Ordinal Models of Bargaining

Abstract

In this section we will investigate the possibility of developing a theory of bargaining which depends only on the ordinal information contained in the players’ utility functions.¹ By ordinal information, we mean information about each player’s preference ordering over riskless alternatives. If A is a set of riskless alternatives over which player i has a preference ordering, then any realvalued function u_i on A such that u_i (a) > u_i (b) if and only if player i prefers a to b is an ordinal utility function for player i. Thus if u_i is a utility function representing player i’s preferences on A, then v_i is also such a utility function if and only if v_i = m_i (u_i)² where m_i is a monotone increasing (i.e., order preserving) function from the set of real numbers to itself. That is, for any real numbers x and y, m_i (x) > m_i (y) if and only if x > y. We will confine our attention to continuous (invertible) order preserving transformations, and for any x in Rⁿ denote (m₁ (x₁), ...,m_n (x_n)) by m(x).