Hi,


How to find, or at least express, the equilibrium of a zero-sum game with an $n\*n$ payoff matrix (each player has $n$ strategies) and the payoff of the entry $(i,j)$ is $u(i,j)$. $u$ a random function of the strategies $i$ and $j$.


What about the case where $n \rightarrow \infty$.


Any reference or code is welcome.


Thanks a lot!



Equilibrium of random zero-sum game,
Assuming the $u(i,j)$ are iid with a continuous distribution, the probability that $(i,j)$ is
a saddle point, i.e. that $u(i,j)$ is the greatest entry in its column and the least in its row, is
$((n-1)!)^2/(2n-1)! \approx 2^{1-n^2} \sqrt{\pi/n}$ as $n \to \infty$. Thus the probability
that the game has a saddle point goes to $0$ (and very rapidly) as $n \to \infty$. Instead, with probability approaching $1$ the optimal strategies are mixed strategies.


