Silverman's game

Last updated May 06, 2021

In game theory, Silverman's game is a two-person zero-sum game played on the unit square. It is named for mathematician 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 $1 < T < \infty$ and $0 < ν < \infty$ . For example, consider $S$ to be the set of integers from $1$ to $n$ , $T = 3$ and $ν = 2$ .

Each player chooses an element of $S$ , $x$ and $y$ . Suppose player A plays $x$ and player B plays $y$ . Without loss of generality, assume player A chooses the larger number, so $x \geq y$ . Then the payoff to A is 0 if $x = y$ , 1 if $1 < x / y < T$ and $- ν$ if $x / y \geq T$ . Thus each player seeks to choose the larger number, but there is a penalty of $ν$ for choosing too large a number.

A large number of variants have been studied, where the set $S$ may be finite, countable, or uncountable. Extensions allow the two players to choose from different sets, such as the odd and even integers.

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References

Evans, Ronald J. (April 1979). "Silverman's game on intervals". American Mathematical Monthly . 86 (4): 277–281. doi:10.1080/00029890.1979.11994788.
Evans, Ronald J.; Heuer, Gerald A. (March 1992). "Silverman's game on discrete sets" (PDF). Linear Algebra and Its Applications. 166: 217–235. doi: 10.1016/0024-3795(92)90279-J .
Heuer, Gerald; Leopold-Wildburger, Ulrike (1995). Silverman's Game. Springer. p. 293. ISBN 978-3-540-59232-7.

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