Philippine Science Letters
vol. 6 | no. 1 | 2013
published online January 31, 2013


A Variation of Discrete Silverman’s Game with Varying Payoffs

by Guido David* and Richele S. De Asis

*Institute of Mathematics, and
Computational Science Research Center
University of the Philippines
Diliman, Quezon City



A discrete form of Silverman's game, a two-player zero sum game, is played with each player choosing a number from 1 to n. Each player's goal is to choose the larger number as long as it is less than three times the opponent's chosen number. Here we consider a variation of Silverman’s game, wherein the payoff to the player choosing the larger number is the difference between the two numbers, but if the larger number is at least three times the smaller number, the payoff to the player choosing the smaller number is twice the difference between the two numbers. Analysis of payoff matrices and the Minimax Theorem are used to solve the game. In both versions of the game, results show that when n = 3 the unique optimal strategy reduces each player’s choices to just three numbers. The difference between the two solutions is in the probabilities each player must select the three choices.

*Corresponding author
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Received: October 28, 2012
Revised: December 9, 2012
Accepted: December 10, 2012
Published: January 31, 2013
Editor-in-charge: Raymond Girard R. Tan
Reviewer: Raymond Girard R. Tan