Silverman's game

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.

References

Evans, Ronald J. (April 1979). "Silverman's game on intervals". American Mathematical Monthly. 86 (4): 277–281. doi:10.2307/23207451979.
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.

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