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**Harvard**

Jonasson, J. (2004) *On the optimal strategy in a random game*.

** BibTeX **

@article{

Jonasson2004,

author={Jonasson, Johan},

title={On the optimal strategy in a random game},

journal={Electronic Communications in Probability},

issn={1083-589X},

volume={9},

pages={132-139},

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. },

year={2004},

}

** RefWorks **

RT Journal Article

SR Electronic

ID 25480

A1 Jonasson, Johan

T1 On the optimal strategy in a random game

YR 2004

JF Electronic Communications in Probability

SN 1083-589X

VO 9

SP 132

OP 139

AB 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.

LA eng

LK http://publications.lib.chalmers.se/records/fulltext/local_25480.pdf

OL 30