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In [[game theory]], '''Silverman's game''' is a [[zero sum game]] played on the [[unit square]]. It is named for [[David Silverman (mathematician)|David Silverman]].


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It is played by two players on a given set ''S'' of [[positive number|positive real numbers]]. Before play starts, a threshold <math>T</math> and penalty <math>\nu</math> are chosen with <math>0<\nu<\infty</math> and 1&nbsp;<&nbsp;''T''&nbsp;<&nbsp;&infin;
Each player chooses an element of ''S''. [[Without loss of generality]], suppose player A plays ''x'' and player B plays ''y'', with ''x''&nbsp;>&nbsp;''y''. Then the payoff to A is 1 if 1&nbsp;<&nbsp;''x''/''y''&nbsp;<&nbsp;''T'' and <math>-\nu</math> if ''x''/''y''&nbsp;>&nbsp;''T''; equal numbers involve zero payoff. Thus each player seeks to choose the larger number, but there is a penalty of <math>\nu</math> for choosing too large a number.
 
==References==
 
* R. J. Evans 1979. ''Silverman's game on intervals''.  [[American Mathematical Monthly]], volume 86, number 4, pp. 277&ndash;281.
 
 
{{gametheory-stub}}
[[Category:Game theory]]

Revision as of 18:48, 3 June 2013

In game theory, Silverman's game is a zero sum game played on the unit square. It is named for David Silverman.

It is played by two players on a given set S of positive real numbers. Before play starts, a threshold T and penalty ν are chosen with 0<ν< and 1 < T < ∞ Each player chooses an element of S. Without loss of generality, suppose player A plays x and player B plays y, with x > y. Then the payoff to A is 1 if 1 < x/y < T and ν if x/y > T; equal numbers involve zero payoff. Thus each player seeks to choose the larger number, but there is a penalty of ν for choosing too large a number.

References


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