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Nash Equilibrium Via Convex Analysis NASH EQUILIBRIUM VIA CONVEX ANALYSIS Nguyen Mau Nam1 Abstract. In this note we present a simple proof of Nash's famous theorem on the existence of Nash Equilibrium. The argument (due to R. T. Rockafellar) uses very basic elements of convex analysis, and the Brouwer Fixed-Point Theorem. Key words. Nash equilibrium, Brouwer Fixed-Point Theorem, Convex Analysis 1 Elements of Convex Analysis In this section we review some elements of convex analysis to be used in the proof of Nash's theorem. The detailed proofs are given for the convenience of the readers. The readers are also referred to the books [2, 6] for more complete study of convex analysis in n finite dimensions. Throughout the note we consider the Euclidean space R with the inner product denoted by h·; ·i and the Euclidean norm denoted by k · k. n For two points a and b in R , the line segment connecting them is [a; b] := λa + (1 − λ)b λ 2 [0; 1] : Note that if a = b, then this interval reduces to a singleton [a; b] = fag. n A subset Ω of R is convex if [a; b] ⊂ Ω whenever a; b 2 Ω. Equivalently, Ω is convex if and only if λa + (1 − λ)b 2 Ω for all a; b 2 Ω and λ 2 [0; 1]. n n Let f : R ! R be an real-valued function. The epigraph of f is a subset of R × R defined by n+1 n epi f := (x; α) 2 R x 2 R and α ≥ f(x) : The function f is called convex if n f λx + (1 − λ)y ≤ λf(x) + (1 − λ)f(y) for all x; y 2 R and λ 2 (0; 1): From the definition, we can show that f is convex if and only if its epigraph is a convex set. n Given a nonempty set Ω ⊂ R , the distance function associated with Ω is defined by n d(x; Ω) := inf kx − !k ! 2 Ω ; x 2 R : (1.1) n For each x 2 R , the Euclidean projection from x to Ω is defined based on the distance function as follows P(x; Ω) := ! 2 Ω kx − !k = d(x; Ω) : (1.2) 1Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR 97202, United States ([email protected]). This note is based on a presentation at the Analysis Seminar at Portland State. The research of Nguyen Mau Nam was partially supported by the Simons Foundation under grant #208785. 1 n n Proposition 1.1 Let Ω be a nonempty closed subset of R . Then for any x 2 R , the Euclidean projection P(x; Ω) is nonempty. Proof. By definition (1.2), for each k 2 N, there exists !k 2 Ω such that 1 d(x; Ω) ≤ kx − ! k < d(x; Ω) + : k k Obviously, f!kg is a bounded sequence. Thus it has a subsequence f!km g that converges to !. Since Ω is closed, ! 2 Ω. Letting m ! 1 in the inequality 1 d(x; Ω) ≤ kx − !km k < d(x; Ω) + ; km we have d(x; Ω) = kx − !k, which ensures that ! 2 P(x; Ω). An interesting consequence of convexity is the following projection property. n n Proposition 1.2 If Ω is a nonempty closed convex subset of R , then for each x 2 R , the Euclidean projection P(x; Ω) is a singleton. Proof. The nonemptiness of the projection P(x; Ω) follows from Proposition 1.1. To prove the uniqueness, suppose that !1;!2 2 P(x; Ω) with !1 6= !2. Then kx − !1k = kx − !2k = d(x; Ω): By the classical parallelogram equality, we have that ! + ! 2 k! − ! k2 2kx − ! k2 = kx − ! k2 + kx − ! k2 = 2 x − 1 2 + 1 2 : 1 1 2 2 2 This directly implies that ! + ! 2 k! − ! k2 x − 1 2 = kx − ! k2 − 1 2 < kx − ! k2 = d(x; Ω)2; 2 1 4 1 ! + ! which is a contradiction due to the inclusion 1 2 2 Ω. 2 n Next we characterized the Euclidean projection to convex sets in R . n Proposition 1.3 Let Ω be a nonempty closed convex subset of R . Then !¯ 2 P(¯x; Ω) if and only if hx¯ − !;¯ ! − !¯i ≤ 0 for all ! 2 Ω: (1.3) Proof. Suppose that! ¯ 2 P(¯x; Ω). For any ! 2 Ω and λ 2 (0; 1), we have! ¯ +λ(! −!¯) 2 Ω. Thus kx¯ − !¯k2 = d(¯x; Ω)2 ≤ kx¯ − !¯ + λ(! − !¯)k2 = kx¯ − !¯k2 − 2λhx¯ − !;¯ ! − !¯i + λ2k! − !¯k2: 2 This readily implies that 2hx¯ − !;¯ ! − !¯i ≤ λk! − !¯k2: Letting λ # 0, we obtain (1.3) Let us now prove the converse by assuming that (1.3) is satisfied. For any ! 2 Ω, the following estimates show that! ¯ 2 P(¯x; Ω): kx¯ − !k2 = kx¯ − !¯ +! ¯ − !k2 = kx¯ − !¯k2 + k!¯ − !k2 + 2hx¯ − !;¯ !¯ − !i = kx¯ − !¯k2 + k!¯ − !k2 − 2hx¯ − !;¯ ! − !¯i ≥ kx¯ − !¯k2: The proof is now complete. Figure 1: A characterization of Euclidean projection n n We know from the above that for any nonempty closed set Ω in R and for any x 2 R , the Euclidean projection P(x; Ω) is a singleton. Now we show that the projection mapping is in fact nonexpansive, i.e., satisfies the Lipschitz property in (1.4), which also implies that it is continuous. n Proposition 1.4 Let Ω be a nonempty closed convex subset of R . Then for any x1; x2 2 n R , we have the estimate 2 P(x1; Ω) − P(x2; Ω) ≤ P(x1; Ω) − P(x2; Ω); x1 − x2 : In particular, it implies the Lipschitz continuity of the projection mapping with constant ` = 1: n P(x1; Ω) − P(x2; Ω) ≤ kx2 − x2k for all x1; x2 2 R : (1.4) Proof. It follows from the preceding proposition that n P(x2; Ω) − P(x1; Ω); x1 − P(x1; Ω) ≤ 0 for all x1; x2 2 R : Changing the role of x1; x2 in the above inequality and summing them up give us P(x1; Ω) − P(x2; Ω); x2 − x1 + P(x1; Ω) − P(x2; Ω) ≤ 0: 3 This implies the first estimate in the proposition. Finally, the nonexpansive property of the Euclidean projection follows directly from 2 P(x1; Ω) − P(x2; Ω) ≤ P(x1; Ω) − P(x2; Ω); x1 − x2 ≤ kP(x1; Ω) − P(x2; Ω)k · kx1 − x2k n for all x1; x2 2 R , which completes the proof of the proposition. n Let Ω be a nonempty, convex subset of R and letx ¯ 2 Ω. The normal cone to Ω atx ¯ is n N(¯x; Ω) := v 2 R hv; x − x¯i ≤ 0 for all x 2 Ω : (1.5) The following proposition establish a useful relation between the normal cone and the projection to convex sets. n Proposition 1.5 Let Ω be a nonempty closed convex subset of R and let x¯ 2 Ω. Then v 2 N(¯x; Ω) if and only if x¯ 2 P(¯x + v; Ω). Proof. By the definition, one has that v 2 N(¯x; Ω) if and only if hv; w − x¯i = hx¯ + v − x;¯ w − x¯i ≤ 0 for all w 2 Ω; which is equivalent to the fact thatx ¯ 2 P(¯x + v; Ω) by Proposition 1.3. Consider the constrained optimization problem (P): minimize f(x) subject to x 2 Ω; n n where f : R ! R is a convex function and Ω is a nonempty closed convex subset of R . Recall that an elementx ¯ 2 Ω is called an optimal solution of problem (P) if and only if f(x) ≥ f(¯x) for all x 2 Ω: n 1 Proposition 1.6 Suppose that f : R ! R is convex and C and that Ω is a nonempty n closed convex subset of R . An element x¯ is an optimal solution of problem (P) if and only if 0 2 rf(¯x) + N(¯x; Ω); or, equivalently, −∇f(¯x) 2 N(¯x; Ω). Proof. Suppose thatx ¯ is an optimal solution of the problem. Then f(x) ≥ f(¯x) for all x 2 Ω: Fix any u 2 Ω and t 2 (0; 1). Since Ω is convex,x ¯ + t(u − x¯) = tu + (1 − t)¯x 2 Ω. Thus, f(¯x + t(u − x¯)) ≥ f(¯x): 4 This implies f(¯x + t(u − x¯)) − f(¯x) ≥ 0: t Taking the limit as t ! 0+, one obtains hrf(¯x); u − x¯i ≥ 0; which can be rewritten as h−∇f(¯x); u − x¯i ≤ 0. Since this inequality holds true for any u 2 Ω, one has that −∇f(¯x) 2 N(¯x; Ω). Let us prove the converse. Suppose that −∇f(¯x) 2 N(¯x; Ω). Then 0 ≤ hrf(¯x); u − x¯i for all u 2 Ω: Since f is a convex function, we always have n hrf(¯x); u − x¯i ≤ f(u) − f(¯x) for all u 2 R : In particular, for every u 2 Ω, 0 ≤ hrf(¯x); u − x¯i ≤ f(u) − f(¯x): Thus, f(¯x) ≤ f(u) for every u 2 Ω, which ensures thatx ¯ is an optimal solution of (P). 2 Nash Equilibrium This section gives a brief introduction to noncooperative game theory and presents a simple proof of the existence of Nash equilibrium as a consequence of convexity and Brouwer's fixed point theorem. For simplicity, we only consider two-person games while more general situations can be treated similarly. m n Definition 2.1 Let Ω1 and Ω2 be nonempty subsets of R and R , respectively. A non- cooperative game in the case of two players I and II consists of two strategy sets Ωi and two real-valued functions ui :Ω1 × Ω2 ! R for i = 1; 2 called the payoff functions.
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