Monotone bargaining is Nash-solvable

نویسندگان

  • Vladimir Gurvich
  • Gleb A. Koshevoy
چکیده

Given two finite ordered sets A = {a1, . . . , am} and B = {b1, . . . , bn}, introduce the set of mn outcomes of the game O = {(a, b) | a ∈ A, b ∈ B} = {(ai, bj) | i ∈ I = {1, . . . ,m}, j ∈ J = {1, . . . , n}. Two players, Alice and Bob, have the sets of strategies X and Y that consist of all monotone non-decreasing mappings x : A → B and y : B → A, respectively. It is easily seen that each pair (x, y) ∈ X × Y produces at least one deal, that is, an outcome (a, b) ∈ O such that x(a) = b and y(b) = a. Denote by G(x, y) ⊆ O the set of all such deals related to (x, y). The obtained mapping G = Gm,n : X × Y → 2 is a game correspondence. Choose an arbitrary deal g(x, y) ∈ G(x, y) to obtained a mapping g : X × Y → O, which is a game form. We will show that each such game form is tight and, hence, Nash-solvable, that is, for any pair u = (uA, uB) of utility functions uA : O → R of Alice and uB : O → R of Bob, the obtained monotone bargaining game (g, u) has at least one Nash equilibrium in pure strategies. Moreover, the same equilibrium can be chosen for all selections g(x, y) ∈ G(x, y). We also obtain an efficient algorithm that determines such an equilibrium in time linear in mn, although the numbers of strategies |X | = ( m+n−1 m ) and |Y | = ( m+n−1 n ) are exponential in mn. Our results show that, somewhat surprising, the players have no need to hide or randomize their bargaining strategies, even in the zero-sum case.

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عنوان ژورنال:
  • CoRR

دوره abs/1711.00940  شماره 

صفحات  -

تاریخ انتشار 2017