Cursed equilibrium | |
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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, and has since become a canonical behavioral solution concept for Bayesian games in behavioral economics.
Preliminaries
Bayesian games
Let be a finite set of players and for each , define their finite set of possible actions and as their finite set of possible types; the sets and are the sets of joint action and type profiles, respectively. Each player has a utility function , and types are distributed according to a joint probability distribution . A finite Bayesian game consists of the data .
Bayesian Nash equilibrium
For each player , a mixed strategy specifies the probability of player playing action when their type is .
For notational convenience, we also define the projections and , and let be the joint mixed strategy of players , where gives the probability that players play action profile when they are of type .
Definition: a Bayesian Nash equilibrium (BNE) for a finite Bayesian game consists of a strategy profile such that, for every , every , and every action played with positive probability , we have
where is player 's beliefs about other players types given his own type .
Definition
Average strategies
First, we define the "average strategy of other players", averaged over their types. Formally, for each and each , we define by putting
Notice that does not depend on . It gives the probability, viewed from the perspective of player when he is of type , that the other players will play action profile when they follow the mixed strategy . More specifically, the information contained in does not allow player to assess the direct relation between and given by .
Cursed equilibrium
Given a degree of mispercetion , we define a -cursed equilibrium for a finite Bayesian game as a strategy profile such that, for every , every , we have
for every action played with positive probability .
For , we have the usual BNE. For , 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 -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.
References
- ^ Eyster, Erik; Rabin, Matthew (2005). "Cursed Equilibrium". Econometrica. 73 (5): 1623–1672. doi:10.1111/j.1468-0262.2005.00631.x.
- Cohen, Shani; Li, Shengwu (2022). "Sequential Cursed Equilibrium". arXiv:2212.06025 .
- Szembrot, Nichole (2017). "Are voters cursed when politicians conceal policy preferences?". Public Chcoice. 173: 25–41. doi:10.1007/s11127-017-0461-9.