Cursed equilibrium explained

Cursed equilibrium
Supersetof:	Bayesian Nash equilibrium
Discoverer:	Erik Eyster, Matthew Rabin

In game theory, a cursed equilibrium is a solution concept for static games of incomplete information. It is a generalization of the usual Bayesian Nash equilibrium, allowing for players to underestimate the connection between other players' equilibrium actions and their types – that is, the behavioral bias of neglecting the link between what others know and what others do. Intuitively, in a cursed equilibrium players "average away" the information regarding other players' types' mixed strategies.

The solution concept was first introduced by Erik Eyster and Matthew Rabin in 2005,^[1] and has since become a canonical behavioral solution concept for Bayesian games in behavioral economics.^[2]

Preliminaries

Bayesian games

Let

I be a finite set of players and for each i \in I , define A _{i} their finite set of possible actions and T _{i} as their finite set of possible types; the sets A = \prod _{i} A _{i} and T = \prod _{i} T _{i} are the sets of joint action and type profiles, respectively. Each player has a utility function u _{i} : A x T → R , and types are distributed according to a joint probability distribution p \in \Delta T . A finite Bayesian game consists of the data G = ((A _{i,} T _{i,} u _{i)} _{i} , p) .

Bayesian Nash equilibrium

For each player

i \in I , a mixed strategy \sigma _{i} : T _{i} → \Delta A _{i} specifies the probability \sigma _{i} ( a _{i} | t _{i)} of player i playing action a _{i} \in A _{i} when their type is t _{i} \in T _{i} .

For notational convenience, we also define the projections

A _{-i} = \prod _{j} A _{j} and T _{-i} = \prod _{j} T _{j} , and let \sigma _{-i} : T _{-i} → \prod _{j} \Delta A _{j} be the joint mixed strategy of players j ≠ i , where \sigma _{-i} (a _{-i} | t _{-i} ) gives the probability that players j ≠ i play action profile a _{-i} when they are of type t _{-i} .

Definition: a Bayesian Nash equilibrium (BNE) for a finite Bayesian game

G = ((A _{i,} T _{i,} u _{i)} _{i} , p) consists of a strategy profile \sigma = (\sigma _{i)} _{i} such that, for every i \in I , every t _{i} \in T _{i} , and every action a ^{*} _{i} played with positive probability \sigma _{i} ( a ^{*} _{i} | t _{i)} > 0 , we havea ^{*} _{i} \in \underset{a _{i} \in A _{i}\operatorname{argmax}} \sum _{t _{-i} \in T _{-i}} p _{i(t} _{-i} | t _{i)} \sum _{a _{-i} \in A _{-i}} \sigma _{-i} (a _{-i} | t _{-i} ) u _{i} (a _{i,} a _{-i} , t _{i,} t _{-i} )

where

p _{i(t} _{-i} | t _{i)} = (p(t _{i,} t _{-i} ))/(\sum _{t _{-i} \in T _{-i}} p(t _{i} | t _{-i} ) p(t _{-i} )) is player i 's beliefs about other players types t _{-i} given his own type t _{i} .

Definition

Average strategies

First, we define the "average strategy of other players", averaged over their types. Formally, for each

i \in I and each t _{i} \in T _{i} , we define \overline{\sigma} _{-i} : T _{i} → \prod _{j} \Delta A _{j} by putting\overline{\sigma} _{-i} (a _{-i} | t _{i)=} \sum _{t _{-i} \in T _{i}} p _{i(t} _{-i} | t _{i)} \sigma _{-i} (a _{-i} | t _{-i} )

Notice that

\overline{\sigma} _{-i} (a _{-i} | t _{i)} does not depend on t _{-i} . It gives the probability, viewed from the perspective of player i when he is of type t _{i} , that the other players will play action profile a _{-i} when they follow the mixed strategy \sigma _{-i} . More specifically, the information contained in \overline{\sigma} _{-i} does not allow player i to assess the direct relation between a _{-i} and t _{-i} given by \sigma _{-i} (a _{-i} | t _{-i} ) .

Cursed equilibrium

Given a degree of mispercetion

\chi \in [ 0 , 1 ] , we define a\chi -cursed equilibrium for a finite Bayesian game G = ((A _{i,} T _{i,} u _{i)} _{i} , p) as a strategy profile \sigma = (\sigma _{i)} _{i} such that, for every i \in I , every t _{i} \in T _{i} , we havea ^{*} _{i} \in \underset{a _{i} \in A _{i}\operatorname{argmax}} \sum _{t _{-i} \in T _{-i}} p _{i(t} _{-i} | t _{i)} \sum _{a _{-i} \in A _{-i}} \left[\chi \overline{\sigma} _{-i} (a _{-i} | t _{i)} + ( 1- \chi)\sigma _{-i} (a _{-i} | t _{-i} ) \right] u _{i} (a _{i,} a _{-i} , t _{i,} t _{-i} )

for every action

a ^{*} _{i} played with positive probability \sigma _{i} ( a ^{*} _{i} | t _{i)} > 0 .

For

\chi = 0 , we have the usual BNE. For \chi = 1 , the equilibrium is referred to as a fully cursed equilibrium, and the players in it as fully cursed.

Applications

Trade with asymmetric information

In bilateral trade with two-sided asymmetric information, there are some scenarios where the BNE solution implies that no trade occurs, while there exist

\chi -cursed equilibria where both parties choose to trade.

Ambiguous political campaigns and cursed voters

In an election model where candidates are policy-motivated, candidates who do not reveal their policy preferences would not be elected if voters are completely rational. In a BNE, voters would correctly infer that if a candidate is ambiguous about their policy position, then it's because such a position is unpopular. Therefore, unless a candidate has very extreme – unpopular – positions, they would announce their policy preferences.

If voters are cursed, however, they underestimate the connection between the non-announcement of policy position and the unpopularity of the policy. This leads to both moderate and extreme candidates concealing their policy preferences.^[3]

Notes and References

1. Eyster . Erik . Rabin . Matthew . Matthew Rabin . Cursed Equilibrium . Econometrica . 2005 . 73 . 5 . 1623–1672. 10.1111/j.1468-0262.2005.00631.x.
2. Cohen . Shani. Li. Shengwu . Li Shengwu . 2022. Sequential Cursed Equilibrium. 2212.06025. econ.TH.
3. Szembrot . Nichole . Are voters cursed when politicians conceal policy preferences? . Public Chcoice . 2017 . 173 . 25–41 . 10.1007/s11127-017-0461-9.