The multiplication game

TL;DR

This paper analyzes a game where players choose numbers to maximize winning chances based on leading digits, revealing that the Benford distribution is optimal and extending results to more general mathematical settings.

Contribution

It demonstrates that the Benford distribution is optimal for the multiplication game and extends the analysis to games on topological groups using Haar measure.

Findings

Benford distribution is optimal for the game.

Optimal strategies exist for the original integer game.

Results extend to games on compact topological groups.

Abstract

The multiplication game is a two-person game in which each player chooses a positive integer without knowledge of the other player's number. The two numbers are then multiplied together and the first digit of the product determines the winner. Rather than analyzing this game directly, we consider a closely related game in which the players choose positive real numbers between 1 and 10, multiply them together, and move the decimal point, if necessary, so that the result is between 1 and 10. The mixed strategies are probability distributions on this interval, and it is shown that for both players it is optimal to choose their numbers from the Benford distribution. Furthermore, this strategy is optimal for any winning set, and the probability of winning is the Benford measure of the player's winning set. Using these results we prove that the original game in which the players choose integers has a well-defined value and that strategies exist that are arbitrarily close to optimal. Finally, we consider generalizations of the game in which players choose elements from a compact topological group and show that choosing them according to Haar measure is an optimal strategy.