Calculating mixed strategies for a $2\times 2\times 2$ three-player game using linear programming

Question

I have a game which is somewhat similar to matching pennies*, but with $3$ players. The normal form representation of this game is as follows: where the payoffs in parentheses are for $A$, $B$ and $C$, in that order, $\alpha$ is a factor which decreases the amplitude of the payoffs for $A$ and $B$ with $0<\alpha<1$, the payoffs for $A$ and $B$ are symmetrical, and $a_i>0, \forall i\in\{1,\cdots,8\}$.

I would like to avoid plugging in numbers for $a_i$ for now as there are multiple scenarios I am studying, but if comments or answers imply it would be helpful then I will edit accordingly. Given the symmetry of payoffs between $A$ and $B$, I believe there is no pure Nash equilibrium as stated. Further, assume there are no dominating strategies either (this would likely require me showing some scenarios by plugging in numbers for $a_i$, but, if possible for now, please take my word for it). I am thus looking for mixed strategies in this general setting, if possible.

Let $p$, $q$ and $r$ be the probabilities of choosing $X$ for $A$, $B$ and $C$ respectively. Given there are only two possible actions per player, the probabilities of choosing $Y$ are $(1-p)$, $(1-q)$ and $(1-r)$. I would like to know if there is a generic solution to this problem, or if not how to write the Linear Programming problem associated to this game. What I need is a general solution which is based on finding best responses.

NB: I already have equations using the "indifference between both outcomes" principle used in $2$-player games, and this is not the approach I am looking for.

*in the sense that pure best responses move circularly in such a way that no pure equilibrium exist.

Answer 1

Apparently, this is not possible using LP, which solves 2-player zero-sum games. 2-player non-zero sum games can be solved using LCP. Beyond 2 players, as far as I can tell, there is no way to use linear programming to achieve this. I do not have a formal reference stating this, but the following are hints that this is in fact the case:

• https://people.csail.mit.edu/costis/zerosum_final3.pdf : an extension to multiplayer polymatrix games (the game in the statement is not polymatrix).
• https://arxiv.org/pdf/1602.03280.pdf : an extension to "multilinear" games, which is apparently the case I am in, in which they use tensor-complementarity to solve such games
• https://link.springer.com/article/10.1007/s10957-014-0616-5 : the referenced article on tensor complementarity for the previous article

=> so the solution is likely not trivial and uses a generalization of linear programming. Elements as to how this works can be found in the above references. As I am new to this field I cannot vouch for these methods / have not implemented them, and this is as far as I plan to go on this issue for now (I have decided to use alternative methods).