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3 3 using the fact that each action in the support of

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Unformatted text preview: ther firm can increase his price and still have 1/3 units of demand due to the capacity constraint on the other firm, thus making positive profits. It can be established using Theorem 2 that there exists a mixed strategy Nash equilibrium. Let us next proceed to construct a mixed strategy Nash equilibrium. 17 Game Theory: Lecture 6 Continuous Games Bertrand Competition with Capacity Constraints We focus on symmetric Nash equilibria, i.e., both firms use the same mixed strategy. We use the cumulative distribution function F (·) to represent the mixed strategy used by either firm. It can be seen that the expected payoff of player 1, when he chooses p1 and firm 2 uses the mixed strategy F (·), is given by p 2 u1 (p1 , F (·)) = F (p1 ) 1 + (1 − F (p1 )) p1 . 3 3 Using the fact that each action in the support of a mixed strategy must yield the same payoff to a player at the equilibrium, we obtain for all p in the support of F (·), p 2 −F (p ) + p = k , 3 3 for some k ≥ 0. From this we obtain: 3k F (p ) = 2 − . p 18 Game Theory: Lecture 6 Continuous Games Bertrand Competition with Capacity Constraints Note next that the upper support of the mixed strategy must be at p = 1, which implies that F (1) = 1. Combining with the preceding, we obtain ⎧ if 0 ≤ p ≤ 1 , ⎨ 0, 2 1, 2− p if 1 ≤ p ≤ 1, F (p ) = 2 ⎩ 1, if p ≥ 1. 19 Game Theory: Lecture 6 Continuous Games Uniqueness of a Pure Strategy Nash Equilibrium in Continuous Games We have shown in the previous lecture the following result: Given a strategic form game �I , (Si ), (ui )�, assume that the strategy sets Si are nonempty, convex, and compact sets, ui (s ) is continuous in s , and ui (si , s−i ) is quasiconcave in si . Then the game �I , (Si ), (ui )� has a pure strategy Nash equilibrium. The next example shows that even under strict convexity assumptions, there may be infinitely many pure strategy Nash equilibria. 20 Game Theory: Lecture 6 Continuous Games Uniqueness of a Pure Strategy Nash Equilibrium Example Consider a game with 2 players, Si = [0,...
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