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Lecture14-new

# Lecture14-new - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 14: Nash Bargaining Solution Asu Ozdaglar MIT March 30, 2010 1

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Game Theory: Lecture 14 Introduction Outline Rubinstein Bargaining Model with Alternating Offers Nash Bargaining Solution Relation of Axiomatic and Strategic Model Reference: Osborne and Rubinstein, Bargaining and Markets. 2
Game Theory: Lecture 14 Strategic Model Introduction In this lecture, we discuss an axiomatic approach to the bargaining problem. In particular, we introduce the Nash bargaining solution and study the relation between the axiomatic and strategic (noncooperative) models. As we have seen in the last lecture, the Rubinstein bargaining model allows two players to offer alternating proposals indefinitely, and it assumes that future payoffs of players 1 and 2 are discounted by δ 1 , δ 2 ( 0, 1 ) . 3

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Game Theory: Lecture 14 Strategic Model Rubinstein Bargaining Model with Alternating Offers We showed that the following stationary strategy profile is a subgame perfect equilibrium for this game. Player 1 proposes x * 1 and accepts offer y if, and only if, y y * 1 . Player 2 proposes y * 2 and accepts offer x if, and only if, x x * 2 , where x * 1 = 1 - δ 2 1 - δ 1 δ 2 , y * 1 = δ 1 ( 1 - δ 2 ) 1 - δ 1 δ 2 , x * 2 = δ 2 ( 1 - δ 1 ) 1 - δ 1 δ 2 , y * 2 = 1 - δ 1 1 - δ 1 δ 2 . Clearly, an agreement is reached immediately for any values of δ 1 and δ 2 . To gain more insight into the resulting allocation, assume for simplicity that δ 1 = δ 2 . Then, we have If 1 moves first, the division will be ( 1 1 + δ , δ 1 + δ ) . If 2 moves first, the division will be ( δ 1 + δ , 1 1 + δ ) . 4
Game Theory: Lecture 14 Strategic Model Rubinstein Bargaining Model with Alternating Offers The first mover’s advantage (FMA) is clearly related to the impatience of the players (i.e., related to the discount factor δ ): If δ 1, the FMA disappears and the outcome tends to ( 1 2 , 1 2 ) . If δ 0, the FMA dominates and the outcome tends to ( 1, 0 ) . More interestingly, let’s assume the discount factor is derived from some interest rates r 1 and r 2 . δ 1 = e - r 1 Δ t , δ 2 = e - r 2 Δ t These equations represent a continuous-time approximation of interest rates. It is equivalent to interest rates for very small periods of time Δ t : e - r i Δ t 1 1 + r i Δ t . Taking Δ t 0, we get rid of the first mover’s advantage. lim Δ t 0 x * 1 = lim Δ t 0 1 - δ 2 1 - δ 1 δ 2 = lim Δ t 0 1 - e - r 2 Δ t 1 - e - ( r 1 + r 2 ) Δ t = r 2 r 1 + r 2 . 5

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Game Theory: Lecture 14 Strategic Model Alternative Bargaining Model: Nash’s Axiomatic Model Bargaining problems represent situations in which: There is a conflict of interest about agreements. Individuals have the possibility of concluding a mutually beneficial agreement. No agreement may be imposed on any individual without his approval. The strategic or noncooperative model involves explicitly modeling the bargaining process (i.e., the game form).
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