$\def\argmax{\mathop{\rm argmax}}$
Recall that potential games are non-cooperative games that allow for a potential function that behaves in a sense "similarly" as the pay-off functions. Inspired by a certain class of optimization algorithms, I met a game that is "almost" potential. It is defined as follows. There are $n$ players. Player $i$ has the strategy set $S_i$ and pay-off function $f_i: S\to\mathbb{R}$, where $S=S_1\times\cdots\times S_n$. There is a potential function $F: S\to\mathbb{R}$ such that for every $i\in\{1,\ldots,n\}$, $x\in S$ and $x'_i\in S_i$, the following three conditions hold:
- $f_i(x)-f_i(x'_i,x_{-i})\ge0\;\Longrightarrow\;F(x)-F(x'_i,x_{-i})\ge0$
- $F(x)-F(x'_i,x_{-i})>0\;\Longrightarrow\;f_i(x)-f_i(x'_i,x_{-i})>0$
- $\displaystyle\argmax_{x_i\in S_i}f_i(x)\subseteq\argmax_{x_i\in S_i}F(x)$
where for $x=(x_1,\ldots,x_n)\in S$ we denoted $x_{-i}=(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n)$, and $\argmax$ is the set of all maximisers.
Note that the game is similar to (but not contained in) two known classes of games:
- generalized ordinal potential games, defined by $f_i(x)-f_i(x'_i,x_{-i})>0\;\Longrightarrow\;F(x)-F(x'_i,x_{-i})>0$
- pseudo-potential games, defined by $\displaystyle\argmax_{x_i\in S_i}f_i(x)\supseteq\argmax_{x_i\in S_i}F(x)$
It would help me to know if my game belongs to any known game class and if anything can be said about its pure Nash equilibria (perhaps under some assumptions on $S_i$, such as compactness). I googled a lot, but found nothing. Many thanks!
Tom