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 of positive real numbers. Before play starts, a threshold and penalty are chosen with and. For example, consider to be the set of integers from to, and.
Each player chooses an element of, and. Suppose player A plays and player B plays. Without loss of generality, assume player A chooses the larger number, so. Then the payoff to A is 0 if, 1 if and if. Thus each player seeks to choose the larger number, but there is a penalty of for choosing too large a number.
A large number of variants have been studied, where the set may be finite, countable, or uncountable. Extensions allow the two players to choose from different sets, such as the odd and even integers.
