Cursed equilibrium

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.

Cursed equilibrium
Solution concept in game theory
Relationship
Superset of	Bayesian Nash equilibrium
Significance
Proposed by	Erik Eyster, Matthew Rabin

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

Bayesian Nash equilibrium

For each player ${\displaystyle i\in I}$, a mixed strategy ${\displaystyle \sigma _{i}:T_{i}\rightarrow \Delta A_{i}}$ specifies the probability ${\displaystyle \sigma _{i}(a_{i}|t_{i})}$ of player ${\displaystyle i}$ playing action ${\displaystyle a_{i}\in A_{i}}$ when their type is ${\displaystyle t_{i}\in T_{i}}$.

For notational convenience, we also define the projections ${\displaystyle A_{-i}=\prod _{j\neq i}A_{j}}$ and ${\displaystyle T_{-i}=\prod _{j\neq i}T_{j}}$, and let ${\displaystyle \sigma _{-i}:T_{-i}\rightarrow \prod _{j\neq i}\Delta A_{j}}$ be the joint mixed strategy of players ${\displaystyle j\neq i}$, where ${\displaystyle \sigma _{-i}(a_{-i}|t_{-i})}$ gives the probability that players ${\displaystyle j\neq i}$ play action profile ${\displaystyle a_{-i}}$ when they are of type ${\displaystyle t_{-i}}$.

Definition: a Bayesian Nash equilibrium (BNE) for a finite Bayesian game ${\displaystyle G=((A_{i},T_{i},u_{i})_{i\in I},p)}$ consists of a strategy profile ${\displaystyle \sigma =(\sigma _{i})_{i\in I}}$ such that, for every ${\displaystyle i\in I}$, every ${\displaystyle t_{i}\in T_{i}}$, and every action ${\displaystyle a_{i}^{*}}$ played with positive probability ${\displaystyle \sigma _{i}(a_{i}^{*}|t_{i})>0}$, we have

${\displaystyle a_{i}^{*}\in {\underset {a_{i}\in A_{i}}{\operatorname {argmax} }}\sum _{t_{-i}\in T_{-i}}p_{i}(t_{-i}|t_{i})\sum _{a_{-i}\in A_{-i}}\sigma _{-i}(a_{-i}|t_{-i})u_{i}(a_{i},a_{-i},t_{i},t_{-i})}$

where ${\displaystyle p_{i}(t_{-i}|t_{i})={\frac {p(t_{i},t_{-i})}{\sum _{t_{-i}\in T_{-i}}p(t_{i}|t_{-i})p(t_{-i})}}}$ is player ${\displaystyle i}$'s beliefs about other players types ${\displaystyle t_{-i}}$ given his own type ${\displaystyle t_{i}}$.

Definition

Average strategies

First, we define the "average strategy of other players", averaged over their types. Formally, for each ${\displaystyle i\in I}$ and each ${\displaystyle t_{i}\in T_{i}}$, we define ${\displaystyle {\overline {\sigma }}_{-i}:T_{i}\rightarrow \prod _{j\neq i}\Delta A_{j}}$ by putting

${\displaystyle {\overline {\sigma }}_{-i}(a_{-i}|t_{i})=\sum _{t_{-i}\in T_{i}}p_{i}(t_{-i}|t_{i})\sigma _{-i}(a_{-i}|t_{-i})}$

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

Cursed equilibrium

Given a degree of mispercetion ${\displaystyle \chi \in [0,1]}$, we define a ${\displaystyle \chi }$-cursed equilibrium for a finite Bayesian game ${\displaystyle G=((A_{i},T_{i},u_{i})_{i\in I},p)}$ as a strategy profile ${\displaystyle \sigma =(\sigma _{i})_{i\in I}}$ such that, for every ${\displaystyle i\in I}$, every ${\displaystyle t_{i}\in T_{i}}$, we have

${\displaystyle a_{i}^{*}\in {\underset {a_{i}\in A_{i}}{\operatorname {argmax} }}\sum _{t_{-i}\in T_{-i}}p_{i}(t_{-i}|t_{i})\sum _{a_{-i}\in A_{-i}}\left[\chi {\overline {\sigma }}_{-i}(a_{-i}|t_{i})+(1-\chi )\sigma _{-i}(a_{-i}|t_{-i})\right]u_{i}(a_{i},a_{-i},t_{i},t_{-i})}$

for every action ${\displaystyle a_{i}^{*}}$ played with positive probability ${\displaystyle \sigma _{i}(a_{i}^{*}|t_{i})>0}$.

For ${\displaystyle \chi =0}$, we have the usual BNE. For ${\displaystyle \chi =1}$, 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 ${\displaystyle \chi }$-cursed equilibria where both parties choose to trade.^[1]

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]