Monotone bargaining is Nash-solvable

Abstract

Given two finite ordered sets $A = \{a_1, \ldots, a_m\}$ and $B = \{b_1, \ldots, b_n\}$, introduce the set of $m n$ outcomes of the game $O = \{(a, b) \mid a \in A, b \in B\} = \{(a_i, b_j) \mid i \in I = \{1, \ldots, m\}, j \in J = \{1, \ldots, n\}$. Two players, Alice and Bob, have the sets of strategies $X$ and $Y$ that consist of all monotone non-decreasing mappings $x: A \rightarrow B$ and $y: B \rightarrow A$, respectively. It is easily seen that each pair $(x,y) \in X \times Y$ produces at least one {\em deal}, that is, an outcome $(a,b) \in O$ such that $x(a) = b$ and $y(b) = a$. Denote by $G(x,y) \subseteq O$ the set of all such deals related to $(x,y)$. The obtained mapping $G = G_{m,n}: X \times Y \rightarrow 2^O$ is a game correspondence. Choose an arbitrary deal $g(x,y) \in G(x,y)$ to obtained a mapping $g : X \times Y \rightarrow O$, which is a game form. We will show that each such game form is tight and, hence, Nash-solvable, that is, for any pair $u = (u_A, u_B)$ of utility functions $u_A : O \rightarrow \mathbb R$ of Alice and $u_B: O \rightarrow \mathbb R$ 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 $g(x,y) \in G(x,y)$. We also obtain an efficient algorithm that determines such an equilibrium in time linear in $m n$, although the numbers of strategies $|X| = \binom{m+n-1}{m}$ and $|Y| = \binom{m+n-1}{n}$ are exponential in $m n$. Our results show that, somewhat surprising, the players have no need to hide or randomize their bargaining strategies, even in the zero-sum case.