The impact of the number of sellers on quantal response equilibrium predictions in Bertrand oligopolies

• Author: Chaohua Dong,Hang Wu,Ralph‐C Bayer
• Date: 01 November 2019
• DOI: http://doi.org/10.1111/jems.12298
• Published date: 01 November 2019

Received: 11 December 2017

|

Revised: 8 November 2018

|

Accepted: 24 November 2018

DOI: 10.1111/jems.12298

ORIGINAL ARTICLE

The impact of the number of sellers on quantal response

equilibrium predictions in Bertrand oligopolies

Ralph‐C Bayer

1

|

Chaohua Dong

2

|

Hang Wu

3

1

School of Economics, University of

Adelaide, Adelaide, South Australia,

Australia

2

School of Statistics and Mathematics,

Zhongnan University of Economics and

Law, Wuhan, China

3

School of Management, Harbin Institute

of Technology, Harbin, China

Correspondence

Hang Wu, School of Management, Harbin

Institute of Technology, Harbin, China.

Email: hang.wu@hit.edu.cn

Funding information

National Natural Science Foundation of

China, Grant/Award Numbers: 71671143,

71831005; Australian Research Council,

Grant/Award Number: DP120101831

Abstract

This paper studies how increasing the number of sellers in a Bertrand oligopoly

with homogenous goods affects the equilibrium price level predicted by logistic

quantal response equilibrium (LQRE) and power‐function QRE (PQRE). We

show that increasing the number of sellers reduces the average posted price in a

PQRE, but can increase the average posted price in an LQRE. Our results

indicate that the comparative‐static predictions of QRE (McKelvey & Palfrey,

1995, Games Econ Behav, 10, 6–38) are not necessarily robust to changes of the

quantal response function.

KEYWORDS

Bertrand Oligopoly, Comparative Statics, Quantal Response Equilibrium

JEL CLASSIFICATION

C7, L11

1

|

INTRODUCTION

Quantal response equilibrium (QRE; McKelvey & Palfrey, 1995) has been successfully used to rationalize decisions

observed in the laboratory. The concept has proven particularly useful when applied to games such as Bertrand

competition (e.g., Baye & Morgan, 2004), auctions (e.g., Goeree, Holt, & Palfrey, 2002), or tournaments (e.g., Gürtler &

Harbring, 2010). QRE extends the Nash equilibrium concept by allowing for bounded rationality. It allows players to

choose suboptimal actions with positive probability. In QRE, players hold accurate expectations about other players’

(mixed) strategies, but instead of playing a best response with certainty, they noisily best‐respond, where the likelihood

of choosing certain actions depends on the relative expected payoffs the actions yield. In practice, the two most

commonly used specifications of QRE are the logistic QRE (LQRE) and the power‐function QRE (PQRE). LQRE

assumes that the choice probability for an action is proportional to an exponential function of the payoff a player can

expect if the action is chosen (McKelvey & Palfrey, 1995). By contrast, PQRE uses a power form of the probabilistic

choice function (Chen, Friedman, & Thisse, 1997). In practice, researchers may prefer one specification to the other for

reasons such as compatibility with the domain of expected payoffs,

1

analytical tractability (e.g., Baye & Morgan, 2004),

or simply because of a better fit to their data (e.g., Goeree et al., 2002).

Using Bertrand competition with homogenous goods as an example, this study investigates whether there are qualitative

differences between the two predominant QRE specifications with regard to the comparative statics of their predictions.

This is particularly important as questions have been raised about the empirical content of QRE. Haile, Hortaçsu, and

Kosenok (2008) showed that QRE without any restrictions is void of empirical content, since any probabilistic choice

function can be rationalized by some distribution of decision errors. This has the implication that any behavior can be

explained perfectly by choosing an appropriate error structure. In response to this criticism, Goeree, Holt, and Palfrey

J Econ Manage Strat. 2019;28:787–793. wileyonlinelibrary.com/journal/jems © 2018 Wiley Periodicals, Inc.

|

787