Answer to Question #253762 in Microeconomics for Ashwin

Question

Advance Nash equilibrium exercise Microeconomics

Consider a two-player game that satisfies the hypotheses of the existence theorem (see below). Let N={1,2}  and suppose the game is symmetric, ie.:

1. A_1 = A_2

2. for all a,b in A

Use Kakutani's Theorem to prove that exists an element a^*₁ in A₁ such that (a^*_1, a^*₁)  is a Nash equilibrium (such an equilibrium is called a symmetric equilibrium).

Theorem (Existence):

Consider the gameIf for all i: A_i is a non-empty, convex, compact subset of R^n:

1.is quasi concave

2.is continuous.

So, there is a Nash equilibrium in pure strategies.

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Answer 1

Note: The symbols used in the question are not standard but derived from some context. We followed the same symbols used in the question even though some particulars in the symbols have nothing to do with the proof.

$Eg: \space In \space a_1^*, *$ has nothing to do with the proof.

 

We are given a 2-player game that satisfies the hypothesis of the Existence theorem of Nash equilibria.

Hypothesis of existence theorem of Nash equilibria are

1.A_i are a sequence of nonempty convex, compact subset of the n-dimensional Euclidian space ℝⁿ. (Remember that is a compact Hausdorff space Y, which will be useful later..! )

2.The game is quasi-concave and continuous.

In our case there is only 2 such sets. $A_1 \space and\space A_2.\space Also, A=A_ 1 =A _2 A=A1=A2.$ Now the domain of our game, say f, is A. Since we have a symmetric game in our hand (by assumption) a game position can be represented by (a_i,a_j). ( first player's i^th option and second players j^th option)

We need to prove the existence of a point $a_1^* \space such\space that\space (a ^∗ _1 , a_1*)$ is a Nash equilibrium.

Definition of Nash Equilibrium

Nash equilibrium of a game is defined as a fixed point of f i.e. ordered pair of strategies where each player's strategy is a best response to the strategies of the other players. 

Working of f

The function f associates with each point in A, a new ordered pair (2 possible response) where each player's strategy is the best response to other players' strategies in A. Equilibrium happens when these points become equal. Or more formally, when function fixes a point. 

Now our task reduced to prove that f has a fixed point in A.

Kakutani's theorem has been used to prove the existence of this fixed point.

Kakutani's theorem

Let A be a compact, convex subset of Rⁿ. Let f: A → 2^A be a set function (a function whose range contains sets as elements) on A with the following properties:

• f has a closed graph.
• f(x) is non-empty and convex for all x ∈ A.( We already have this property)

Then f has a fixed point.

Now it is enough to prove that f has a closed graph. It can be deduced from the closed graph theorem.

Closed graph theorem

 If f is a map from a topological space X (Rⁿ in our case) into a compact Hausdorff space Y (again, Rⁿ in our case) then the graph of f is closed if and only if f is continuous (which is given).

Now it satisfies all the conditions for Kakutani's theorem to apply. 

So, f has a fixed point.

That is there exist a point a₁* such that (a₁*,a₁*) is a Nash equilibrium.

This completes the proof.