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 Template:Mvar of positive real numbers. Before play starts, a threshold Template:Mvar and penalty Template:Mvar are chosen with 1 < T < ∞Script error: No such module "Check for unknown parameters". and 0 < ν < ∞Script error: No such module "Check for unknown parameters".. For example, consider Template:Mvar to be the set of integers from 1Script error: No such module "Check for unknown parameters". to Template:Mvar, T = 3Script error: No such module "Check for unknown parameters". and ν = 2Script error: No such module "Check for unknown parameters"..
Each player chooses an element of Template:Mvar, Template:Mvar and Template:Mvar. Suppose player A plays Template:Mvar and player B plays Template:Mvar. Without loss of generality, assume player A chooses the larger number, so x ≥ yScript error: No such module "Check for unknown parameters".. Then the payoff to A is 0 if x = yScript error: No such module "Check for unknown parameters"., 1 if 1 < x/y < TScript error: No such module "Check for unknown parameters". and −νScript error: No such module "Check for unknown parameters". if x/y ≥ TScript error: No such module "Check for unknown parameters".. Thus each player seeks to choose the larger number, but there is a penalty of Template:Mvar for choosing too large a number.
A large number of variants have been studied, where the set Template:Mvar may be finite, countable, or uncountable. Extensions allow the two players to choose from different sets, such as the odd and even integers.
References
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