Silverman's game: Difference between revisions
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* {{Cite journal |title=Silverman's game on discrete sets |first1=Ronald J. |last1=Evans |first2=Gerald A. |last2=Heuer |journal=Linear Algebra and its Applications |volume=166 |pages=217–235 |date=March 1992 |url=http://math.ucsd.edu/~revans/Silverman.pdf |doi=10.1016/0024-3795(92)90279-J}} |
* {{Cite journal |title=Silverman's game on discrete sets |first1=Ronald J. |last1=Evans |first2=Gerald A. |last2=Heuer |journal=Linear Algebra and its Applications |volume=166 |pages=217–235 |date=March 1992 |url=http://math.ucsd.edu/~revans/Silverman.pdf |doi=10.1016/0024-3795(92)90279-J}} |
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[[Category:Non-cooperative games]] |
[[Category:Non-cooperative games]] |
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* {{cite book |last1=Heuer |first1=Gerald |last2=Leopold-Wildburger |first2=Ulrike |date=1995 |title=Silverman's Game |url=https://www.springer.com/us/book/9783540592327 |isbn=978-3-540-59232-7 |publisher=Springer |page=293 }} |
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Revision as of 11:39, 29 October 2018
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 S of positive real numbers. Before play starts, a threshold T and penalty ν are chosen with 1 < T < ∞ and 0 < ν < ∞. For example, consider S to be the set of integers from 1 to n, T = 3 and ν = 2.
Each player chooses an element of S, x and y. Suppose player A plays x and player B plays y. Without loss of generality, assume player A chooses the larger number, so x ≥ y. Then the payoff to A is 0 if x = y, 1 if 1 < x/y < T and −ν if x/y ≥ T. Thus each player seeks to choose the larger number, but there is a penalty of ν for choosing too large a number.
A large number of variants have been studied, where the set S may be finite, countable, or uncountable. Extensions allow the two players to choose from different sets, such as the odd and even integers.
References
- Evans, Ronald J. (April 1979). "Silverman's game on intervals". American Mathematical Monthly. 86 (4): 277–281. doi:10.2307/23207451979.
- Evans, Ronald J.; Heuer, Gerald A. (March 1992). "Silverman's game on discrete sets" (PDF). Linear Algebra and its Applications. 166: 217–235. doi:10.1016/0024-3795(92)90279-J.
- Heuer, Gerald; Leopold-Wildburger, Ulrike (1995). Silverman's Game. Springer. p. 293. ISBN 978-3-540-59232-7.