Ingame theory, acursed equilibrium is asolution concept forstatic games ofincomplete information. It is a generalization of the usualBayesian Nash equilibrium, allowing for players to underestimate the connection between other players' equilibrium actions and their types – that is, thebehavioral 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 byErik Eyster andMatthew Rabin in 2005,^[1] and has since become a canonicalbehavioral solution concept for Bayesian games inbehavioral 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${\displaystyle u_i:A\times T\to \mathbb{R}}$, and types are distributed according to a joint probability distribution${\displaystyle p\in \mathrm{\Delta }T}$. 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${\displaystyle {\sigma }_i:T_i\to \mathrm{\Delta }{A}_i}$ 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${\displaystyle A_{-i}=\prod _{j\ne i}{A}_j}$ and${\displaystyle T_{-i}=\prod _{j\ne i}{T}_j}$, and let${\displaystyle {\sigma }_{-i}:T_{-i}\to \prod _{j\ne i}\mathrm{\Delta }{A}_j}$ be the joint mixed strategy of players${\displaystyle j\ne i}$, where${\displaystyle {\sigma }_{-i}(a_{-i}|t_{-i})}$ gives the probability that players${\displaystyle j\ne i}$ play action profile${\displaystyle a_{-i}}$ when they are of type${\displaystyle t_{-i}}$.

Definition: aBayesian 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${\displaystyle a_i^{\ast }}$ played with positive probability${\displaystyle {\sigma }_i(a_i^{\ast }|t_i)>0}$, we have

${\displaystyle a_i^{\ast }\in \underset{a_i\in A_i}{\mathrm{argmax}}\sum _{t_{-i}\in T_{-i}}{p}_i(t_{-i}|t_i)\sum _{a_{-i}\in A_{-i}}{\sigma }_{-i}(a_{-i}|t_{-i})u_i(a_i,a_{-i},t_i,t_{-i})}$

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${\displaystyle {\overline{\sigma }}_{-i}:T_i\to \prod _{j\ne i}\mathrm{\Delta }{A}_j}$ 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

${\displaystyle a_i^{\ast }\in \underset{a_i\in A_i}{\mathrm{argmax}}\sum _{t_{-i}\in T_{-i}}{p}_i(t_{-i}|t_i)\sum _{a_{-i}\in A_{-i}}\left[\chi {\overline{\sigma }}_{-i}(a_{-i}|t_i)+(1-\chi ){\sigma }_{-i}(a_{-i}|t_{-i})\right]u_i(a_i,a_{-i},t_i,t_{-i})}$

for every action${\displaystyle a_i^{\ast }}$ played with positive probability${\displaystyle {\sigma }_i(a_i^{\ast }|t_i)>0}$.

For${\displaystyle \chi =0}$, we have the usual BNE. For${\displaystyle \chi =1}$, the equilibrium is referred to as afully cursed equilibrium, and the players in it asfully 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]

• Game theory equilibrium concepts
• Behavioral economics

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