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 mixed strategies.

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

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

Bayesian Nash equilibrium

For each player $i\in I$, a mixed strategy $\sigma _{i}:T_{i}\rightarrow \Delta A_{i}$ 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 $A_{-i}=\prod _{j\neq i}A_{j}$ and $T_{-i}=\prod _{j\neq i}T_{j}$, and let $\sigma _{-i}:T_{-i}\rightarrow \prod _{j\neq i}\Delta A_{j}$ be the joint mixed strategy of players $j\neq i$, where $\sigma _{-i}(a_{-i}|t_{-i})$ gives the probability that players $j\neq i$ play action profile $a_{-i}$ when they are of type $t_{-i}$.

Definition: a Bayesian Nash equilibrium (BNE) for a finite Bayesian game $G=((A_{i},T_{i},u_{i})_{i\in I},p)$ consists of a strategy profile $\sigma =(\sigma _{i})_{i\in I}$ 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 have

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

where $p_{i}(t_{-i}|t_{i})={\frac {p(t_{i},t_{-i})}{\sum _{t_{-i}\in T_{-i}}p(t_{i}|t_{-i})p(t_{-i})}}$ is player $i$'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 ${\overline {\sigma }}_{-i}:T_{i}\rightarrow \prod _{j\neq i}\Delta A_{j}$ 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 $G=((A_{i},T_{i},u_{i})_{i\in I},p)$ as a strategy profile $\sigma =(\sigma _{i})_{i\in I}$ such that, for every $i\in I$, every $t_{i}\in T_{i}$, we have

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

for every action $a_{i}^{*}$ 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 assymetric information

In bilateral trade with two-sided assymetric 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 a election model where candidates are policy-motivated, a candidate which does 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 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 preferrences.