75 Consider the following integer programming formulation of the sta ble

75 consider the following integer programming

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7.5 Consider the following integer programming formulation of the sta- ble matching problem. To describe the program, we use the follow- ing notation. Let m be a particular man and w a particular women. Then j > m w represents the set of all women j that m prefers over w , and i > w m represents the set of all men i that w prefers over m . In Section 2.7 we introduced linear programming. Integer programming is linear programming in which the variables are required to take integer values.
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Exercises 145 In the following program the variable x ij will be selected to be 1 if man i and woman j are matched in the matching selected: maximize i,j x ij subject to j x m,j 1 for all men m (E7.1) i x i,w 1 for all women w j> m w x m,j + i> w m x i,w + x m,w 1 for all pairs ( m, w ) x m,w { 0 , 1 } for all pairs ( m, w ) Prove that this integer program is a correct formulation of the stable matching problem. Consider the relaxation of the integer program that allows frac- tional stable matchings. It is identical to the above program, ex- cept that instead of each x m,w being either 0 or 1, x m,w is allowed to take any real value in [0 , 1]. Show that the following program is the dual program to the relaxation of E7.1. minimize i α i + j β j i,j γ ij subject to α m + β w j< m w γ m,j i< w m γ i,w γ m,w 1 for all pairs ( m, w ) α i , β j , γ i,j 0 for all i and j . Use complementary slackness (Theorem ?? ) to show that every feasible fractional solution to the relaxation of E7.1 is optimal and that setting α m = j x m,j for all m , β w = i x i,w for all w and γ ij = x ij for all i, j is optimal forthe dual program.
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8 Coalitions and Shapley value In this chapter, we consider cooperative game theory , in which players form coalitions to work toward a common goal. In these settings, there is a set n > 2 of players that can achieve a common goal yielding an overall payo ff of v , if they all cooperate with each other. However, subsets of these players, so-called coalitions , have the option of going o ff on their own and collaborating only with each other, rather than working as part of a grand coalition. Questions addressed by this theory include: How should rewards be shared among the players so as to discourage subgroups from defecting? What power or influence does a player have in the game? 8.1 The Shapley value and the glove market We review the example discussed in the Chapter (1). Suppose that three people are selling their wares in a market. Two of them are selling a single, left-handed glove, while the third is selling a right-handed one. A wealthy tourist arrives at the market in dire need of a pair of gloves, willing to pay $100 for a pair of gloves. She refuses to deal with the glove-bearers individually, and thus, these sellers have to come to some agreement as to how to make a sale of a left- and right-handed glove to her and how to then split the $100 amongst themselves. Clearly, the third player has an advantage, because his commodity is in scarcer supply. This means that
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  • Game Theory, Two-person zero-sum games

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