Introduction to Game Theory
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1 Introduction to Game Theory Part 1. Static games of complete information Chapter 3. Mixed strategies and existence of equilibrium Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
2 Topics covered 1 Mixed strategies 2 Existence of Nash equilibrium V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
3 Non-existence: Matching pennies Consider the following game There are two players I = {i 1, i 2 } Each player s strategy space is S i = {Heads, Tails} The payoff of the game is as follows: Each player has a penny and must choose whether to display it with heads or tails facing up If the two pennies match then player i2 wins player i 1 s penny If the pennies do not match then i1 wins i 2 s penny Player i 1 Player i 2 Heads Tails Heads 1, 1 1, 1 Tails 1, 1 1, 1 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
4 Non-existence: Matching pennies There is no Nash equilibrium Player i 1 Player i 2 Heads Tails Heads 1, 1 1, 1 Tails 1, 1 1, 1 If the players strategies match then player i 1 prefers to switch strategies If the players strategies do not match then i 2 prefers to switch This situation occurs in many games Poker, battle To overcome this difficulty, we introduce the notion of a mixed strategy V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
5 Mixed strategies A mixed strategy is a probability measure (distribution) over the strategies in S i A strategy in S i is called a pure strategy The set of mixed strategies is denoted by Prob(S i ) or (S i ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
6 Mixed strategies A mixed strategy p = (p(s i )) si S i of player i is a vector in R S i satisfying s i S i, p si = p(s i ) 0 and p(s i ) = 1 s i S i If the mixed strategy p is such that there exists ŝ i S i satisfying { 0 if si ŝ s i S i, p(s i ) = i 1 if s i = ŝ i then p is denoted Dirac(ŝ i ) or 1ŝi and (abusing notations) is assimilated with the pure strategy ŝ i V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
7 Mixed strategies: interpretation A family p i = (p j ) j i of mixed strategies p j (S j ) can represent agent i s uncertainty about which strategy each other agent j will play Notation The expected value of agent i s payoff if he plays s i believing that the other players will play according to p i is denoted by and is defined by u i (s i, p i ) E p i [u i (s i )] = u i (s i, p i ) s i S i p j (s j ) u i (s i, s i ) j i } {{ } p i (s i ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
8 Mixed strategies: interpretation Notation If p i is a mixed strategy in (S i ) we let p = (p j ) j I and the expected value E p [u i ] = p j (s j ) u i (s i1,..., s in ) s S j J = s S p i1 (s i1 )... p in (s in )u i (s i1,..., s in ) is denoted by Observe that u i (p) = u i (p) s i S i p i (s i )u i (s i, p i ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
9 Mixed strategies: interpretation Definition We say that there is no belief that player i could hold about the strategies the other players will choose such that it would be optimal to play s i when p i j i (S j ), s i argmax { E p i [u i (s i)] : s i S i } In other words, for every belief p i that agent i could hold about the others, there exists a pure strategy ŝ i S i such that E p i [u i (s i )] < E p i [u i (ŝ i )] Be careful, the strategy ŝ i may depend on the belief p i V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
10 Mixed strategies: interpretation Proposition Assume that the pure strategy s i is strictly dominated by the pure strategy σ i s i S i, u i (s i, s i ) < u i (σ i, s i ) Then there is no belief that player i could hold about the strategies the other players will choose such that it would be optimal to play s i More precisely, for every family p i = (p j ) j i of mixed strategies p j (S j ), we have E p i [u i (s i )] < E p i [u i (σ i )] In this case, the strategy σ i improves the expected payoff independently of the belief p i agent i holds about the other players actions V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
11 Mixed strategies: interpretation The converse may not be true Consider the following game Player i 1 Player i 2 L R T 3, 0, M 0, 3, B 1, 1, For any belief p i2 agent i 1 may have about i 2 s strategies The strategy B is never a best response If pi2 (L) > 1/2 then i 1 s best response is T If p i2 (L) < 1/2 then i 1 s best response is M If p i2 (L) = 1/2 then i 1 s best response is either T or M However, the strategy B is not strictly dominated by another pure strategy V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
12 Mixed strategies: interpretation Player i 1 Player i 2 L R T 3, 0, M 0, 3, B 1, 1, Consider the mixed strategy p i1 defined by p i1 (T) = 1/2, p i1 (M) = 1/2 and p i1 (B) = 0 Such a probability will be denoted by 1 p i1 = (1/2, 1/2, 0) For any belief p i2 agent i 1 may have about i 2 s strategies, we have u i1 (B, p i2 ) = u i1 (1 B, p i2 ) = 1 < 3/2 = u i1 (p i1, p i2 ) The strategy B is strictly dominated by the mixed strategy p i1 = (1/2, 1/2, 0) 1 Sometimes one my find the notations: p i1 = 1/2 Dirac(L) + 1/2 Dirac(M) or p i1 = L M. V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
13 Mixed strategies: interpretation A given pure strategy can be a best response to a mixed strategy Even if the pure strategy is not a best response to any other pure strategy Player i 1 Player i 2 L R T 3, 0, M 0, 3, B 2, 2, The pure strategy B is not a best response for player i 1 to either L or R by player i 2 But B is the best response for player i 1 to the mixed strategy p i2 by player i 2 provided that 1 3 < p i 2 (L) < 2 3 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
14 Nash equilibrium with mixed strategies We fix a game G = (S i, u i ) i I Definition A profile of mixed strategies p = (p i ) i I is a Nash equilibrium of the game G if each player s mixed strategy is a best response to the other players mixed strategies, i.e., i I, p i argmax{u i (p i, p i) : p i (S i )} The family p i = (p j ) j i represents player i s uncertainty about which strategy each player j will choose Remark Fix three players i, j and k. What player j believes about the possible strategies played by player i coincides with what player k believes V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
15 Nash equilibrium with mixed strategies Consider an abstract game G = (S i, u i ) i I Fix a family p i i = (pi j ) j i of mixed strategies representing player i s beliefs about player j s strategies Denote by Si (pi i ) the set of pure strategies best response of player i defined by S i (p i i) argmax{u i (s i, p i i) : s i S i } Assume that S i is finite, then Si (pi i ) is non-empty If p i is a mixed strategy in (S i ), we denote by supp p i its support defined by supp p i = {p i > 0} = {s i S i : p i (s i ) > 0} V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
16 Nash equilibrium with mixed strategies Proposition A mixed strategy p i is a best response to pi i, i.e., p i argmax{u i (p i, p i i) : p i (S i )} if and only if the support of p i best response to p i i, i.e., is a subset of all pure strategies that are In other words the set {s i S i : p i (s i ) > 0} supp p i S i (p i i) argmax{u i (p i, p i i) : p i (S i )} of best responses to p i i coincides with Prob(S i (p i i)) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
17 Nash equilibrium with mixed strategies: An equivalent definition Theorem A profile of mixed strategies p = (p i ) i I is a Nash equilibrium of the game G if and only if for every player i every pure strategy in the support of p i is a best response to the other players mixed strategies, i.e., i I, supp p i argmax{u i (s i, p i) : s i S i } Interpretation Players have identical beliefs about other players possible actions or strategies Players choose best response strategies consistent with these beliefs V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
18 Nash equilibrium with mixed strategies: interpretation A mixed-strategy Nash equilibrium does not rely on any player flipping coins, rolling dice, or otherwise choosing a strategy at random Player j s mixed strategy is interpreted as a common statement of any other player i s uncertainty about player j s choice of a pure strategy Under this interpretation each player chooses a single action rather than a mixed strategy A Nash equilibrium (p i ) i I in mixed strategy is such that any pure strategy in the support of p i is a best response given that player i s beliefs about player j s actions is p j The idea is to endow player j with a small amount of private information such that, depending on the realization of the private information, player j slightly prefers one of the relevant pure strategies Since player i does not observe j s private information, i remains uncertain about j s choice V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
19 Nash equilibrium with mixed strategies: controversial interpretations See the nice discussion in Osborne and Rubinstein (1994) Mixed strategies as objects of choice Mixed strategy Nash equilibrium as a steady state Mixed strategies as pure strategies in an extended game Mixed strategies as pure strategies in a perturbated game Mixed strategies as beliefs V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
20 Nash equilibrium: Matching pennies Player i 1 Player i 2 Heads Tails Heads 1, 1 1, 1 Tails 1, 1 1, 1 Suppose that player i 1 believes that player i 2 will play Heads with probability q and Tails with probability 1 q Given this belief we have u i1 (Heads, (q, 1 q)) = 1 2q and u i1 (Tails, (q, 1 q)) = 2q 1 Player i 1 s best response is Heads if q < 1/2 and Tails if q > 1/2 Player i 1 is indifferent between Heads and Tails if q = 1/2 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
21 Nash equilibrium: Matching pennies Player i 1 Player i 2 Heads Tails Heads 1, 1 1, 1 Tails 1, 1 1, 1 Fix now a mixed strategy p i1 = (r, 1 r) for player i 1, i.e., p i1 (Heads) = r and p i1 (Tails) = 1 r V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
22 Nash equilibrium: Matching pennies If agent i 1 believes that i 2 is playing the mixed strategy p i2 = (q, 1 q) Then we can compute the set of best responses β i1 (p i2 ) argmax{u i1 (p i1, p i2 ) : p i1 (S i1 )} (1) Remember that we must have β i1 (p i2 ) = Prob(S i 1 (p i2 )) Since S i1 = {Head, Tails}, there are only three possibilities β i1 (p i2 ) = {Heads}, β i1 (p i2 ) = {Tails} or β i1 (p i2 ) = (S i1 ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
23 Nash equilibrium: Matching pennies Observe that u i1 (p i1, p i2 ) = (2q 1) + r(2 4q) The mixed strategy p i1 = (r, 1 r) solves p i1 argmax{u i1 (q i1, p i2 ) : q i1 Prob(S i1 )} if and only if r belongs to the set r (q) = argmax{(2q 1) + r(2 4q) : r [0, 1]} V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
24 Nash equilibrium: Matching pennies if q < 1/2 then r (q) = {1} and i 1 s best response is to play the pure strategy Heads if q > 1/2 then r (q) = {0} and i 1 s best response is to play the pure strategy Tails if q = 1/2 then r (q) = [0, 1] and any mixed strategy is a best response, i.e., i 1 is indifferent between Heads and Tails The object q r (q) is called a correspondence V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
25 Nash equilibrium: Matching pennies Player i 1 s best response (r (q), 1 r (q)) to i 2 s strategy (q, 1 q) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
26 Nash equilibrium: Matching pennies Player i 1 Player i 2 Heads Tails Heads 1, 1 1, 1 Tails 1, 1 1, 1 Assume now that player i 2 plans to choose a mixed strategy p i2 = (q, 1 q) i.e., p i2 (Heads) = q and p i2 (Tails) = 1 q If agent i 2 believes that i 1 is playing the mixed strategy p i1 = (r, 1 r) Then we can compute the set of best responses argmax{u i2 (p i1, p i2 ) : p i2 (S i2 )} (2) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
27 Nash equilibrium: Matching pennies Observe that u i2 (p i1, p i2 ) = q(4r 2) (r + 1) A mixed strategy p i2 = (q, 1 q) solves (2) if q belongs to the set q (r) = argmax{q(4r 2) (r + 1) : q [0, 1]} if r < 1/2 then q (r) = {0} and i 2 s best response is to play the pure strategy Tails if r > 1/2 then q (r) = {1} and i 2 s best response is to play the pure strategy Heads if r = 1/2 then q (r) = [0, 1] and any mixed strategy is a best response, i.e., i 2 is indifferent between Heads and Tails V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
28 Nash equilibrium: Matching pennies Player i 2 s best response (q (r), 1 q (r)) to i 1 s strategy (r, 1 r) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
29 Nash equilibrium: Matching pennies Permuting q and r we get the following graph V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
30 Nash equilibrium: Matching pennies We can draw in the same picture the best response correspondence of each player V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
31 Nash equilibrium: Matching pennies A Nash equilibrium is a pair (p i 1, p i 2 ) such that p i argmax{u i (p i, p j) : p i (S i )} The pair defined by p i 1 = ( r, 1 r) and p i 2 = ( q, 1 q) is a Nash equilibrium if and only if r r ( q) and q r ( r) The unique Nash equilibrium of the Matching Pennies is then p i1 = (1/2, 1/2) and p i2 = (1/2, 1/2) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
32 The battle of sexes Chris Pat Opera Fight Opera 2, 1 0, 0 Fight 0, 0 1, 2 Denote by (q, 1 q) the mixed strategy in which Pat plays Opera with probability q Denote by (r, 1 r) the mixed strategy in which Chris plays Opera with probability r V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
33 The battle of sexes: 3 NE with mixed strategies 1 Pat and Chris play the pure strategy Opera 2 Pat and Chris play the pure strategy Fight 3 Pat plays the mixed strategy where Opera is chosen with probability 1/3 and Chris plays the mixed strategy where Opera is chosen with probability 2/3 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
34 2 players with 2 pure strategies Consider the problem of defining player i 1 s best response (r, 1 r) when player i 2 plays (q, 1 q) Player i 1 Player i 2 Left Right Up x, y, Down z, w, We discuss the four following cases (i) x > z and y > w (ii) x < z and y < w (iii) x > z and y < w (iv) x < z and y > w Then we turn to the remaining cases involving x = z or y = w V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
35 2 players with 2 pure strategies In case (i), the pure strategy Up strictly dominates Down In case (ii), the pure strategy Down strictly dominates Up V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
36 2 players with 2 pure strategies In cases (iii) and (iv), neither Up nor Down is strictly dominated Let q = (w y)/(x z + w y) In case (iii) Up is optimal for q > q and Down for q < q, whereas in case (iv) the reverse is true In both cases, any value of r is optimal when q = q V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
37 2 players with 2 pure strategies Observe that q = 1 if x = z and q = 0 if y = w For cases involving x = z or y = w the best response correspondences are L-shaped (two adjacent sides of the unit square) If we add arbitrary payoffs for player i 2 Then we can perform analogous computations and get the same 4 best-response correspondences V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
38 2 players with 2 pure strategies V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
39 2 players with 2 pure strategies Fix any of the four best response correspondence for player i 1 Fix any of the four best response correspondence for player i 2 Checking all 16 possible pairs, there is always at least one intersection We obtain the following qualitative features that can result: There can be a single pure strategy Nash equilibrium a single mixed strategy equilibrium 2 pure strategy equilibria and a single mixed strategy equilibrium V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
40 General existence result Theorem Consider a game G = (S i, u i ) i I and assume that for each player i, (1) the set S i is a compact, convex and non-empty subset of R n i for some n i N (2) the payoff function s u i (s) is continuous on S = i I S i (3) for each s i S i, the function s i u i (s i, s i ) is quasi-concave in the sense that s i S i, {s i S i : u i (s i, s i ) u i (s i, s i )} is convex Then there exists at least one pure strategy Nash equilibrium V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
41 Mathematics Definition A subset C R n is convex if it is stable under convex combination, i.e., for every x and y in C the segment is a subset of C [x, y] {αx + (1 α)y : α [0, 1]} Definition Consider a function u : C R where C is a convex and non-empty subset of R n. The function u is said concave if u(αx + (1 α)y) αu(x) + (1 α)u(y) for any pair (x, y) in C and any α [0, 1]. V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
42 Mathematics Proposition Consider a function u : C R where C is a convex and non-empty subset of R n. If u is concave then it is quasi-concave. Proposition Consider a function u : C R where C is a convex and non-empty subset of R n. The function u is quasi-concave if and only if u(αx + (1 α)y) min{u(x), u(y)} for any pair (x, y) in C and any α [0, 1]. V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
43 Mathematics Definition A subset K R n is compact if it is bounded, i.e., M > 0, x K, x M and closed, i.e., for every sequence (x n ) n N in K converging to some x R n the limit vector x still belongs to K. Proposition If (x n ) n N is a sequence in a compact set K, then there exists ϕ : N N strictly increasing such that lim x ϕ(n) = x n V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
44 Mathematics: examples Proposition A subset C of R is compact convex and non-empty if and only if there exists a b such that C = [a, b]. Example If (a k, b k ) 1 k n is a family of pairs with a k b k then the set C defined by n C [a k, b k ] = [a 1, b 1 ]... [a n, b n ] k=1 is a compact, convex and non-empty subset of R n. V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
45 Mathematics: examples Consider a finite set S (e.g. pure strategies) Denote by (S) the set of probabilities on S defined by { } (S) = p = (p s ) s S R S + : p s = 1 s S Proposition The set (S) is a compact convex and non-empty set of R S. V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
46 Mathematics: Closed graph Definition Consider a non-empty subset X of R n and F : X X a correspondence. The graph of F is said closed when the following set is closed in X X gph F {(x, y) K K : y F (x)} The graph of F is closed if and only for all sequences (x n ) n N and (y n ) n N in K satisfying lim n x n = x and lim n y n = y if y n F (x n ) for each n then in the limit we get y F (x) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
47 Mathematics: Kakutani s Fixed-point theorem Theorem (Kakutani) Consider a convex compact and non-empty set K of R n and F : K K a correspondence. If (1) for every x K the set F (x) is non-empty convex and closed (and hence compact) (2) the graph of F is closed Then the correspondence F admits a fixed point x, i.e., x F (x ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
48 Mathematics: Brouwer s Fixed-point theorem Corollary Consider a convex compact and non-empty set K of R n and f : K K a continuous function. Then there exists a fixed point x K, i.e., f(x ) = x Proof. Let F : K K be the correspondence defined by x K, F (x) = {x} Show that F has a closed graph if and only if f is continuous V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
49 Mathematics: Fixed-point theorem V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
50 Mathematics: Fixed-point theorem V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
51 General existence result Theorem Consider a game G = (S i, u i ) i I and assume that for each player i, (1) the set S i is a compact, convex and non-empty subset of R n i for some n i N (2) the payoff function s u i (s) is continuous on S = i I S i (3) for each s i S i, the function s i u i (s i, s i ) is quasi-concave in the sense that s i S i, {s i S i : u i (s i, s i ) u i (s i, s i )} is convex Then there exists at least one pure strategy Nash equilibrium V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
52 General existence result Proof. Based on an application of Berge s Maximum Theorem and Kakutani s Fixed-point Theorem Theorem If u : X Y [, ) is continuous and K is a non-empty compact subset of Y then the correspondence F : x F (x) argmax{u(x, y) : y K} has a closed graph and the function is continuous f : x f(x) max{u(x, y) : y K} V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
53 Nash existence result Theorem (Nash) Consider a game G = (S i, u i ) i I. If for each player i the set of pure strategies S i is finite then there exists at least one Nash equilibrium with mixed strategies V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, / 53
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