metadata: authors: 0: author: Basu, Kaushik
1: author: Pattanaik, Prasanta K.
count: World
docna: 0: docna: Nash equilibria of games when players'
            preferences are quasi-transitive
docty: Policy Research Working Paper
subtopic: Human Rights,Disability,Services & Transfers to Poor,Economic Assistance,Access of Poor to Social Services,Crime and Society
abstracts: cdata!: 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 as erroneous
            by the same individual. 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 to provide a formal structure for such analysis 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.'
repnme: repnme: Nash equilibria of games when players'
            preferences are quasi-transitive
display_title: Nash equilibria of games when
            players' preferences are quasi-transitive
listing_relative_url: /research/2014/09/20211385/nash-equilibria-games-players-preferences-quasi-transitive
topicv3: Criminal Justice m1326993 1014,Social Assistance m1327201 1466,Discrimination m1505707 3379
publishtoextweb_dt: 2014-09-01T00:00:00Z
no_of_pages: 12
model_extractions: WPS7037 Policy Research Working Paper 7037 Nash Equilibria of Games When Players’ Preferences Are Quasi-Transitive _Kaushik Basu Prasanta K. Pattanaik_ Development Economics Vice Presidency Office of the Chief Economist September 2014
Policy Research Working Paper 7037

# **Abstract**

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 as erroneous by the same individual. 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 to provide a formal structure for such analysis 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.’ This paper is a product of the Office of the Chief Economist, Development Economics Vice Presidency. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at kbasu@worldbank.org and prasanta.pattanaik@ucr.edu.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **Nash Equilibria of Games When Players’ Preferences Are Quasi-Transitive***

## Kaushik Basu World Bank

1818 H Street NW, Mail Stop MC4-404 Washington, DC 20433 USA Tel: (202) 458-1076; Fax: (202) 522-0906; E-mail: <u>kbasu@worldbank.org;</u> Department of Economics, Cornell University, Ithaca, NY 14853 Prasanta K. Pattanaik Department of Economics, University of California, Riverside, CA 92521 Telephone & Fax: (951) 653 0979; E-mail: <u>prasanta.pattanaik@ucr.edu</u>

- For helpful comments, we are grateful to Rajat Deb, Roberto Veneziani, and Naoki Yoshihara.

1
# **1. Introduction**

Much of game theory is founded on the assumption that individuals are endowed with welldefined 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's preferences is that her preferences are necessarily transitive. A little bit of introspection makes it clear that this assumption is untenable when it comes to the indifference relation. Most of us are indifferent between having � and �+1 grains of sugar with coffee, for all �, but have a strict preference one way or the other between no sugar and a spoonful of sugar. There is a substantial analytical literature on this (see, among others, Georgescu-Roegen, 1936; Armstrong, 1939, 1951; Luce 1956; Mazumdar, 1957; Fishburn, 1970; Sen, 1970; Pattanaik, 1970; Quinn 1990; and Anand 1993); and the recognition of this problem actually goes back to ancient Greece in the form of the sorites paradox or heap paradox.

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 everybody worse off, thereby making room for some restrictions of voluntary transactions (Parfit, 1984; Basu, 2003; 2007).

The present paper is concerned not with these ethical matters but some foundational gametheoretic questions. Suppose we have a game where the players’ preferences are quasi-transitive (i.e., a game where the strict preference of each player is transitive) but the preferences of some of the players are not transitive, so that indifference relations are intransitive for some players. When can we be sure that a Nash equilibrium of such a game exists? In itself, this is an abstract exercise, but we hope that it will provide the groundwork for further investigation concerning government interventions, policy making and the ethics of individual autonomy.

The plan of the paper is as follows. In Section 2, we introduce finite games with quasi-transitive but not necessarily transitive preferences of players, and their mixed extensions. In Section 3, we prove a result, which shows that, if an agent’s quasi-transitive preferences over lotteries over mixed strategy profiles satisfy certain axioms, then there exists a utility function of the von Neumann-Morgenstern type, which preserves the agent’s strict preferences. Utilizing this result, in Section 4 we demonstrate the existence of a Nash equilibrium in a mixed strategy game where the players’ preferences satisfy our axioms.

2
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