Silverman's game

In game theory, Silverman's game is a zero sum game played on the unit square. It is named for 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 $\nu$ are chosen with $0<\nu<\infty$ and 1 < T < ∞ Each player chooses an element of S. Without loss of generality, suppose player A plays x and player B plays y, with x > y. Then the payoff to A is 1 if 1 < x/y < T and $-\nu$ if x/y > T; equal numbers involve zero payoff. Thus each player seeks to choose the larger number, but there is a penalty of $\nu$ for choosing too large a number.