I have this simple market game:
- $I=\{1,2...,n\}$ players
- $S_i$ strategy space of each player $i\in I$
- $u_i(s_i,s_{-i})=R_i(s_i)-C(s_i,s_{-i})$
There's only one type of resource. The resource is limited.
$s_i$ indicates the portion of the entire resource being used by player $i$.
The utility of player $i$ is the difference between the reward coming from using $s_i$ and the cost of buying said amount, given that the price of the single unit of resource increases as the demand increases, in other words: the more resources are used, the more they are expensive.
I want to classify this game as a submodular game. For this purpose, consider the strategy spaces to be complete lattices and $u_i$ continuous $\forall i \in I$.
My understanding is that, in this case, if:
- $u_i$ is submodular in own strategies for each player
- $u_i$ verifies decreasing differences in the strategies of the other players
the game is submodular.
Since the utilities are functions of one variable fixed the other players strategies, they are always submodular. Supposing they are also of class $C^2$, then the decreasing difference property is true if the second order mixed partial derivatives are all non-positive. This is true by definition of the cost function, since an increase in the demand always leads to an increase of the price and since the cost function is multiplied by $-1$ in the utility definition.
The best response of player $i$ can be written as $f_i=\arg\max_{s_i}\{u_i(s_i,s_{-i})\}$
which is a maximization of a submodular function. What puzzles me is that in literature (including Topkins 1979 which is the paper that first presents submodular games) all the important results of this class of games is presented and proofed in the case in which either the game is submodular and the best response is a minimization or the game is supermodular and the best response is a maximization.
It seems to me that my game is submodular with a maximization as best response. I see no obstacles in applying the Tarski's fixed point theorem to prove that there exists at least one N.E., but I am not so confident since this is not my area of expertise at all. I was not able to find any proof of this specific case in literature.
Can you please help me in validating or refute what I wrote above? Thanks for your interest