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On the optimal strategy in a random game

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1.	Title	Title of document	On the optimal strategy in a random game
2.	Creator	Author's name, affiliation, country	Johan Jonasson; Chalmers University of Technology
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	two-person game, mixed strategy, equalizing strategy, saddle point
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	Swedish Research Council
7.	Date	(YYYY-MM-DD)	2004-10-13
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/1100
10.	Identifier	Digital Object Identifier	10.1214/ECP.v9-1100
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 9
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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