An analysis of a war-like card game

TL;DR

This paper analyzes the probability of winning strategies in a specific two-player card game, proving that the likelihood depends on the ratio of cards dealt and converges to zero or one based on the golden ratio.

Contribution

It provides a rigorous proof that the probability of a winning strategy in the game converges to zero or one depending on the card ratio, resolving an open problem posed by Winkler.

Findings

Probability of a winning strategy is zero when the ratio is below the golden ratio.

Probability of a winning strategy is one when the ratio exceeds the golden ratio.

The probability converges to zero or one as the number of cards increases.

Abstract

In his book "Mathematical Mind-Benders", Peter Winkler poses the following open problem, originally due to the first author: "[In the game Peer Pressure,] two players are dealt some number of cards, initially face up, each card carrying a different integer. In each round, the players simultaneously play a card; the higher card is discarded and the lower card passed to the other player. The player who runs out of cards loses. As the number of cards dealt becomes larger, what is the limiting probability that one of the players will have a winning strategy?" We show that the answer to this question is zero, as Winkler suspected. Moreover, assume the cards are dealt so that one player receives r >= 1 cards for every one card of the other. Then if r < phi = (1+sqrt 5)/2, the limiting probability that either player has a winning strategy is still zero, while if r > phi, it is one.