sswp1465_updated_041123.pdf

DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES

CALIFORNIA INSTITUTE OF TECHNOLOGY

PASADENA, CALIFORNIA 91125

Cursed Sequential Equilibrium

Meng

-Jhang Fong

California Institute of Technology

-Hsuan Lin

California Institute of

Technology

Thomas R. Palfrey

California Institute of Technology

SOCIAL SCIENCE WORKING PAPER 1465

April

2023

Cursed Sequential Equilibrium

*

Meng-Jhang Fong



Po-Hsuan Lin

Thomas R. Palfrey

§

April 11, 2023

Abstract

This paper develops a framework to extend the strategic form analysis of cursed

equilibrium (CE) developed by Eyster and Rabin (2005) to multi-stage games. The

approach uses behavioral strategies rather than normal form mixed strategies, and

imposes sequential rationality. We define cursed sequential equilibrium (CSE) and

compare it to sequential equilibrium and standard normal-form CE. We provide a

general characterization of CSE and establish its properties. We apply CSE to five

applications in economics and political science. These applications illustrate a wide

range of differences between CSE and Bayesian Nash equilibrium or CE: in signaling

games; games with preplay communication; reputation building; sequential voting;

and the dirty faces game where higher order beliefs play a key role. A common theme

in several of these applications is showing how and why CSE implies systematically

different behavior than Bayesian Nash equilibrium in dynamic games of incomplete

information with private values, while CE coincides with Bayesian Nash equilibrium

for such games.

JEL Classification Numbers: C72, D83

Keywords: Multi-stage Games, Private Information, Cursed Equilibrium, Learning

Grants from the National Science Foundation (SES-0617820) and the Gordon and Betty Moore Foun-

dation (1158) supported this research. We are grateful to Shengwu Li and Shani Cohen for recent corre-

spondence that helped to clarify the differences between the CSE and SCE approaches to the generalization

of cursed equilibrium for dynamic games. We thank participants of the Caltech Theory Seminar and Colin

Camerer for comments and also thank Matthew Rabin for earlier discussions on the subject during his visit

at Caltech as a Moore Distinguished Scholar.

Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, CA 91125

USA. mjfong@caltech.edu

Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, CA 91125

USA. plin@caltech.edu

Corresponding Author: Division of the Humanities and Social Sciences, California Institute of Technol-

ogy, Pasadena, California 91125 USA. trp@hss.caltech.edu. Fax: +16263958967 Phone: +16263954088

1 Introduction

Cursed equilibrium (CE) proposed by Eyster and Rabin (2005) is a leading behavioral equi-

librium concept that was developed to explain the “winner’s curse” and related anomalies

in applied game theory. The basic idea behind CE is that individuals do not fully take

account of the dependence of other players’ strategic actions on private information. Cursed

behavior of this sort has been detected in a variety of contexts. Capen et al. (1971) first

noted that in oil-lease auctions, “the winner tends to be the bidder who most overestimates

the reserves potential” (Capen et al. (1971), p. 641). Since then, this observation of overbid-

ding relative to the Bayesian equilibrium benchmark, which can result in large losses for the

winning bidder, has been widely documented in laboratory auction experiments (Bazerman

and Samuelson, 1983; Kagel and Levin, 1986; Kagel et al., 1989; Forsythe et al., 1989; Dyer

et al., 1989; Lind and Plott, 1991; Kagel and Levin, 2009; Ivanov et al., 2010; Camerer et al.,

2016). In addition, the neglect of the connection between the opponents’ actions and private

information is also found in non-auction environments, such as bilateral bargaining games

(Samuelson and Bazerman, 1985; Holt and Sherman, 1994; Carrillo and Palfrey, 2009, 2011),

zero-sum betting games with asymmetric information (Rogers et al., 2009; Søvik, 2009), and

voting and jury decisions (Guarnaschelli et al., 2000).

While CE provides a tractable alternative to Bayesian Nash equilibrium and can explain

some anomalous behavior in games with a winner’s-curse structure, a significant limitation is

that it is only developed as a strategic form concept for simultaneous-move Bayesian games.

Thus, when applying the standard CE to dynamic games, the CE analysis is carried out on

the strategic form representation of the game, implying that CE cannot distinguish behavior

across dynamic games that differ in their timing of moves but have the same strategic form.

That is, players are assumed to choose type-dependent contingent strategies simultaneously

and

not

update their beliefs as the history of play unfolds. A further limitation implied

by the strategic form approach is that CE and standard Bayesian Nash equilibrium make

identical predictions in games with a private-values information structure (Eyster and Rabin

(2005), Proposition 2). In this paper we extend the CE in a simple and natural way to

multi-stage games of incomplete information. We call the new equilibrium concept

Cursed

Sequential Equilibrium

(CSE).

In Section 2, we present the framework and our extension of cursed equilibrium to dy-

namic games. We consider the framework of multi-stage games with observed actions,

introduced by Fudenberg and Tirole (1991b), where players’ private information is repre-

sented by types, with the assumption that the set of available actions is independent of their

types at each public history. Our new solution concept is in the same spirit of the cursed

equilibrium—in our model, at each stage, players will (partially) neglect the dependence of

the other players’ behavioral strategies on their types, by placing some weight on the incor-

rect belief that all types adopt the average

behavioral

strategy. Specifically, at each public

history, this corresponds to the average distribution of actions

given the current belief about

others’ types at that stage

. Therefore, as players update their beliefs about others’ private

information via Bayes’ rule, but with incorrect beliefs about the other players’ behavioral

strategies, in later stages this can lead them to have

incorrect

beliefs about the other players’

average distribution of actions.

Following Eyster and Rabin (2005)’s notion of cursedness, we parameterize the model by

a single parameter

∈

1] which captures the degree of cursedness and define fully cursed

(

= 1) CSE analogously to fully cursed (

= 1) CE. Recall that in a fully cursed (

= 1) CE,

each type of each player chooses a best reply to expected (cursed) equilibrium distribution

of other players’ actions, averaged over the type-conditional strategies of the other players,

with this average distribution calculated using the prior belief on types. Loosely speaking,

a player best responds to the average CE strategy of the others. In a

-CE, players are

only partially cursed, in the sense that each player best responds to a

-weighted linear

combination of the

average

-CE strategy of the others and the

true

(type-dependent)

-CE

strategy of the others.

The extension of this definition to multi-stage games with observed actions is different

from

-CE in two essential ways: (1) the game is analyzed with behavioral strategies; and

(2) we impose sequential rationality and Bayesian updating. In a fully cursed (

= 1) CSE,

(1) implies at every stage

and each public history at

, each type of each player

chooses a

best reply to the expected (cursed) equilibrium distribution of other players’ stage-

actions,

averaged over the type-conditional stage-

behavioral strategies of other players, with this

average distribution calculated using

’s current belief about types at stage

. That is, player

best responds to the average stage-

CSE strategy of others. Moreover, (2) requires that

each player’s belief at each public history is derived by Bayes’ rule wherever possible, and

best replies are with respect to the continuation values computed by using the fully cursed

beliefs about the behavioral strategies of the other players in current and future stages.

-CSE, for

χ <

1, is then defined in analogously to

-CE, except for using a

-weighted

linear combination of the

average

-CSE

behavioral strategies

of others and the

true

(type-

dependent)

-CSE

behavioral strategies

of others. Thus, similar to the fully cursed CE, in

a fully cursed (

= 1) CSE, each player believes other players’ actions at each history are

independent

of their private information. On the other hand,

= 0 corresponds to the

standard sequential equilibrium where players have correct perceptions about other players’

behavioral strategies and are able to make correct Bayesian inferences.

After defining the equilibrium concept, in Section 3 we explore some general properties of

the model. We first prove the existence of a cursed sequential equilibrium in Proposition 1.

Intuitively speaking, CSE mirrors the standard sequential equilibrium. The only difference is

that players have incorrect beliefs about the other players’ behavioral strategies at each stage

since they fail to fully account for the correlation between others’ actions and types at every

history. We prove in Proposition 2 that the set of CSE is upper hemi-continuous with respect

. Consequently, every limit point of a sequence of

-CSE points as

converges to 0 is a

sequential equilibrium. This result bridges our behavioral solution concept with the standard

equilibrium theory. Finally, we also show in Proposition 4 that

-CSE is equivalent to

-CE

for one-stage games, demonstrating the connection between the two behavioral solutions.

In multi-stage games, cursed beliefs about behavioral strategies will distort the evolution

of a player’s beliefs about the other players’ types. As shown in Proposition 3, a direct

consequence of the distortion is that in

-CSE players tend to update their beliefs about

others’ types too passively. That is, there is some persistence in beliefs in the sense that

at each stage

, each

-cursed player’s belief about any type profile is at least

times the

belief about that type profile at stage

−

1. Among other things, this implies that if the

prior

belief about the types is full support and

χ >

0, the full support property will persist

at all histories, and players will (possibly incorrectly) believe every profile of others’ types is

possible at every history.

This dampened updating property plays an important role in our framework. Not only

does it contribute to the difference between CSE and the standard CE through the updating

process, but it also implies additional restrictions on off-path beliefs. The effect of dampened

updating is starkly illustrated in the pooling equilibria of signaling games where every type

of sender behaves the same everywhere. In this case, Proposition 5 shows if an assessment

associated with a pooling equilibrium is a

-CSE, then it also a

′

-CSE for all

′

≤

, but it

is not necessarily a pooling equilibrium for all

′

> χ

. This contrasts with one of the main

results about CE, that if a pooling equilibrium is a

-CE for some

, then it is a

′

-CE for

all

′

∈

1] (Eyster and Rabin (2005), Proposition 3).

This suggests that perhaps the dampened updating property is an equilibrium selection

device that eliminates some pooling equilibrium, but actually this is

not

a general property.

As we demonstrate later, the

-CE and

-CSE sets can be non-overlapping, which we il-

1

For the off-path histories, similar to the idea of Kreps and Wilson (1982), we impose the

-consistency

requirement (see Definition 2) so the assessment is approachable by a sequence of totally mixed behavioral

strategies. The only difference is that players’ beliefs are incorrectly updated by assuming others play the

-cursed behavioral strategies. Hence, in our approach if

= 0, a CSE is a sequential equilibrium.

lustrate with a variety of applications. The intuition is that in CSE, players generally do

not have correct beliefs about the opponents’ average behavioral strategies. The pooling

equilibrium is just a special case where players have correct beliefs.

In Section 4 we explore the implications of cursed sequential equilibrium with five ap-

plications in economics and political science. Section 4.1 analyzes the

-CSE of signaling

games. Besides studying the theoretical properties of pooling

-CSE, we also analyze two

simple signaling games that were studied in a laboratory experiment (Brandts and Holt,

1993). We show how varying the degree of cursedness can change the set of

-CSE in these

two signaling games in ways that are consistent with the reported experimental findings.

Next, we turn to the exploration of how sequentially cursed reasoning can influence strategic

communication. To this end, we analyze the

-CSE for a public goods game with communi-

cation (Palfrey and Rosenthal, 1991; Palfrey et al., 2017) in Section 4.2, finding that

-CSE

predicts there will be less effective communication when players are more cursed.

Next, in Section 4.3 we apply

-CSE to the centipede game studied experimentally by

McKelvey and Palfrey (1992) where one of the players believes the other player might be

an “altruistic” player who always passes. This is a simple reputation-building game, where

selfish types can gain by imitating altruistic types in early stages of the game. The public

goods application and the centipede game are both private-values environments, so these

two applications clearly demonstrate how CSE departs from CE and the Bayesian Nash

equilibrium, and shows the interplay between sequentially cursed reasoning and the learning

of types in private-value models.

In strategic voting applications, conditioning on “pivotality”—the event where your vote

determines the final outcome—plays a crucial role in understanding equilibrium voting be-

havior. To illustrate how cursedness distorts the pivotal reasoning, in Section 4.4 we study

the three-voter two-stage agenda voting game introduced by Ordeshook and Palfrey (1988).

Since this is a private value game, the predictions of the

-CE and the Bayesian Nash equilib-

rium coincide for all

. That is, cursed equilibrium predicts no matter how cursed the voters

are, they are able to correctly perform pivotal reasoning. On the contrary, our CSE predicts

that cursedness will make the voters less likely to vote strategically. This is consistent with

the empirical evidence about the prevalence of sincere voting over sequential agendas when

inexperienced voters have incomplete information about other voters’ preferences (Levine

and Plott, 1977; Plott and Levine, 1978; Eckel and Holt, 1989).

Finally, in Section 4.5 we study the relationship between cursedness and epistemic rea-

soning by considering the two-person dirty faces game previously studied by Weber (2001)

and Bayer and Chan (2007). In this game,

-CSE predicts cursed players are, to some extent,

playing a “coordination” game where they coordinate on a specific learning speed about their

face types. Therefore, from the perspective of CSE, the non-equilibrium behavior observed

in experiments can be interpreted as possibly due to a coordination failure resulting from

cognitive limitations.

The cursed sequential equilibrium extends the concept of cursed equilibrium from static

Bayesian games to multi-stage games with observed actions. This generalization preserves

the spirit of the original cursed equilibrium in a simple and tractable way, and provides

additional insights about the effect of cursedness in dynamic games. A contemporaneous

working paper by Cohen and Li (2023) is closely related to our paper. That paper adopts an

approach based on the coarsening of information sets to define sequential cursed equilibrium

(SCE) for extensive form games with perfect recall. The SCE model captures a different

kind of cursedness

that arises if a player neglects the dependence of other players’ unob-

served (i.e., either future or simultaneous) actions on the history of play in the game, which

is different from the dependence of other players’ actions on their type (as in CE and CSE).

In the terminology of Eyster and Rabin (2005) (p. 1665), the cursedness is with respect to

endogenous information, i.e., what players observe about the path of play. The idea is to

treat the unobserved actions of other players in response to different histories (endogenous

information) similarly to how cursed equilibrium treats players’ types. A two-parameter

model of partial cursedness is developed, and a series of examples demonstrate that for

plausible parameter values, the model is consistent with some experimental findings related

to the failure of subjects to fully take account of unobserved hypothetical events, whereas

behavior is “more rational” if subjects make decisions after directly observing such events.

At a more conceptual level, our paper is related to several other behavioral solution concepts

developed for dynamic games, such as agent quantal response equilibrium (AQRE) (McK-

elvey and Palfrey, 1998), dynamic cognitive hierarchy theory (DCH) (Lin and Palfrey, 2022;

Lin, 2022), and the analogy-based expectation equilibrium (ABEE) (Jehiel, 2005; Jehiel and

Koessler, 2008), all of which modify the requirements of sequential equilibrium in different

ways than cursed sequential equilibrium.

2 The Model

Since CSE is a solution concept for dynamic games of incomplete information, in this pa-

per we will focus on the framework of multistage games with observed actions (Fudenberg

and Tirole, 1991b). Section 2.1 defines the formal structure of multi-stage games with ob-

2

We illustrate some implications of these differences in the application to signaling games in Section 4.1.

For a more detailed discussion of the differences between CSE and SCE, see Fong et al. (2023)

served actions, followed by Section 2.2, where the

-cursed sequential equilibrium is formally

developed.

2.1 Multi-Stage Games with Observed Actions

Let

{

,...,n

}

be a finite set of players. Each player

∈

has a

type

drawn from

a finite set Θ

. Let

∈

≡ ×

be the type profile and

−

∈

−

≡ ×

be the

type profile without player

. All players share a common (full support) prior distribution

(

) : Θ

→

1). Therefore, for every player

, the belief of other players’ types conditional

on his own type is

(

−

) =

(

−

,θ

)

′

−

∈

−

(

′

−

,θ

)

At the beginning of the game, players observe their own types, but not the other players’

types. That is, each player’s type is his own private information.

The game is played in stages

= 1

,...,T

. In each stage, players simultaneously choose

actions, which will be revealed at the end of the stage. The feasible set of actions can vary

with histories, so games with alternating moves are also included. Let

−

be the set of

all possible histories at stage

, where

{

∅

}

and

is the set of terminal histories.

Let

∪

be the set of all possible histories of the game, and

H\H

be the set of

non-terminal histories.

For every player

, the available information at stage

is in Θ

×H

−

. Therefore, player

’s

information sets can be specified as

∈Q

{

(

h,θ

) :

∈H\H

,θ

∈

}

. That is, a type

player

’s information set at the public history

can be defined as

−

∈

−

(

,θ

−

With a slight abuse of notation, it will be denoted as (

). For the sake of simplicity, we

assume that, at each history, the feasible set of actions for every player is independent of

their type and use

(

−

) to denote the feasible set of actions for player

at history

−

Let

∈H\H

(

) denote player

’s feasible actions in all histories of the game and

×···×

. In addition, we assume

is finite for all

∈

and

(

)

|≥

1 for all

∈

and any

∈H\H

A behavioral strategy for player

is a function

→

∆(

) satisfying

(

−

,θ

)

∈

∆(

(

−

)). Furthermore, we use

(

−

,θ

) to denote the probability player

chooses

∈

(

−

). We use

= (

,...,a

)

∈ ×

(

−

)

≡

(

−

) to denote the action

profile at stage

and

−

to denote the action profile at stage

without player

. If

the action profile realized at stage

, then

= (

−

). Finally, each player

has a payoff

function

→

and we let

= (

,...,u

) be the profile of payoff functions. A

multi-stage game with observed actions, Γ, is defined by the tuple Γ =

⟨

T,A,N,

⟩

2.2 Cursed Sequential Equilibrium

In a multi-stage game with observed actions, a solution is defined by an “assessment,” which

consists of a (behavioral) strategy profile

, and a belief system

. Since action profiles will

be revealed to all players at the end of each stage, the belief system specifies, for each player,

a conditional distribution over the set of type profiles conditional on each history. Consider

an assessment (

μ,σ

). Following the spirit of the cursed equilibrium, for player

at stage

we define the

average behavioral strategy profile of the other players

as:

−

(

−

,θ

) =

−

∈

−

(

−

,θ

)

−

(

−

,θ

−

)

for any

∈

and

−

∈H

−

In CSE, players have incorrect perceptions about other players’ behavioral strategies.

Instead of thinking they are using

−

, a

-cursed

type

player

would believe the other

players are using a

-weighted average of the average behavioral strategy and the true

behavioral strategy:

−

(

−

,θ

−

,θ

) =

−

(

−

,θ

) + (1

−

)

−

(

−

,θ

−

)

The beliefs of player

about

−

are updated in the

-CSE via Bayes’ rule, whenever

possible, assuming other players are using the

-cursed behavioral strategy rather than the

true behavioral strategy. We call this updating rule the

-cursed Bayes’ rule

. Specifically, an

assessment satisfies the

-cursed Bayes’ rule if the belief system is derived from the Bayes’

rule while perceiving others are using

−

rather than

−

Definition 1.

(

μ,σ

)

satisfies

-cursed Bayes’ rule if the following rule is applied to update

the posterior beliefs whenever

′

−

∈

−

(

′

−

,θ

)

−

(

−

,θ

′

−

,θ

)

(

−

,θ

) =

(

−

,θ

)

−

(

−

,θ

−

,θ

)

′

−

∈

−

(

′

−

,θ

)

−

(

−

,θ

′

−

,θ

)

Let Σ

be the set of totally mixed behavioral strategy profiles, and let Ψ

be the set of

assessments (

μ,σ

) such that

∈

and

is derived from

using

-cursed Bayes’ rule.

3

We assume throughout the paper that all players are equally cursed, so there is no

subscript on

. The

framework is easily extended to allow for heterogeneous degrees of cursedness.

4

= 0, players have correct beliefs about the other players’ behavioral strategies at every stage.

5

In the following, we will use

(

) to denote the belief system derived under

-cursed Bayes’ Rule. Also,

note that both

−

and

are induced by

; that is,

−

(

) =

−

[

](

) and

(

) =

[

](

). For the ease

of exposition, we drop [

] when it does not cause confusion.

Lemma 1 below shows that another interpretation of the

-cursed Bayes’ rule is that players

have correct perceptions about

−

but are unable to make perfect Bayesian inference when

updating beliefs. From this perspective, player

’s cursed belief is simply a linear combination

of player

’s cursed belief at the beginning of that stage (with

weight) and the Bayesian

posterior belief (with 1

−

weight). Because

is totally mixed, there are no off-path histories.

Lemma 1.

For any

(

μ,σ

)

∈

= (

−

)

∈H\H

and

∈

(

−

,θ

) =

χμ

(

−

,θ

) + (1

−

)

(

−

,θ

)

−

(

−

,θ

−

)

′

−

(

′

−

,θ

)

−

(

−

,θ

′

−

)

Proof.

See Appendix A.

This is analogous to Lemma 1 of Eyster and Rabin (2005). Another insight provided

by Lemma 1 is that even if player types are independently drawn, i.e.,

(

) = Π

(

players’ cursed beliefs about other players’ types are generally

not

independent across players.

That is, in general,

(

−

,θ

)

= Π

(

,θ

)

The belief system will preserve the

independence only when the players are either fully rational (

= 0) or fully cursed (

= 1).

Finally, we place a consistency restriction, analogous to consistent assessments in sequen-

tial equilibrium, on how

-cursed beliefs are updated off the equilibrium path, i.e., when

′

−

∈

−

(

′

−

,θ

)

−

(

−

,θ

′

−

,θ

) = 0

An assessment satisfies

-consistency if it is in the closure of Ψ

Definition 2.

(

μ,σ

)

satisfies

-consistency if there is a sequence of assessments

{

(

,σ

)

}⊆

such that

lim

→∞

(

,σ

) = (

μ,σ

)

For any

∈

1],

, and

∈

Θ, let

(

,θ,σ

−

,σ

) be player

’s perceived

conditional realization probability of terminal history

∈H

at history

∈H\H

if the

type profile is

and player

uses the behavioral strategy

whereas perceives other players’

using the cursed behavioral strategy

−

. At every non-terminal history

, a

-cursed player

-CSE will use

-cursed Bayes’ rule (Definition 1) to derive the posterior belief about the

other players’ types. Accordingly, a type

player

’s conditional expected payoff at history

is given by:

(

,θ

) =

−

∈

−

∈H

(

−

,θ

)

(

,θ,σ

−

,σ

)

(

,θ

−

)

Definition 3.

An assessment

(

∗

,σ

∗

)

is a

-cursed sequential equilibrium if it satisfies

consistency and

∗

(

,θ

)

maximizes

(

∗

,θ

)

for all

∈H\H

3 General Properties of

-CSE

In this section, we characterize some general theoretical properties of

-CSE. The first result

is the existence of the

-CSE. The definition of

-CSE mirrors the definition of the sequential

equilibrium by Kreps and Wilson (1982)—the only difference is that players in

-CSE update

their beliefs by

-cursed Bayes’ rule and best respond to

-cursed (behavioral) strategies.

Therefore, one can prove the existence of

-CSE in a similar way as in the standard argument

of the existence of sequential equilibrium.

Proposition 1.

For any

∈

and any finite multi-stage game with observed actions,

there is at least one

-CSE.

Proof.

We briefly sketch the proof here, and the details can be found in Appendix A.

Fix any

∈

1]. For any

∈

and any information set

= (

−

,θ

), player

has

to choose every action

∈

(

−

) with probability at least

. Since there are no off-path

histories, the belief system is uniquely pinned down by

-cursed Bayes’ rule and a

-CSE

exists in this

-constrained game. We denote this

-CSE as (

,σ

). By compactness, there

is a converging sub-sequence of assessments such that (

,σ

)

→

(

∗

,σ

∗

) as

→

0, which is

-CSE, as desired.

Let Φ(

) be the correspondence that maps

∈

1] to the set of

-CSE. Proposition 1

guarantees Φ(

) is non-empty for any

∈

1]. Because

-cursed Bayes’ rule changes con-

tinuously in

, we can further prove in Proposition 2 that Φ(

) is an upper hemi-continuous

correspondence.

Proposition 2.

Φ(

)

is upper hemi-continuous with respect to

Proof.

The proof follows a standard argument. See Appendix A for details.

As shown in Corollary 1, a direct consequence of upper hemi-continuity is that every

limit point of a sequence of

-CSE when

→

0 is a sequential equilibrium. This result

bridges our behavioral equilibrium concept with standard equilibrium theory.

Corollary 1.

Every limit point of a sequence of

-CSE with

converging to 0 is a sequential

equilibrium.

Proof.

By Proposition 2, we know Φ(

) is upper hemi-continuous at 0. Consider of a se-

quence of

-CSE. As

→

0, the limit point remains a CSE, which is a sequential equilibrium

= 0. This completes the proof.

Finally, by a similar argument to Kreps and Wilson (1982), for any

∈

1],

-CSE is

also upper hemi-continuous with respect to payoffs. In other words, our

-CSE preserves

the continuity property of sequential equilibrium.

The next result is the characterization of a necessary condition for

-CSE. As seen

from Lemma 1, players update their beliefs more passively in

-CSE than in the stan-

dard equilibrium—they put

-weight on their beliefs formed in previous stage. To formalize

this, we define the

-dampened updating property

in Definition 4. An assessment satisfies

this property if at

any

non-terminal history, the belief puts at least

weight on the belief

in previous stage—both on and off the equilibrium path. In Proposition 3, we show that

-consistency implies the

-dampened updating property.

Definition 4.

An assessment

(

μ,σ

)

satisfies the

-dampened updating property if for any

∈

and

= (

−

)

∈H\H

(

−

,θ

)

≥

χμ

(

−

,θ

)

Proposition 3.

-consistency implies

-dampened updating for any

∈

Proof.

See Appendix A.

It follows that if assessment (

μ,σ

) satisfies the

-dampened updating property, then for

any player

, any history

and any type profile

, player

’s belief about

−

is bounded by

χμ

(

−

,θ

)

≤

(

−

,θ

)

≤

−

′

−

(

′

−

,θ

)

One can see from this condition that when

increases, the feasible range of

(

−

,θ

)

shrinks, and the restriction on the belief system becomes more stringent. Moreover, if the

history

is an off-path history of (

μ,σ

), then this condition characterizes the feasible set of

off-path beliefs, which shrinks as

increases.

An important implication of this observation is that Φ(

) is not lower hemi-continuous

with respect to

. The intuition is that for some

-CSE that contains off-path histories, the

off-path beliefs to support the equilibrium might not be

-consistent for sufficiently large

In this case, the

-CSE is not attainable by a sequence of

-CSE where

converges to

from above, causing the lack of lower hemi-continuity.

Lastly, another implication of

-dampened updating property is that for each player

history

and type profile

, the belief

(

−

,θ

) has a lower bound that is

independent

of the strategy profile. The lower bound is characterized in Corollary 2. This result implies

that when

χ >

(

−

)

0 implies

(

−

,θ

)

0 for all

, so that if prior beliefs are

bounded away from zero, beliefs are always bounded away from 0 as well. In other words,

when

χ >

0, because of the

-dampened updating, beliefs will always have full support even

if at off-path histories.

Corollary 2.

For any

-consistent assessment

(

μ,σ

)

∈

and

∈H\H

(

−

,θ

)

≥

(

−

)

Proof.

See Appendix A.

If the game has only one stage, then the dampened updating property has no effect, in

which case

-CSE and

-CE are equivalent solution concepts. This is formally stated and

proved in Proposition 4.

Proposition 4.

For any one-stage game and for any

-CSE and

-CE are equivalent.

Proof.

For any one-stage game, the only public history is the initial history

∅

. Thus, in any

-CSE, for each player

∈

and type profile

∈

Θ, player

’s belief about other players’

types at this history is

(

−

∅

,θ

) =

(

−

)

Since the game has only one stage, the outcome is simply

= (

,...,a

), the action profile

at stage 1. Moreover, given any behavioral strategy profile

, player

believes

will be the

outcome with probability

(

∅

,θ

)

−

(

−

∅

,θ

) + (1

−

)

−

(

−

∅

,θ

−

)

Therefore, if

is the behavioral strategy profile of a

-CSE in an one-stage game, then for

6

An example is provided in Section 4.1 (see Footnote 7).

each player

, type

∈

and each

∈

(

∅

) such that

(

∅

,θ

)

∈

argmax

′

∈

(

∅

)

−

∈

−

(

−

)







−

∈

−

(

∅

)

−

(

−

∅

,θ

) + (1

−

)

−

(

−

∅

,θ

−

)







(

′

−

,θ

−

)

which coincides with the maximization problem of

-CE. This completes the proof.

From the proof of Proposition 4, one can see that in one-stage games players have

correct

perceptions about the average strategy of others. Therefore, the maximization problem

-CSE coincides with the problem of

-CE. For general multi-stage games, because of

the

-dampened updating property, players will update beliefs incorrectly and thus their

perceptions about other players’ future moves can also be distorted.

4 Applications

In this section, we will explore

-CSE in five applications of multi-stage games with observed

actions, in order to illustrate the range of effects it can have and to show how it is different

from the

-CE and sequential equilibrium.

Our first application is the sender-receiver signaling game, which is practically the sim-

plest possible multi-stage game. From our analysis, we will see both the theoretical and

empirical implications of our

-CSE.

4.1 Pooling Equilibria in Signaling Games

We first make a general observation about pooling equilibria in multi-stage games. Player

follows a

pooling strategy

if for every non-terminal history,

, all types of player

take

the same action

∈

(

). Conceptually, since every type of player

takes the same

action, players other than

cannot make any inference about

’s type from

’s actions. A

pooling

-CSE

is a

-CSE where every player follows a pooling strategy. Hence, every player

has correct beliefs about any other player’s future move because every type of every player

chooses the same action.

Since in any pooling

-CSE, players can correctly anticipate other players’ future moves

no matter how cursed they are, one may naturally conjecture that a pooling

-CSE is also a

′

-CSE for any

′

∈

1]. As shown by Eyster and Rabin (2005), this is true for one-stage

Bayesian games: if a pooling strategy profile is a

-cursed equilibrium, then it is also a

′

-cursed equilibrium for any

′

∈

1]. Surprisingly, this result does not extend to multi-

stage games. Proposition 5 shows if a pooling behavioral strategy profile is a

-CSE, then it

remains a

′

-CSE only for

′

≤

, which is a weaker result than Eyster and Rabin (2005).

This result is driven by the

-dampened updating property which restricts the set of

off-path beliefs. As discussed above, when

gets larger, the set of feasible off-path beliefs

shrinks, eliminating some pooling

-CSE.

Proposition 5.

A pooling

-CSE is a

′

-CSE for

′

≤

Proof.

See Appendix B.

The proof strategy is similar to the one in Eyster and Rabin (2005) Proposition 3. Given

-CSE behavioral strategy profile, we can separate the histories into on-path and off-path

histories. For on-path histories in a pooling equilibrium, since all types of players make the

same decisions, players cannot make any inference about other players’ types. Therefore,

for on-path histories, their beliefs are the prior beliefs, which are independent of

. On the

other hand, for off-path histories, as shown in Proposition 3, a necessary condition for

-CSE

is that the belief system has to satisfy the

-dampened updating property. When

gets

larger, this requirement becomes more stringent, and hence some pooling

-CSE may break

down.

Example 1 is a signaling game where the sender has only two types and two messages, and

the receiver has only two actions. This example demonstrates the implication of Proposition

5 and shows the lack of lower hemi-continuity; i.e., it is possible for a pooling behavioral

strategy profile to be a

-CSE, but not a

′

-CSE for

′

> χ

. We will also use this example to

illustrate how the notion of cursedness in sequential cursed equilibrium proposed by Cohen

and Li (2023) departs from our CSE.

Example 1.

The sender has two possible types drawn from the set Θ =

{

,θ

}

with

Pr(

) = 1

4. The receiver does not have any private information. After the sender’s type

is drawn, the sender observes his type and decides to send a message

∈ {

A,B

}

, or any

mixture between the two. After that, the receiver decides between action

∈{

L,R

}

or any

mixture between the two, and the game ends. The game tree is illustrated in Figure 1.

If we solve for the

-CE of the game (or the sequential equilibria), we find that there

are two pooling equilibria for every value of

. In the first pooling

-CE, both sender types

choose

; the receiver chooses

in response to

and

at the off-path history

. In

the second pooling

-CE, both sender types pool at

and the receiver chooses

at both

−

1

[

1

4

]

−

2

[

3

4

]

Nature

Figure 1: Game Tree for Example 1

histories. By Proposition 3 of Eyster and Rabin (2005), these two equilibria are in fact

pooling

-CE for all

∈

1]. The intuition is that in a pooling

-CE, players are not

able to make any inference about other players’ types from their actions because the average

normal form strategy is the same as the type-conditional normal form strategy. Therefore,

their beliefs are independent of

, and hence a pooling

-CE will still be an equilibrium for

any

∈

1].

However, as summarized in Claim 1 below, the

-CSE imposes

stronger

restrictions than

-CE in this example, in the sense that when

is sufficiently large, the second pooling equi-

librium cannot be supported as a

-CSE. The key reason is that when the game is analyzed

in its normal form, the

-dampened updating property shown in Proposition 3 does not have

any bite, allowing both pooling equilibria to be supported as a

-CE for any value of

. Yet,

in the

-CSE analysis, the additional restriction of

-dampened updating property eliminates

some extreme off-path beliefs, and hence, eliminates the second pooling

-CSE equilibrium

for sufficiently large

. For simplicity, we use a four-tuple [(

(

)

(

)); (

(

)

(

))] to

denote a behavioral strategy profile.

Claim 1.

In this example, there are two pure pooling

-CSE, which are:

[(

A,A

); (

L,R

)]

is a pooling

-CSE for any

∈

[(

B,B

); (

R,R

)]

with

(

)

∈

−

is a pooling

-CSE if and only if

≤

Proof.

See Appendix B.

From previous discussion, we know in general, the sets of

-CSE and

-CE are non-

overlapping because of the nature of sequential distortion of beliefs in

-CSE. Yet, a pooling

-CSE is an exception. In a pooling

-CSE, players can correctly anticipate others’ future

moves, so a pooling

-CSE will mechanically be a pooling

-CE. In cases such as this, we

can find that

-CSE is a

refinement

-CE.

Remark.

This game is useful for illustrating some of the differences between the notions

of “cursedness” in

-CSE and the sequential cursed equilibrium ((

,ψ

)-SCE) proposed

by Cohen and Li (2023). The first distinction is that the

and

parameters capture

substantively different sources of distortion in a player’s beliefs about the other players’

strategies. In

-CSE, the degree of cursedness,

, captures how much a player neglects the

dependence of the other players’ behavioral strategies on those players’ (exogenous)

private

information

, i.e, types, drawn by nature, and as a result, mistakenly treats different types

as behaving the same with probability

. In contrast, in (

,ψ

)-SCE, the cursedness

parameter,

, captures how much a player neglects the dependence of the other players’

strategies on future moves of the others, or current moves that are unobserved because of

simultaneous play. Thus, it is a neglect related to endogenous information. If player

observes

a previous move by some other player

, then player

correctly accounts for the

dependence of player

’s chosen action on player

’s private type, as would be the case in

-CSE only at the boundary where

= 0.

In the context of pooling equilibria in sender-receiver signaling games, if

= 1, then

in SCE the sender believes the receiver will respond the same way both on and off the

equilibrium path. This distorts how the sender perceives the receiver’s future action in

response to an off-equilibrium path message. In

-CSE, cursedness does not hinder the

sender from correctly perceiving the receiver’s strategy since the receiver only has one type.

Take the strategy profile [(

A,A

); (

L,R

)] for example, which is a pooling

-CSE equilibrium

for all

∈

1]. However, with (

,ψ

)-SCE, a sender misperceives that the receiver, upon

receiving the off-path message

, will, with probability

, take the same action (

) as

when receiving the on-path message

. If

is sufficiently high, the sender will deviate to

send

, which implies that [(

A,A

); (

L,R

)] cannot be supported as an equilibrium when

is sufficiently large (

3). The distortion induced by

also creates an additional SCE

is sufficiently large: [(

B,B

); (

L,R

)]. To see this, if

= 1, then a sender incorrectly

believes that the receiver will continue to choose

if the sender deviates to

, rather than

switching to

, and hence

is optimal for both sender types. However, [(

B,B

); (

L,R

)] is

not

-CSE equilibrium for any

∈

1], or a

-CE in the sense of Eyster and Rabin

7

Note that the

-CSE correspondence Φ(

) is not lower hemi-continuous with respect to

. To see

this, we consider a sequence of

{

}

where

8

9

1

9

for

≥

1. From the analysis of Claim 1, we

know [(

B,B

); (

R,R

)]

̸∈

Φ(

) for any

≥

1. However, in the limit where

→

9, [(

B,B

); (

R,R

)] with

2

(

1

) = 1

3 is indeed a CSE. That is, [(

B,B

); (

R,R

)] is not approachable by this sequence of

-CSE.

(2005), or a sequential equilibrium.

In the two possible pooling equilibria analyzed in the last paragraph, the second SCE

parameter,

, does not have any effect, but the role of

can be illustrated in the context of

the [(

B,B

); (

R,R

)] sequential equilibrium. This second SCE parameter,

, is introduced to

accommodate a player’s possible failure to fully account for the informational content from

observed

events. The larger (1

−

) is, the greater extent a player neglects the informational

content of observed actions. Although the parameter

has a similar flavor to 1

−

CSE, it is different in a number of ways. In particular this parameter only has an effect

via its interaction with

and thus does not independently arise. In the two parameter

model, the overall degree of cursedness is captured by the product,

−

), and thus

any cursedness effect of

is shut down when

= 0. For instance, under our

-CSE, the

strategy profile [(

B,B

); (

R,R

)] can only be supported as an equilibrium when

is sufficiently

small. However, [(

B,B

); (

R,R

)] can be supported as a (

,ψ

)-SCE even when (1

−

) = 1

as long as

is sufficiently small. In fact, when

= 0, a (

,ψ

)-SCE is equivalent to

sequential equilibrium regardless of the value of

100

1

[

1

2

]

2

[

1

2

]

BH 3

BH 4

Nature

Figure 2: Game Tree for

BH 3

and

BH 4

in Brandts and Holt (1993)

Example 2.

Here we analyze two signaling games that were studied experimentally by

Brandts and Holt (1993) (

BH 3

and

BH 4

) and show that

-CSE can help explain some

of their findings. In both Game

BH 3

and Game

BH 4

, the sender has two possible types

{

,θ

}

which are equally likely. There are two messages

∈{

I,S

}

available to the sender.

After seeing the message, the receiver chooses an action from

∈{

C,D,E

}

. The game tree

and payoffs for both games are summarized in Figure 2.

8

stands for “

ntuitive” and

stands for “

equential but not intuitive”, corresponding to the two pooling

sequential equilibria of the two games.

In both games, there are two pooling sequential equilibria. In the first equilibrium, both

sender types send message

, and the receiver will choose

in response to

and choose

in response to

. In the second equilibrium, both sender types send message

, and the

receiver will choose

in response to

while choose

in response to

. Both are sequential

equilibria, in both games, but only the first equilibrium where the sender sends

satisfies

the intuitive criterion proposed by Cho and Kreps (1987).

Since the equilibrium structure is similar in both games, the sequential equilibrium and

the intuitive criterion predict the behavior should be the same in both games. However, this

prediction is strikingly rejected by the data. Brandts and Holt (1993) report that in the

later rounds of the experiment, almost all type

senders send

in Game

BH 3

(97 %), and

yet all type

senders send

in Game

BH 4

(100%). In contrast, type

senders behave

similarly in both games—46

2% and 44

1% of type

senders send

in Games

BH 3

and

BH 4

, respectively. Qualitatively speaking, the empirical pattern reported by Brandts and

Holt (1993) is that

sender type

is more likely to send

in Game BH 3 than Game BH 4

while sender type

’s behavior is insensitive to the change of games.

To explain this finding, Brandts and Holt (1993) propose a descriptive story based on

naive receivers

. A naive receiver will think both sender types are equally likely, regardless of

which message is observed. This naive reasoning will lead the receiver to choose

in both

games. Given this naive response, a type

sender has an incentive to send

in Game

and choose

in Game

BH 4

. (Brandts and Holt (1993), p. 284 – 285)

In fact, their story of naive reasoning echoes the logic of

-CSE. When the receiver is

fully cursed (or naive), he will ignore the correlation between the sender’s action and type,

causing him to not update the belief about the sender’s type. Proposition 6 characterizes

the set of

-CSE of both games. Following the notation in Example 1, we use a four-tuple

[(

(

)

(

)); (

(

)

(

))] to denote a behavioral strategy profile.

Proposition 6.

The set of

-CSE of Game BH 3 and BH 4 are characterized as below.



In Game BH 3, there are three pure

-CSE:

[(

I,I

); (

C,D

)]

is a pooling

-CSE if and only if

≤

[(

S,S

); (

D,C

)]

is a pooling

-CSE if and only if

≤

[(

I,S

); (

C,C

)]

is a separating

-CSE if and only if

≥



In Game BH 4, there are three pure

-CSE:

[(

I,I

); (

C,D

)]

is a pooling

-CSE if and only if

≤

[(

S,S

); (

D,C

)]

is a pooling

-CSE if and only if

≤

[(

S,S

); (

C,C

)]

is a pooling

-CSE for any

∈

Proof.

See Appendix B.

As noted earlier for Example 1, by Proposition 3 of Eyster and Rabin (2005), pooling

equilibria (1) and (2) in games BH 3 and BH 4 survive as

-CE for

all

∈

1]. Hence,

Proposition 6 implies that

-CSE refines the

-CE pooling equilibria for larger values of

Moreover,

-CSE actually eliminates

all

pooling equilibria in BH 3 if

χ >

3. Proposition

6 also suggests that for any

∈

1], sender type

will behave similarly in both games,

which is qualitatively consistent with the empirical pattern. In addition,

-CSE predicts that

a highly cursed (

χ >

3) type

sender will send different messages in different games—

highly cursed type

senders will send

and

in Games

BH 3

and

BH 4

, respectively. This

is consistent with the empirical data.

4.2 A Public Goods Game with Communication

Our second application is a threshold public goods game with private information and pre-

play communication, variations of which have been studied in laboratory experiments (Pal-

frey and Rosenthal, 1991; Palfrey et al., 2017). Here we consider the “unanimity” case where

there are

players and the threshold is also

Each player

has a private cost parameter

, which is independently drawn from a

uniform distribution on [0

] where

K >

1. After each player’s

is drawn, each player ob-

serves their own cost, but not the others’ costs. Therefore,

is player

’s private information

and corresponds to

in the general formulation.

The game consists of two stages. After

the profile of cost parameters is drawn, the game will proceed to stage 1 where each player

simultaneously broadcasts a public message

∈ {

}

without any cost or commitment.

After all players observe the message profile from this first stage, the game proceeds to stage

2 which is a unanimity threshold public goods game. Player

has to pay the cost

if he

contributes, but the public good will be provided only if all players contribute. The public

good is worth a unit of payoff for every player. Thus, if the public good is provided, each

player’s payoff will be 1

−

If there is no communication stage, the unique Bayesian Nash equilibrium is that no

player contributes, which is also the unique

-CE for any

∈

1]. In contrast, with the

communication stage, there exists an efficient sequential equilibrium where each player

9

This application has a continuum of types. The framework of analysis developed for finite types is

applied in the obvious way.

sends

= 1 if and only if

≤

1 and contributes if and only if all players send 1 in the

first stage.

Since this is a private value game, the standard cursed equilibrium has no bite,

and this efficient sequential equilibrium is also a

-CE for all values of

, by Proposition 2

of Eyster and Rabin (2005). In the following, we demonstrate that the prediction of

-CSE

is different from CE (and sequential equilibrium).

To analyze the

-CSE, consider a collection of “cutoff” costs,

{

,...,C

}

. In

the communication stage, each player communicates the message

= 1 if and only if

≤

. In the second stage, if there are exactly 0

≤

players sending

= 1 in the

first stage, then such a player would contribute in the second stage if and only if

≤

-CSE is a collection of these cost cutoffs such that the associated strategies are a

-CSE

for the public goods game with communication. The most efficient sequential equilibrium

identified above for

= 0 corresponds to cutoffs with

···

−

= 0 and

= 1.

There are in fact multiple equilibria in this game with communication. In order to

demonstrate how the cursed belief can distort players’ behavior, here we will focus on the

-CSE that is similar to the most efficient sequential equilibrium identified above, where

···

−

= 0 and

. The resulting

-CSE is given in Proposition 7.

Proposition 7.

In the public goods game with communication, there is a

-CSE where

···

−

= 0

, and

2. there is a unique

∗

(

N,K,χ

)

≤

s.t.

∗

(

N,K,χ

)

that solves:

∗

(

N,K,χ

)

−

∗

(

N,K,χ

)

−

= 1

−

χ.

Proof.

See Appendix B.

To provide some intuition, we sketch the proof by analyzing the two-person game, where

the

-CSE is characterized by four cutoffs

{

}

, with

= 0 and

. If players use the strategy that they would send message 1 if and only if the cost is

less than

, then by Lemma 1, at the history where both players send 1, player

’s cursed

10

One can think of the first stage as a poll, where players are asked the following question: “Are you

willing to contribute if everyone else says they are willing to contribute?”. The message

= 1 corresponds

to a “yes” answer and the message

= 0 corresponds to a “no” answer.