Perfect regular equilibrium

• Published date: 01 December 2020
• DOI: http://doi.org/10.1111/ijet.12199
• Date: 01 December 2020

doi: 10.1111/ijet.12199

Perfect regular equilibrium

Hanjoon Michael Jung∗

We extend the solution concept of perfect Bayesian equilibrium to general games that allow a

continuum of types and strategies. In finite games, a perfect Bayesian equilibrium is weakly con-

sistent and a subgame perfect Nash equilibrium. In general games, however,it might not satisfy

these criteria. To solve this problem, we revise the definition of perfect Bayesian equilibrium

by replacing Bayes’ rule with regular conditional probability. The revised solution concept is

referred to as perfect regular equilibrium. Wepresent the conditions that ensure the existence of

this equilibrium. Then we show that every perfect regular equilibrium is always weakly consistent

and a subgame perfect Nashequilibr ium, and is equivalentto a simple version of perfect Bayesian

equilibrium in a finite game.

Key wor ds Bayes’rule, general game, perfect Bayesian equilibrium, perfect regular equilibrium,

regular conditional probability, solution concept

JEL classification C72

Accepted 13 January 2018

1 Introduction

We propose a revised version of perfect Bayesian equilibrium in general multi-period games with

observed actions, which allow a continuum of types and strategies. Fudenberg and Tirole (1991)

formulated the perfect Bayesian equilibrium in the setting of finite multi-period games with observed

actions, which allow only a finite number of types and strategies. In the finite games, every perfect

Bayesian equilibrium satisfies weak consistencyand the subgame perfect Nash equilibrium condition.

However, it might not satisfy these criteria in general games. To solve this problem, we revise its

definition by replacing Bayes’ rule with regular conditional probability. We refer to this revised

version as perfect regular equilibrium. Weshow that it satisfies these criteria in general multi-period

games with observed actions.

In game theory, most of the solution concepts were developed as refinements of the Nash equi-

librium introduced by Nash (1951). A Nash equilibrium consists of a set of strategies for each player

such that each strategy is the best response to the other strategies. This Nash equilibrium became

known as a compelling condition for rational strategies, and thus it became a necessary condition for

rational solution concepts. However, the Nash equilibrium was defined in strategic form games in

which in which all players must choose their strategies once and simultaneously.Hence, this solution

concept may not properlypredict players’ behavior in multi-period games where players choose their

actions in each period after observing actions taken before. In the multi-period games, players can

∗Ma Yinchu School of Economics,Tianjin University, Tianjin, China. Email: junghanjoon73@hanmail.net

I am very grateful to the anonymous referee for his or her valuable comments that have improvedthis dr aft. I gratefully

acknowledge research support from the NationalScience Council of the Republic of China (NSC 99-2410-H-001-114).

International Journal of Economic Theory (2018) 1–19 © IAET 1

International Journal of Economic Theory 16 (2020) 380–398 © IAET

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International Journal of Economic Theory

Perfect regular equilibrium Hanjoon Michael Jung

have different incentives in different periods. Since the Nash equilibrium requires all the players to

choose their strategies once and simultaneously, it might not reflect those various incentives in the

multi-period games. As a result, a Nash equilibrium may include incredible threats.

The subgame perfect Nash equilibrium by Selten (1975) improved the Nash equilibrium. The

basic idea behind this solution concept was to break a whole game into subgames and to find Nash

equilibria in every subgame. When we analyze each of the subgames separately,we are able to consider

players’ incentives within those subgames separately. Thus, the subgame perfect Nash equilibrium

can reflect different incentives in different subgames, and so it mayexclude some incredible threats.1

In games with incomplete information, however,subgames are too large to catch each of the players’

incentivesseparately. So, the subgame perfect Nash equilibrium might fail to reflect players’incentives

in different periods, which means that it may yet include incredible threats.

The perfect Bayesian equilibrium and the sequential equilibrium introduced by Kreps and Wil-

son (1982) improved the subgame perfect Nash equilibrium. These solution concepts breaka w hole

game into information sets and search strategies that satisfy sequential rationality for each of the

information sets. Sequential rationality is a condition for the strategies of rational players, and re-

quires that each strategy be the best response to the other strategies for each of the information

sets. This sequential rationality, therefore, inherits the spirit of the Nash equilibrium condition. In-

formation sets are small enough to represent each of the incentives separately. Consequently, these

solution concepts can reflect different incentives in different periods, and thus they can exclude

incredible threats in finite games. In general games that allow a continuum of types and strate-

gies, however, these solution concepts may cause more serious problems than including incredible

threats.

Information sets can be viewed as the smallest units of analysis. In games, players cannot distin-

guish decision points in a common information set. So, whatever action they choose, the same action

must be applied to all decision points in a common information set. That is, players can choose only

one action for each of their information sets. Hence, information sets would be the smallest units

that we can analyze in order to find players’ rational strategies.

These smallest units of analysis, however,might be too small for us to find rational strategies, and

sometimes we might need more information to find rational strategies for each of the information

sets. Sufficient information for finding rational strategies is constituted by rational beliefs, which

are probability distributions over each of the information sets and are also consistent with the given

strategies. For this reason, both of the solution concepts, perfect Bayesianequilibr ium and sequential

equilibrium, require specification of rational beliefs.

These solution concepts employ Bayes’ rule when they define the conditions for rational beliefs.

In general games, however,Bayes’ rule has limited application in practice, and this limited application

could result in their inability to satisfy weak consistency and the subgame perfect Nash equilibrium

condition.

Weak consistency is a criterion of rational beliefs, and it places restrictions only on the beliefs

on the equilibrium path. Weak consistency is viewed as a necessary condition for rational beliefs.

However, it is weak in that it does not locate any restriction on the beliefs off the equilibrium

path. The subgame perfect Nash equilibrium condition, on the other hand, is a criterion of the

rational strategies. It places restrictions on all actions on the equilibrium path, and furthermore it

sets restrictions on some of the actions off the equilibrium path. In this way, it can indirectly inspect

1Tofind a subgame perfect Nash equilibrium in practice, it is convenient to analyze subgames from back to front. This is

because, by analyzing backward, we can naturally consider players’future incentives in any period. In this sense, we may

view the subgame perfect Nash equilibrium as a combination of the Nash equilibrium and backward induction.

2International Journal of Economic Theory (2018) 1–19 © IAET

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