lecture5 notes

# 2 p p 0 let x b e the corresponding ow allocation if

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 0: A small increase in the price of provider 1 will generate 1 2 positive proﬁts, thus provider 1 has an incentive to deviate. 2 p = p > 0: Let x b e the corresponding ﬂow allocation. If x = 1, 1 2 1 then provider 2 has an incentive to decrease its price. If x1 < 1, then provider 1 has an incentive to decrease its price. 3 0 ≤ p < p : Player 1 has an incentive to increase its price since its 1 2 ﬂow allocation remains the same. 4 0 ≤ p < p : For � suﬃciently small, the proﬁt function of player 2, 2 1 given p1 , is strictly increasing as a function of p2 , showing that provider 2 has an incentive to increase its price. 9 Game Theory: Lecture 5 Existence Results Existence Results 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 Game Theory: Lecture 5 Existence Results Approach Recall that a mixed strategy proﬁle σ∗ is a NE if ui (σi∗ , σ∗ i ) ≥ ui (σi , σ∗ i ), − − for all σi ∈ Σi . ∗ In other words, σ∗ is a NE if and only if σi∗ ∈ B−i (σ∗ i ) for all i , − ∗ ( σ ∗ ) is the b est response of player i , given that the other where B−i −i players’ strategies are σ∗ i . − We deﬁne the correspondence B : Σ � Σ such that for all σ ∈ Σ, we have (1) B (σ) = [Bi (σ−i )]i ∈I 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 Game Theory: Lecture 5 Existence Results Deﬁnitions A set in a Euclidean space is compact if and only if it is bounded and closed. A set S is convex...
View Full Document

## This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

Ask a homework question - tutors are online