -2-10, -1 Prisoner 1 DC 0, -2-10, -7 C -1, -10-5, -5 C -1, -10-5, -11 Microeconomics 2 Dr. F. Fatemi Page 49

24 Extensive form: Nature µ 1-µ Prisoner 1 C DC C DC Prisoner 2 Prisoner 2 C DC C DC C DC C DC Microeconomics 2 Dr. F. Fatemi Page 50

25 Definition: A Bayesian game is characterized by 1) A set of players I, 2) A set of strategies for each player S i 3) A set of random variables for each player chosen by the nature Θ i, 4) The joint probability distribution function of the θ i s F(θ 1,, θ I ); which is assumed to be common knowledge, and 5) The payoff that each player receives at any foreseeable outcome u i (. ) Microeconomics 2 Dr. F. Fatemi Page 51

26 So, a Bayesian game can be represented by [I, {S i }, {u i (. )}, Θ, F(. )] Therefore a pure strategy for player i is a function of his type s i (θ i ); the collection of all such a strategies for player i is his strategy set S i. Also, Each player s expected payoff for a given profile of pure strategies is defined by: u i s 1 (. ),, s I (. ) = E θ [u i (s 1 (θ 1 ),, s I (θ I ), θ i )] Microeconomics 2 Dr. F. Fatemi Page 52

27 Definition (MWG 8.E.1): A pure strategy Bayesian Nash equilibrium for the Bayesian game [I, {S i }, {u i (. )}, Θ, F(. )] is a profile of decision rules s 1 (. ),, s I (. ) that constitutes a NE of game Γ N = [I, {S i }, {u i (. )}]. That is for every = 1,, I, u i s i (. ), s i (. ) u i s i (. ), s i (. ) for s i (. ) S i Like before, in a pure strategy BNE each player, for each type that he might have, must be playing the best response to a conditional distribution of his rivals strategies. Microeconomics 2 Dr. F. Fatemi Page 53

28 Proposition (MWG 8.E.1): A profile of decision rules s 1 (. ),, s I (. ) is a BNE in Bayesian game [I, {S i }, {u i (. )}, Θ, F(. )] if and only if, for every i = 1,, I and every θ i Θ i occurring with positive probability, E θ i u i (s i (θ i), s i (θ i ), θ i) θ i E θ i u i (s i, s i (θ i ), θ i) θ i for s i (. ) S i The expectation is taken over types of other players conditional on player i s own type. Microeconomics 2 Dr. F. Fatemi Page 54

29 Example: Then a NE of our example should have the form s 1, (s 1 2, s 2 2 ) ; s 1 2 : prisoner 2 s action if he knows that the state is 1. State 1 (prob. μ) State 2 (prob. 1 μ) Prisoner 2 Prisoner 2 DC C DC C Prisoner 1 DC 0, -2-10, -1 Prisoner 1 DC 0, -2-10, -7 C -1, -10-5, -5 C -1, -10-5, -11 If State=1 then P2 has a dominant action of C and If State=2 then P2 has a dominant action of DC. (So, P1 only needs to consider these two) u 1 DC, (C, DC) = 10μ + 0(1 μ) = 10μ u 1 C, (C, DC) = 5μ 1(1 μ) = 1 4μ Then if < 1, the BNE is DC, (C, DC) and if μ > 1, the BNE is C, (C, DC). 6 6 Microeconomics 2 Dr. F. Fatemi Page 55

30 Example: R&D race Two firms can independently invest in R&D for a new invention. If one of the firms develops the new invention, the other firm would also benefit from it (spill-over effect). The cost of invention is the same for both firms c (0, 1). The benefit of the invention to firm i is (θ i ) 2 where θ i is his type and is private information firm i. It is common knowledge that θ i is uniformly distributed on [0, 1]. Formulate the situation as a game and find the BNE. Microeconomics 2 Dr. F. Fatemi Page 56

31 Players: I = {1, 2} Strategies: S i = {D, N} States: θ i [0, 1] Prob. Dist.: θ i is uniformly distributed on [0, 1] (common knowledge) Payoff function: u i (s i = N; s i = N) = 0 u i (s i = N; s i = D) = (θ i ) 2 u i (s i = D; s i ) = (θ i ) 2 c The BNE should have the form s 1 (θ 1 ); s 2 (θ 2 ) Microeconomics 2 Dr. F. Fatemi Page 57

32 Now let s check for firm i s incentive to spend on R&D: u i (s i = N) = Prob s j θ j = D (θ i ) 2 u i (s i = D; s i ) = (θ i ) 2 c Then firm i would prefer D if: (θ i ) 2 c Prob s j θ j = D (θ i ) 2 or c θ i 1 Prob s j θ j = D Microeconomics 2 Dr. F. Fatemi Page 58

33 Then firm 1 knows that firm 2 would invest in R&D if θ 2 is bigger than a threshold θ 2. So since θ 2 is uniformly distributed: Prob (s 2 (θ 2 ) = D) = Prob θ 2 θ 2 = 1 θ 2 θ 1 = c 1 1 θ 2 or θ 1 2 θ 2 = c Then because the game is symmetric: θ 1 2 θ 2 = c θ 2 2 θ 1 = c By solving simultaneously the cut-off types are (in eq. s i θ i θ i = D): 3 θ 1 = θ 2 = c Microeconomics 2 Dr. F. Fatemi Page 59