Cursed equilibrium

Cursed equilibrium
Solution concept in game theory
Relationship
Superset of	Bayesian Nash equilibrium
Significance
Proposed by	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]

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 ${\displaystyle G=((A_{i},T_{i},u_{i})_{i\in I},p)}$.

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 ${\displaystyle t_{i}}$.

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 ${\displaystyle \sigma _{-i}(a_{-i}|t_{-i})}$.

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.

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]

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]