Nash equilibria of games when players’ preferences are quasi‐transitive

• Published date: 01 March 2018
• Author: Kaushik Basu,Prasanta K. Pattanaik
• DOI: http://doi.org/10.1111/ijet.12143
• Date: 01 March 2018

doi: 10.1111/ijet.12143

Nash equilibria of games when players’

preferences are quasi-transitive

Kaushik Basu∗and Prasanta K. Pattanaik†

Much of game theory is founded on the assumption that individual players are endowed with

preferences that can be represented by a real-valued utility function. However, in reality human

preferences are often not transitive. This is especially true for the indifference relation, which

can lead an individual to make a series of choices which in their totality would be viewed by

the same individual as erroneous. There is a substantial literature that raises intricate questions

about individual liberty and the role of government intervention in such contexts. The aim of

this paper is not to go into these ethical matters but toprovide a formal st ructure for suchanalysis

by characterizing games where individual preferences are quasi-transitive. The paper identifies a

set of axioms which are sufficient for the existence of Nash equilibria in such “games.”

Key wor ds Nash equilibrium, quasi-transitive preference, partial representation

JEL classification C720

Accepted 15 June2017

1 Introduction

Much of game theory is founded on the assumption that individuals are endowed with well-defined

payoff or utility functions. This is not quite as innocuous as may appear at first sight. One implication

of having a real-valued function representing a person’spreferences is that her preferences are neces-

sarily transitive. A little introspection makes it clear that this assumption is untenable when it comes

to the indifference relation. Most of us are indifferentbetween having kand k+1 g rains of sugar with

coffee, for all k, but have a strict preference one way or the other between no sugar and a spoonful of

sugar. There is a substantialanaly tical literature on this (see, among others, Georgescu-Roegen1936;

Armstrong 1939, 1951; Luce 1956; Majumdar 1957; Fishburn 1970; Sen 1970; Pattanaik 1970, 1971;

Quinn 1990; Anand 1993), and the recognition of this problem actually goes back to ancient Greece

in the form of the sorites paradox or heap paradox.1

The recognition of this feature of human preference has important implications in interactive

and game-theoretic situations. It raises important ethical questions about individual autonomy and

collective norms. It can be argued, for instance, that each transaction between consenting adults

should be permitted on grounds of Pareto improvement, but a set of such transactions may leave

∗Department of Economics, Cornell University,Ithaca, NY, USA. Email: kb40@cornell.edu

†Department of Economics, University of California, Riverside, CA, USA.

For helpful comments, we are grateful toRajat D eb,John Hudson, Somdeb Lahiri, Shasi Nandeibam, Roberto Veneziani,

YongshengXu, Naoki Yoshihara, and an anonymousreferee.

1For an interesting application of the concept of perception threshold in the context of price changes, see Carlsonand

Parkin (1975); we are grateful to John Hudsonfor drawing our attention to Carlson and Parkin’s contribution.

International Journal of Economic Theory 14 (2018) 61–69 © IAET 61

International Journal of Economic Theory