05.SimultaneousMoveContinuous

05.SimultaneousMoveContinuous - SIMULTANEOUS-MOVE GAMES...

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SIMULTANEOUS-MOVE GAMES --- CONTINUOUS STRATEGIES AT LEAST ONE PLAYER HAS AN INFINITE NUMBER OF STRATEGIES (such as choosing a number from an interval) Player 1 chooses S 1 and Player 2 chooses S 2 simultaneously. The resulting payoffs, U 1 ( S 1 , S 2 ) and U 2 ( S 1 , S 2 ), are functions of S 1 and S 2 . EXAMPLE: NASH DEMAND GAME THERE IS 100 TL ON THE TABLE AND TWO PLAYERS, A AND B . EACH PLAYER SIMULTANEOUSLY ANNOUNCES “ DEMANDS ”, HOW MUCH HE WANTS FROM 100.TL. IF A’S AND B’S DEMANDS EXCEED 100, EACH GET ZERO. IF TOTAL DEMANDS IS LESS THAN OR EQUAL TO 100, THEY ARE PAID THEIR DEMANDS. WHAT IS THE NASH EQUILIBRIUM? STRATEGY FOR PLAYER A ( S A ) AND PLAYER B ( S B ) IS TO CHOOSE A NONNEGATIVE NUMBER. PAYOFFS : U A ( S A , S B ) = S A IF S A + S B ≤ 100, U B ( S A , S B ) = S B IF S A + S B ≤ 100, = 0 IF S A + S B > 100. = 0 IF S A + S B > 100. 1
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THIS GAME HAS MANY NASH EQUILIBRIA : ANY PAIR OF DEMANDS THAT SUM UP TO 100.TL. IS A NASH EQUILIBRIUM ( S* A + S* B = 100 ). WHY? EXAMPLE: PUBLIC GOOD GAME Two players, Süleyman and Çetin, can each contribute and produce a public good G. Q S = Süleyman’s contribution (a positive real number) Q C = Çetin’s contribution (a positive real number) G = Amount of public good produced = Q S + Q C . B i (G) = The benefit that player i gets if the quantity G of public good is produced (a function of G). C i (Q i ) = The cost of contributing Q i for player i (a function of Q i ). Payoff (Utility) of player i = B i (G) - C i (Q i ). Assume: C i (Q i ). = Q i and B i (G) is a strictly concave function. Benefits, costs B S (G) 45 degree line (Süleyman’s cost) C S (Q S ) = Q S Q S Q C total contributions. Given that Çetin contributes Q C , the difference B S (G) - Q S shows Suleyman’s payoff for each contribution Q S he may choose. 2
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Suleyman’s best response to Q C , is the Q S that maximizes the difference between B S (G) and Q S . Mathematically, Suleyman chooses a nonnegative Q S that maximizes B S ( Q C + Q S ) - Q S GIVEN Q C . The first-order condition to maximize the payoff is B S ( Q C + Q S ) /∂Q S = 1. Marginal Benefit from = Marginal cost from increasing Q S increasing Q S It says: Given the contribution by Cetin, the best-response contribution of Suleyman, if positive, is such that the slope of his benefit function is equal to the slope of his marginal cost (1). If Cetin’s contribution is “too large” (if Q C G C ) then Suleyman’s best response is Q S = 0, because for Q C G C the marginal benefit is smaller than the marginal cost, 1. Benefits, costs slope = 1 B S (G) 45 degree line (Süleyman’s cost) Utility Q S Q C total contributions. Q S G C 3
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By changing the vertical line at Q C and finding Suleyman’s corresponding best response, we get “points” on Suleyman’s best-response function. The best response contribution of Suleyman decreases if Çetin
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05.SimultaneousMoveContinuous - SIMULTANEOUS-MOVE GAMES...

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