Consider the following bargaining situation. Three individuals are considering u

Question

Consider the following bargaining situation. Three individuals are considering undertaking a business venture that will earn them $100 in profit, but they must agree how to split the $100 (in whole dollars). Bargaining works as follows: The three individuals simultaneously make demands for how much they each want d1, d2, and d3 in {0, 1, . . . , 100}. If their demands sum to more than $100 (d1+d2+d3 > 100), then they fail to agree and each gets nothing and so their payoffs are 0. If their demands sum to less than $100 (d1 + d2 + d3 100), they do the project, each gets his or her demand (and then their payoffs are their respective di’s), and the remainder is wasted.

Describe all of the pure strategy Nash equilibria of this game.

Explanation / Answer

If player 1 demands d1>=100, then any strategy of player 2 with a demand of d2>=0 and player 3 with a demand of d3>=0 is a payoff equivalent. Therefore, there are no strictly dominated strategies.
Any strategy demanding more than 100 $ is weakly dominated. To show this note that if player 3 demands d3 > 100, then any strategy of player 1 with d1> 0 and strategy of player 2 with d2>0 is payo equivalent. If player 3 demands 0<= d3 < 100, then player 1 could demand d1 = 100 d2- d3 and would obtain a payo of 100d2-d3. Demanding d1 > 100 will give player 1 a payo of 0. Therefore, any strategy demanding more than 100 dollars is weakly dominated.
Any pair (100-d2-d3,d2,d3) with 100>= d2-d3>=0 is a pure-strategy. Nash equilibrium of this game. There also exist equilibria of the form (d1,d2,d3) where d1,d2,d3 >= 100.

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