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 $1 < T < ∞$ and $0 < ν < ∞$. 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 $x ≥ y$. Then the payoff to A is 0 if $x = y$, 1 if $1 < x/y < T$ and $&minus;ν$ if $x/y ≥ T$. 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.