Random-Turn Hex and Other Selection Games

Authors: Peres, Yuval; Schramm, Oded; Sheffield, Scott; Wilson, David B.

Source: American Mathematical Monthly, Volume 114, Number 5, May 2007 , pp. 373-387(15)

Publisher: Mathematical Association of America

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Abstract:

The game of Hex has two players who take turns placing stones of their respective colors on the hexagons of a rhombus-shaped hexagonal grid. Black wins by completing a crossing between two opposite edges, while White wins by completing a crossing between the other pair of opposite edges. Although ordinary Hex is famously difficult to analyze, Random-Turn Hex—in which players toss a coin before each turn to decide who gets to place the next stone—has a simple optimal strategy. It belongs to a general class of random-turn games—called selection games—in which the expected payoff when both players play the random-turn game optimally is the same as when both players play randomly. We also describe the optimal strategy and study the expected length of the game under optimal play for Random-Turn Hex and several other selection games.

Document Type: Research article

Publication date: 2007-05-01

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American Mathematical Monthly publishes articles, notes, and other features about mathematics and the profession. AMM readers span a broad spectrum of mathematical interests, and include professional mathematicians as well as students of mathematics at all collegiate levels.
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