Cursed equilibrium

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

For each player $i\; \backslash in\; I$, a mixed strategy $\backslash sigma\_i\; :\; T\_i\; \backslash rightarrow\; \backslash Delta\; A\_i$ specifies the probability $\backslash sigma\_i\; (\; a\_i\; |\; t\_i)$ of player $i$ playing action $a\_i\; \backslash in\; A\_i$ when their type is $t\_i\; \backslash in\; T\_i$.

For notational convenience, we also define the projections $A\_\{-i\}=\; \backslash prod\_\{j\; \backslash neq\; i\}\; A\_j$ and $T\_\{-i\}\; =\; \backslash prod\_\{j\; \backslash neq\; i\}\; T\_j$, and let $\backslash sigma\_\{-i\}\; :\; T\_\{-i\}\; \backslash rightarrow\; \backslash prod\_\{j\; \backslash neq\; i\}\; \backslash Delta\; A\_j$ be the joint mixed strategy of players $j\; \backslash neq\; i$, where $\backslash sigma\_\{-i\}\; (a\_\{-i\}\; |\; t\_\{-i\})$ gives the probability that players $j\; \backslash neq\; i$ play action profile $a\_\{-i\}$ when they are of type $t\_\{-i\}$.

Definition: a Bayesian Nash equilibrium (BNE) for a finite Bayesian game $G\; =\; ((A\_i,\; T\_i,\; u\_i)\_\{i\; \backslash in\; I\},\; p)$ consists of a strategy profile $\backslash sigma\; =\; (\backslash sigma\_i)\_\{i\; \backslash in\; I\}$ such that, for every $i\; \backslash in\; I$, every $t\_i\; \backslash in\; T\_i$, and every action $a\_i^*$ played with positive probability $\backslash sigma\_i\; (\; a\_i^*\; |\; t\_i)\; >\; 0$, we have

:$a\_i^*\; \backslash in\; \backslash underset\{a\_i\; \backslash in\; A\_i\}\backslash operatorname\{argmax\}\; \backslash sum\_\{t\_\{-i\}\; \backslash in\; T\_\{-i\}\}\; p\_i(t\_\{-i\}\; |\; t\_i)\; \backslash sum\_\{a\_\{-i\}\; \backslash in\; A\_\{-i\}\}\; \backslash sigma\_\{-i\}\; (a\_\{-i\}\; |\; t\_\{-i\})\; u\_i\; (a\_i,\; a\_\{-i\},\; t\_i,\; t\_\{-i\})$

where $p\_i(t\_\{-i\}\; |\; t\_i)\; =\; \backslash frac\{p(t\_i,\; t\_\{-i\})\}\{\backslash sum\_\{t\_\{-i\}\; \backslash in\; T\_\{-i\}\}\; p(t\_\{i\}\; |\; t\_\{-i\})\; p(t\_\{-i\})\; \}$ is player $i$'s beliefs about other players types $t\_\{-i\}$ given his own type $t\_i$.

First, we define the "average strategy of other players", averaged over their types. Formally, for each $i\; \backslash in\; I$ and each $t\_i\; \backslash in\; T\_i$, we define $\backslash overline\{\backslash sigma\}\_\{-i\}\; :\; T\_i\; \backslash rightarrow\; \backslash prod\_\{j\; \backslash neq\; i\}\; \backslash Delta\; A\_\{j\}$ by putting

:$\backslash overline\{\backslash sigma\}\_\{-i\}\; (a\_\{-i\}\; |\; t\_i)=\; \backslash sum\_\{t\_\{-i\}\; \backslash in\; T\_i\}\; p\_i(t\_\{-i\}\; |\; t\_i)\; \backslash sigma\_\{-i\}\; (a\_\{-i\}\; |\; t\_\{-i\})$

Notice that $\backslash overline\{\backslash sigma\}\_\{-i\}\; (a\_\{-i\}\; |\; t\_i)$ 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 $\backslash sigma\_\{-i\}$. More specifically, the information contained in $\backslash overline\{\backslash sigma\}\_\{-i\}$ does not allow player $i$ to assess the direct relation between $a\_\{-i\}$ and $t\_\{-i\}$ given by $\backslash sigma\_\{-i\}\; (a\_\{-i\}\; |\; t\_\{-i\})$.

Given a degree of mispercetion $\backslash chi\; \backslash in\; [0,\; 1]$, we define a

$\backslash chi$-cursed equilibrium for a finite Bayesian game $G\; =\; ((A\_i,\; T\_i,\; u\_i)\_\{i\; \backslash in\; I\},\; p)$ as a strategy profile $\backslash sigma\; =\; (\backslash sigma\_i)\_\{i\; \backslash in\; I\}$ such that, for every $i\; \backslash in\; I$, every $t\_i\; \backslash in\; T\_i$, we have

:$a\_i^*\; \backslash in\; \backslash underset\{a\_i\; \backslash in\; A\_i\}\backslash operatorname\{argmax\}\; \backslash sum\_\{t\_\{-i\}\; \backslash in\; T\_\{-i\}\}\; p\_i(t\_\{-i\}\; |\; t\_i)\; \backslash sum\_\{a\_\{-i\}\; \backslash in\; A\_\{-i\}\}\; \backslash left[\backslash chi\; \backslash overline\{\backslash sigma\}\_\{-i\}\; (a\_\{-i\}\; |\; t\_i)\; +\; (1-\; \backslash chi)\backslash sigma\_\{-i\}\; (a\_\{-i\}\; |\; t\_\{-i\})\; \backslash right]\; u\_i\; (a\_i,\; a\_\{-i\},\; t\_i,\; t\_\{-i\})$

for every action $a\_i^*$ played with positive probability $\backslash sigma\_i\; (\; a\_i^*\; |\; t\_i)\; >\; 0$.

For $\backslash chi\; =\; 0$, we have the usual BNE. For $\backslash chi\; =\; 1$, 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 $\backslash chi$-cursed equilibria where both parties choose to trade.{{cite journal |last1=Szembrot |first1=Nichole |title=Are voters cursed when politicians conceal policy preferences? |journal=Public Chcoice |date=2017 |volume=173 |pages=25–41 |doi=10.1007/s11127-017-0461-9}}

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Category:Game theory equilibrium concepts

Category:Behavioral economics