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 {{mvar|S}} of positive real numbers. Before play starts, a threshold {{mvar|T}} and penalty {{mvar|ν}} are chosen with {{math|1 < T < ∞}} and {{math|0 < ν < ∞}}. For example, consider {{mvar|S}} to be the set of integers from {{math|1}} to {{mvar|n}}, {{math|1=T = 3}} and {{math|1=ν = 2}}.

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

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