Rationality, strategic uncertainty, and belief diversity in non‐cooperative games

• Published date: 01 December 2018
• Date: 01 December 2018
• DOI: http://doi.org/10.1111/ijet.12161

doi: 10.1111/ijet.12161

Rationality, strategic uncertainty, and belief diversity in

non-cooperative games

Eduardo Zambrano∗

I investigate the existence of epistemic models of complete information games that satisfy the

following properties: players do not rule out their opponent’suse of rational ex ante strategies for

deriving their choices; they do not rule out, ex ante, that they can come to know the action profile

that is ultimately played; and they do not rule out strategic uncertainty. In this paper I showthat

for a large class of games there are no epistemic models that satisfy these three properties.

Key wor ds interactive epistemology, belief diversity in games

JEL classification D83, C72

Accepted 25 June 2016

1 Introduction

In a series of papers, Nachbar (1997, 2001, 2005) identifed a tension between predictability,

rationality, and belief diversity in two-player, discounted, infinitely repeated games. Nachbar shows

that, for a large class of games, there are no beliefs about the opponent’s repeated game strate-

gies that (i) allow the players to predict the choices of their opponents; (ii) have sufficiently

rich supports; and (iii) are consistent with rationality. “Loosely, if players learn to forecast the

path of play whenever each plays a strategy that the other anticipates and if the set of antici-

pated strategies are sufficiently rich, then neither anticipates any of his opponent’s best responses”

(Nachbar 2005, p. 459).

In this paper I show that analogs of the issues that arise in learning in a repeated games environ-

ment, as identified by Nachbar,can also ar ise in an epistemicgame-theoretic environment.

To do this I focus on complete information games and study each player’s decision problem

at the ex ante stage of the strategic interaction, that is, before the vector of (epistemic) types of the

players has been fixed.After the vector of types is selected, which happens at the inter im stage, players

independently choose an action from their action set.

As in a Bayesian game, an ex ante strategy for a player is a rule that maps types to actions. Players

have beliefs about the ex ante strategies chosen by their opponents and about their opponents’ types.

*Department of Economics, Orfalea College of Business, California Polytechnic State University, San Luis Obispo,

California, USA. Email: ezambran@calpoly.edu

I would like to thank John Nachbar,Adam Brandenburger, Amanda Friedenberg, Francisco Rodr´

ıguez, Muhamet Yildiz,

Oliver Board, Don Saari, Marek Kaminsky, David Levine, David Easley, Larry Blume, an anonymous reviewer and

participants at the Minnesota Summer Meeting of the Econometric Society, the Vanderbilt Spring Midwest Economic

Theory Conference, a UC-Irvine IMBS colloquium, and two Notre Dame seminars for their comments on previous

versions of this research. This work is dedicated to the memory of my first Economics Professor,In´

ırida Le ´

on Sandoval,

who passed away while I was starting this research.

International Journal of Economic Theory 14 (2018) 309–321 © IAET 309

International Journal of Economic Theory