6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium


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1 6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18,
2 Introduction Outline PricingCongestion Game Example Existence of a Mixed Strategy Nash Equilibrium in Finite Games Existence in Games with Infinite Strategy Spaces Reading: Fudenberg and Tirole, Chapter 1. 2
3 Example Introduction In this lecture, we study the question of existence of a Nash equilibrium in both games with finite and infinite pure strategy spaces. We start with an example, pricingcongestion game, where players have infinitely many pure strategies. We consider two instances of this game, one of which has a unique pure Nash equilibrium, and the other does not have any pure Nash equilibria. 3
4 Example PricingCongestion Game Consider a price competition model studied in [Acemoglu and Ozdaglar 07]. 1 unit of traffic Reservation utility R Consider a parallel link network with I links. Assume that d units of flow is to be routed through this network. We assume that this flow is the aggregate flow of many infinitesimal users. Let l i (x i ) denote the latency function of link i, which represents the delay or congestion costs as a function of the total flow x i on link i. Assume that the links are owned by independent providers. Provider i sets a price p i per unit of flow on link i. The effective cost of using link i is p i + l i (x i ). Users have a reservation utility equal to R, i.e., if p i + l i (x i ) > R, then no traffic will be routed on link i. 4
5 Example 1 Game Theory: Lecture 5 Example We consider an example with two links and latency functions l 1 (x 1 ) = 0 and l 2 (x 2 ) = 3 x 2 2. For simplicity, we assume that R = 1 and d = 1. Given the prices (p 1, p 2 ), we assume that the flow is allocated according to Wardrop equilibrium, i.e., the flows are routed along minimum effective cost paths and the effective cost cannot exceed the reservation utility. Definition A flow vector x = [x i ] i=1,...,i is a Wardrop equilibrium if I i=1 x i d and p i + l i (x i ) = min{p j + l j (x j )}, for all i with x i > 0, j p i + l i (x i ) R, for all i with x i > 0, with I = d if min j {p j + l j (x j )} < R. i=1 x i 5
6 Example 1 (Continued) Example We use the preceding characterization to determine the flow allocation on each link given prices 0 p 1, p 2 1: x 2 (p 1, p 2 ) = { 2 3 (p 1 p 2 ), 0, p 1 p 2, otherwise, and x 1 (p 1, p 2 ) = 1 x 2 (p 1, p 2 ). The payoffs for the providers are given by: u 1 (p 1, p 2 ) = p 1 x 1 (p 1, p 2 ) u 2 (p 1, p 2 ) = p 2 x 2 (p 1, p 2 ) We find the pure strategy Nash equilibria of this game by characterizing the best response correspondences, B i (p i ) for each player. The following analysis assumes that at the Nash equilibria (p 1, p 2 ) of the game, the corresponding Wardrop equilibria x satisfies x 1 > 0, x 2 > 0, and x 1 + x 2 = 1. For the proofs of these statements, see [Acemoglu and Ozdaglar 07]. 6
7 Example 1 (Continued) Example In particular, for a given p 2, B 1 (p 2 ) is the optimal solution set of the following optimization problem maximize 0 p1 1, 0 x 1 1 p 1 x 1 3 subject to p 1 = p 2 + (1 x 1 ) 2 Solving the preceding optimization problem, we find that { } B 1 (p 2 ) = min 1, 3 + p Similarly, B 2 (p 1 ) = p
8 Example 1 (Continued) Example 1 Best Response Functions p p 1 The figure illustrates the best response correspondences as a function of p 1 and p 2. The correspondences intersect at the unique point (p 1, p 2 ) = (1, 1 2 ), which is the unique pure strategy equilibrium. 8
9 Example 2 Game Theory: Lecture 5 Example We next consider a similar example with latency functions given by { 0 if 0 x 1/2 l 1 (x) = 0, l 2 (x) = x 1/2 x 1/2, for some sufficiently small ɛ > 0. The following list considers all candidate Nash equilibria (p 1, p 2 ) and profitable unilateral deviations for ɛ sufficiently small, thus establishing the nonexistence of a pure strategy Nash equilibrium: 1 p 1 = p 2 = 0: A small increase in the price of provider 1 will generate positive profits, thus provider 1 has an incentive to deviate. 2 p 1 = p 2 > 0: Let x be the corresponding flow allocation. If x 1 = 1, then provider 2 has an incentive to decrease its price. If x 1 < 1, then provider 1 has an incentive to decrease its price. 3 0 p 1 < p 2 : Player 1 has an incentive to increase its price since its flow allocation remains the same. 4 0 p 2 < p 1 : For ɛ sufficiently small, the profit function of player 2, given p 1, is strictly increasing as a function of p 2, showing that provider 2 has an incentive to increase its price. 9 ɛ
10 We start by analyzing existence of a Nash equilibrium in finite (strategic form) games, i.e., games with finite strategy sets. Theorem (Nash) Every finite game has a mixed strategy Nash equilibrium. Implication: matching pennies game necessarily has a mixed strategy equilibrium. Why is this important? Without knowing the existence of an equilibrium, it is difficult (perhaps meaningless) to try to understand its properties. Armed with this theorem, we also know that every finite game has an equilibrium, and thus we can simply try to locate the equilibria. 10
11 Approach Game Theory: Lecture 5 Recall that a mixed strategy profile σ is a NE if u i (σ i, σ i ) u i (σ i, σ i ), for all σ i Σ i. In other words, σ is a NE if and only if σi B (σ ) for all i, (σ i i where B i i ) is the best response of player i, given that the other players strategies are σ i. We define the correspondence B : Σ Σ such that for all σ Σ, we have B(σ) = [B i (σ i )] i I (1) The existence of a Nash equilibrium is then equivalent to the existence of a mixed strategy σ such that σ B(σ): i.e., existence of a fixed point of the mapping B. We will establish existence of a Nash equilibrium in finite games using a fixed point theorem. 11
12 Definitions Game Theory: Lecture 5 A set in a Euclidean space is compact if and only if it is bounded and closed. A set S is convex if for any x, y S and any λ [0, 1], λx + (1 λ)y S. convex set not a convex set 12
13 Weierstrass s Theorem Theorem (Weierstrass) Let A be a nonempty compact subset of a finite dimensional Euclidean space and let f : A R be a continuous function. Then there exists an optimal solution to the optimization problem minimize f (x) subject to x A. There exists no optimal that attains it 13
14 Kakutani s Fixed Point Theorem Theorem (Kakutani) Let A be a nonempty subset of a finite dimensional Euclidean space. Let f : A A be a correspondence, with x A f (x) A, satisfying the following conditions: A is a compact and convex set. f (x) is nonempty for all x A. f (x) is a convexvalued correspondence: for all x A, f (x) is a convex set. f (x) has a closed graph: that is, if {x, y } {x, y } with y n f (x n ), then y f (x). Then, f has a fixed point, that is, there exists some x A, such that x f (x). n n 14
15 Kakutani s Fixed Point Theorem Graphical Illustration is not convexvalued does not have a closed graph 15
16 Proof of Nash s Theorem 1 2 The idea is to apply Kakutani s theorem to the best response correspondence B : Σ Σ. We show that B(σ) satisfies the conditions of Kakutani s theorem. Σ is compact, convex, and nonempty. By definition Σ = Σ i i I where each Σ i = {x j x j = 1} is a simplex of dimension S i 1, thus each Σ i is closed and bounded, and thus compact. Their product set is also compact. B(σ) is nonempty. By definition, B i (σ i ) = arg max u i (x, σ i ) x Σ i where Σ i is nonempty and compact, and u i is linear in x. Hence, u i is continuous, and by Weirstrass s theorem B(σ) is nonempty. 16
17 Proof (continued) Game Theory: Lecture 5 3. B(σ) is a convexvalued correspondence. Equivalently, B(σ) Σ is convex if and only if B i (σ i ) is convex for all i. Let σ i, σ i B i (σ i ). Then, for all λ [0, 1] B i (σ i ), we have u i (σ i, σ i ) u i (τ i, σ i ) for all τ i Σ i, u i (σ i, σ i ) u i (τ i, σ i ) for all τ i Σ i. The preceding relations imply that for all λ [0, 1], we have λu i (σ i, σ i ) + (1 λ)u i (σ i, σ i ) u i (τ i, σ i ) for all τ i Σ i. By the linearity of u i, u i (λσ i + (1 λ)σ i, σ i ) u i (τ i, σ i ) for all τ i Σ i. Therefore, λσ i + (1 λ)σ i B i (σ i ), showing that B(σ) is convexvalued. 17
18 Proof (continued) Game Theory: Lecture 5 4. B(σ) has a closed graph. Suppose to obtain a contradiction, that B(σ) does not have a closed graph. Then, there exists a sequence (σ n, σˆ n) (σ, σˆ ) with σˆ n B(σ n ), but σˆ / B(σ), i.e., there exists some i such that σˆ i / B i (σ i ). This implies that there exists some σ i Σ i and some ɛ > 0 such that u i (σ i, σ i ) > u i (σˆ i, σ i ) + 3ɛ. By the continuity of u i and the fact that σ n i σ i, we have for sufficiently large n, u i (σ i, σn i ) u i (σ i, σ i ) ɛ. 18
19 Proof (continued) Game Theory: Lecture 5 [step 4 continued] Combining the preceding two relations, we obtain u i (σ i, σ n i ) > u i (σˆ i, σ i ) + 2ɛ u i (σˆ in, σ n i ) + ɛ, where the second relation follows from the continuity of u i. This contradicts the assumption that σˆ n i B i (σ n i ), and completes the proof. The existence of the fixed point then follows from Kakutani s theorem. If σ B (σ ), then by definition σ is a mixed strategy equilibrium. 19
20 Existence of Equilibria for Infinite Games A similar theorem to Nash s existence theorem applies for pure strategy existence in infinite games. Theorem (Debreu, Glicksberg, Fan) Consider a strategic form game I, (S i ) i I, (u i ) i I such that for each i I S i is compact and convex; u i (s i, s i ) is continuous in s i ; u i (s i, s i ) is continuous and concave in s i [in fact quasiconcavity suffices]. Then a pure strategy Nash equilibrium exists. 20
21 Definitions Game Theory: Lecture 5 Suppose S is a convex set. Then a function f : S R is concave if for any x, y S and any λ [0, 1], we have f (λx + (1 λ)y ) λf (x) + (1 λ)f (y). concave function not a concave function 21
22 Proof Game Theory: Lecture 5 Now define the best response correspondence for player i, B i : S i S i, { } B i (s i ) = s i S i u i (s i, s i ) u i (s i, s i ) for all s i S i. Thus restriction to pure strategies. Define the set of best response correspondences as B (s) = [B i (s i )] i I. and B : S S. 22
23 Proof (continued) We will again apply Kakutani s theorem to the best response correspondence B : S S by showing that B(s) satisfies the conditions of Kakutani s theorem. 1 2 S is compact, convex, and nonempty. By definition S = S i i I since each S i is compact [convex, nonempty] and finite product of compact [convex, nonempty] sets is compact [convex, nonempty]. B(s) is nonempty. By definition, B i (s i ) = arg max u i (s, s i ) s S i where S i is nonempty and compact, and u i is continuous in s by assumption. Then by Weirstrass s theorem B(s) is nonempty. 23
24 Proof (continued) Game Theory: Lecture 5 3. B(s) is a convexvalued correspondence. This follows from the fact that u i (s i, s i ) is concave [or quasiconcave] in s i. Suppose not, then there exists some i and some s i S i such that B i (s i ) arg max s Si u i (s, s i ) is not convex. This implies that there exists s i, s i S i such that s i, s i B i (s i ) and λs i + (1 λ)s i / B i (s i ). In other words, λu i (s i, s i ) + (1 λ)u i (s i, s i ) > u i (λs i + (1 λ) s i, s i ). But this violates the concavity of u i (s i, s i ) in s i [recall that for a concave function f (λx + (1 λ)y ) λf (x) + (1 λ)f (y )]. Therefore B(s) is convexvalued. 4. The proof that B(s) has a closed graph is identical to the previous proof. 24
25 Remarks Nash s theorem is a special case of this theorem: Strategy spaces are simplices and utilities are linear in (mixed) strategies, hence they are concave functions of (mixed) strategies. Continuity properties of the Nash equilibrium set : Consider strategic form games with finite pure strategy sets S i and utilities u i (s, λ), where u i is a continuous function of λ. Let G (λ) = I, (S i ), (u i (s, λ)) and let E (λ) denote the Nash correspondence that associates with each λ, the set of (mixed) Nash equilibria of G (λ). Proposition Assume that λ Λ, where Λ is a compact set. Then E (λ) has a closed graph. Proof similar to the proof of closedness of B(σ) in Nash s theorem. This does not imply continuity of the Nash equilibrium set E (λ)!! 25
26 Existence of Nash Equilibria Can we relax (quasi)concavity? Example: Consider the game where two players pick a location s 1, s 2 R 2 on the circle. The payoffs are u 1 (s 1, s 2 ) = u 2 (s 1, s 2 ) = d(s 1, s 2 ), where d(s 1, s 2 ) denotes the Euclidean distance between s 1, s 2 R 2. No pure Nash equilibrium. However, it can be shown that the strategy profile where both mix uniformly on the circle is a mixed Nash equilibrium. 26
27 A More Powerful Theorem Theorem (Glicksberg) Consider a strategic form game I, (S i ) i I, (u i ) i I such that for each i I 1 S i is a nonempty and compact metric space; 2 u i (s i, s i ) is continuous in s. Then a mixed strategy Nash equilibrium exists. With continuous strategy spaces, space of mixed strategies infinite dimensional! We will prove this theorem in the next lecture. 27
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