On the optimal strategy in a random game
Johan Jonasson, Chalmers University of Technology
Abstract
Consider a two-person zero-sum game played on a random n by
n matrix
where the entries are iid normal random variables.
Let Z be the number of rows in the support of the
optimal strategy for player I given the realization of the matrix.
(The optimal strategy is a.s. unique and Z a.s. coincides
with the number of columns of the support of the
optimal strategy for player II.)
Faris an Maier (see the references) make simulations
that suggest that as n gets large Z
has a distribution close to
binomial with parameters n and 1/2
and prove that P(Z=n) < 2-(k-1).
In this paper a few more theoretically rigorous steps are taken towards
the limiting distribution of Z: It is shown that there
exists a<1/2 (indeed a<0.4) such that
P((1/2-a)n< Z <(1/2+a)n)
tends to 1 as n
increases.
It is also shown that the expectation of Z is
(1/2+o(1))n.
We also prove that the value of the game with probability
1-o(1) is at most Cn-1/2
for some finite C independent of n.
The proof suggests that an upper bound is in fact given by
f(n)/n, where f(n) is any sequence tending to infinity as
n increases, and it is
pointed out that if this is true, then the variance of Z is
o(n^2)
so that any a>0 will do in the bound on Z above.
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