In
game theory
Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...
, Silverman's game is a two-person
zero-sum game
Zero-sum game is a Mathematical model, mathematical representation in game theory and economic theory of a situation that involves two competition, competing entities, where the result is an advantage for one side and an equivalent loss for the o ...
played on the
unit square
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and .
Cartesian coordinates
In a Cartesian coordinat ...
. It is named for mathematician
David Silverman.
It is played by two players on a given set of
positive real numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
. 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
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
, 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
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
, or
uncountable
In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger tha ...
. Extensions allow the two players to choose from different sets, such as the odd and even integers.
References
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Non-cooperative games
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