Satisfaction Equilibrium

An action profile is a tuple ${\displaystyle {\boldsymbol {a}}=\left(a_{1},\ldots ,a_{K}\right)\in {\mathcal {A}}_{1}\times \ldots \times {\mathcal {A}}_{K}}$. The action profile in which all players are satisfied is an equilibrium of the corresponding game in satisfaction form. At a satisfaction equilibrium, players do not exhibit a particular interest in changing its current action.

Definition [Satisfaction Equilibrium in Pure Strategies^[2]]
The action profile ${\displaystyle {\boldsymbol {a}}}$ is a satisfaction equilibrium in pure strategies for the game ${\displaystyle \left({\mathcal {K}},\left\lbrace {\mathcal {A}}_{k}\right\rbrace _{k\in {\mathcal {K}}},\left\lbrace f_{k}\right\rbrace _{k\in {\mathcal {K}}}\right),}$ if for all ${\displaystyle k\in {\mathcal {K}}}$,

${\displaystyle a_{k}\in f_{k}\left(a_{1},\ldots ,a_{k-1},a_{k+1},\ldots ,a_{K}\right)}$.

Satisfaction Equilibrium in Mixed Strategies

For all ${\displaystyle k\in {\mathcal {K}}}$, denote the set of all possible probability distributions over the set ${\displaystyle {\mathcal {A}}_{k}=\lbrace A_{k,1},A_{k,2},\ldots ,A_{k,N_{k}}\rbrace }$ by ${\displaystyle \triangle \left({\mathcal {A}}_{k}\right)}$, with ${\displaystyle N_{k}=|{\mathcal {A}}_{k}|}$. Denote by ${\displaystyle {\boldsymbol {\pi }}_{k}=\left(\pi _{k,1},\pi _{k,2},\ldots ,\pi _{k,N_{k}}\right)}$ the probability distribution (mixed strategy) adopted by player ${\displaystyle k}$ to choose its actions. For all ${\displaystyle j\in \lbrace 1,\ldots ,N_{k}\rbrace }$, ${\displaystyle \pi _{k,j}}$ represents the probability with which player ${\displaystyle k}$ chooses action ${\displaystyle A_{k,j}\in {\mathcal {A}}_{k}}$. The notation ${\displaystyle {\boldsymbol {\pi }}_{-k}}$ represents the mixed strategies of all players except that of player ${\displaystyle k}$.

Definition [Extension to Mixed Strategies of the Satisfaction Form ^[2]] The extension in mixed strategies of the game ${\displaystyle \left({\mathcal {K}},\left\lbrace {\mathcal {A}}_{k}\right\rbrace _{k\in {\mathcal {K}}},\left\lbrace f_{k}\right\rbrace _{k\in {\mathcal {K}}}\right)}$ is described by the tuple ${\displaystyle \left({\mathcal {K}},\left\lbrace {\mathcal {A}}_{k}\right\rbrace _{k\in {\mathcal {K}}},\left\lbrace {\bar {f}}_{k}\right\rbrace _{k\in {\mathcal {K}}}\right)}$, where the correspondence

${\displaystyle {\bar {f}}_{k}:\prod _{j\in {\mathcal {K}}\setminus \lbrace k\rbrace }\triangle \left({\mathcal {A}}_{j}\right)\rightarrow 2^{\triangle \left({\mathcal {A}}_{k}\right)}}$

determines the set of all possible probability distributions that allow player ${\displaystyle k}$ to choose an action that satisfies its individual conditions with probability one, that is,

${\displaystyle {\bar {f}}_{k}\left({\boldsymbol {\pi }}_{-k}\right)=\left\lbrace {\boldsymbol {\pi }}_{k}\in \triangle \left({\mathcal {A}}_{k}\right):\mathrm {Pr} \left(a_{k}\in f_{k}\left({\boldsymbol {a}}_{-k}\right)|a_{k}\sim {\boldsymbol {\pi }}_{k},{\boldsymbol {a}}_{-k}\sim {\boldsymbol {\pi }}_{-k}\right)=1\right\rbrace .}$

A satisfaction equilibrium in mixed strategies is defined as follows.

Definition [Satisfaction Equilibrium in Mixed Strategies^[2]]
The mixed strategy profile ${\displaystyle {\boldsymbol {\pi }}^{*}\in \triangle \left({\mathcal {A}}_{1}\right)\times \ldots \times \triangle \left({\mathcal {A}}_{K}\right)}$ is an SE in mixed strategies if for all ${\displaystyle k\in {\mathcal {K}}}$,

${\displaystyle {\boldsymbol {\pi }}_{k}^{*}\in {\bar {f}}_{k}\left({\boldsymbol {\pi }}_{-k}^{*}\right)}$.

Let the ${\displaystyle j}$-th action of player ${\displaystyle k}$, i.e., ${\displaystyle A_{k,j}}$, be associated with the unitary vector ${\displaystyle {\boldsymbol {e}}_{j}=\left(e_{1},e_{2}\ldots ,e_{N_{k}}\right)\in \mathrm {R} ^{N_{k}}}$, where, all the components are zero except its ${\displaystyle j}$-th component, which is equal to one. The vector ${\displaystyle {\boldsymbol {e}}_{j}}$ represents a degenerated probability distribution, where the action ${\displaystyle A_{k,j}}$ is deterministically chosen. Using this argument, it becomes clear that every satisfaction equilibrium in pure strategies of the game ${\displaystyle \left({\mathcal {K}},\left\lbrace {\mathcal {A}}_{k}\right\rbrace _{k\in {\mathcal {K}}},\left\lbrace f_{k}\right\rbrace _{k\in {\mathcal {K}}}\right)}$ is also a satisfaction equilibrium in mixed strategies of the game ${\displaystyle \left({\mathcal {K}},\left\lbrace {\mathcal {A}}_{k}\right\rbrace _{k\in {\mathcal {K}}},\left\lbrace {\bar {f}}_{k}\right\rbrace _{k\in {\mathcal {K}}}\right)}$.

At an SE of the game ${\displaystyle \left({\mathcal {K}},\left\lbrace {\mathcal {A}}_{k}\right\rbrace _{k\in {\mathcal {K}}},\left\lbrace f_{k}\right\rbrace _{k\in {\mathcal {K}}}\right)}$, players choose their actions following a probability distribution such that only action profiles that allow all players to simultaneously satisfy their individual conditions with probability one are played with positive probability. Hence, in the case in which one SE in pure strategies does not exist, then, it does not exist a SE in mixed strategies in the game ${\displaystyle \left({\mathcal {K}},\left\lbrace {\mathcal {A}}_{k}\right\rbrace _{k\in {\mathcal {K}}},\left\lbrace {\bar {f}}_{k}\right\rbrace _{k\in {\mathcal {K}}}\right)}$.

ε-Satisfaction Equilibrium

Under certain conditions, it is always possible to build mixed strategies that allow players to be satisfied with probability ${\displaystyle 1-\epsilon }$, for some ${\displaystyle \epsilon >0}$. This observation leads to the definition of a solution concept known as ${\displaystyle \epsilon }$-satisfaction equilibrium (${\displaystyle \epsilon }$-SE).

Definition: [ε-Satisfaction Equilibrium^[2]]
Let ${\displaystyle \epsilon }$ satisfy ${\displaystyle \epsilon \in \left]0,1\right]}$. The mixed strategy profile ${\displaystyle {\boldsymbol {\pi }}^{*}\in \triangle \left({\mathcal {A}}_{1}\right)\times \triangle \left({\mathcal {A}}_{2}\right)\times \ldots \times \triangle \left({\mathcal {A}}_{K}\right)}$ is an epsilon-satisfaction equilibrium (${\displaystyle \epsilon }$-SE) of the game ${\displaystyle \left({\mathcal {K}},\left\lbrace {\mathcal {A}}_{k}\right\rbrace _{k\in {\mathcal {K}}},\left\lbrace {\bar {f}}_{k}\right\rbrace _{k\in {\mathcal {K}}}\right)}$, if for all ${\displaystyle k\in {\mathcal {K}}}$, it follows that

${\displaystyle {\boldsymbol {\pi }}_{k}^{*}\in {\bar {\bar {f}}}_{k}\left({\boldsymbol {\pi }}_{-k}^{*}\right)}$,

where

${\displaystyle {\bar {\bar {f}}}_{k}\left({\boldsymbol {\pi }}_{-k}^{*}\right)=\left\lbrace {\boldsymbol {\pi }}_{k}\in \triangle \left({\mathcal {A}}_{k}\right):\mathrm {Pr} \left(a_{k}\in f_{k}\left({\boldsymbol {a}}_{-k}\right)|a_{k}\sim {\boldsymbol {\pi }}_{k},{\boldsymbol {a}}_{-k}\sim {\boldsymbol {\pi }}_{-k}^{*}\right)\geqslant 1-\epsilon \right\rbrace .}$

From the definition above, it can be implied that if the mixed strategy profile ${\displaystyle {\boldsymbol {\pi }}^{*}}$ is an ${\displaystyle \epsilon }$-SE, it holds that,

${\displaystyle \mathrm {Pr} \left(a_{k}\in f_{k}\left({\boldsymbol {a}}_{-k}\right)|a_{k}\sim {\boldsymbol {\pi }}_{k}^{*},{\boldsymbol {a}}_{-k}\sim {\boldsymbol {\pi }}_{-k}^{*}\right)\geqslant 1-\epsilon .}$

That is, players are unsatisfied with probability ${\displaystyle \epsilon }$. The relevance of the ${\displaystyle \epsilon }$-SE is that it models the fact that players can be tolerant a certain unsatisfaction level. At a given ${\displaystyle \epsilon }$-SE, none of the players is interested in changing its mixed strategy profile as long as it is satisfied with a probability higher than or equal to ${\displaystyle 1-\epsilon }$, for some ${\displaystyle \epsilon >0}$.

In contrast to the conditions for the existence of a SE in either pure or mixed strategies, the conditions for the existence of an ${\displaystyle \epsilon }$-SE are mild.

Proposition [Existence of an ${\displaystyle \epsilon }$-SE^[2]]
Let ${\displaystyle \left({\mathcal {K}},\left\lbrace {\mathcal {A}}_{k}\right\rbrace _{k\in {\mathcal {K}}},\left\lbrace f_{k}\right\rbrace _{k\in {\mathcal {K}}}\right)}$, be a finite game in satisfaction form. Then, if for all ${\displaystyle k\in {\mathcal {K}}}$, there always exists an action profile ${\displaystyle {\boldsymbol {a}}\in {\mathcal {A}}}$ such that

${\displaystyle a_{k}\in f_{k}\left({\boldsymbol {a}}_{-k}\right)}$,

then there always exists a strategy profile ${\displaystyle {\boldsymbol {\pi }}^{*}\in \triangle \left({\mathcal {A}}_{1}\right)\times \triangle \left({\mathcal {A}}_{2}\right)\times \ldots \times \triangle \left({\mathcal {A}}_{K}\right)}$ and a real ${\displaystyle \epsilon }$, with ${\displaystyle 1>\epsilon >0}$, such that, ${\displaystyle {\boldsymbol {\pi }}^{\star }}$ is an ${\displaystyle \epsilon }$-SE.