Solution of Peter Winkler's Pizza Problem

TL;DR

This paper proves a conjecture about a pizza-sharing game, showing that Alice can guarantee at least 4/9 of the pizza using a specific strategy, and provides algorithms for optimal play.

Contribution

It proves a conjecture by Peter Winkler, characterizes Alice's optimal strategies, and offers efficient algorithms for strategy computation in the pizza problem.

Findings

Alice can guarantee at least 4/9 of the pizza.

Optimal strategies depend on the number of slices and allowed jumps.

Algorithms for strategy computation run in linear and quadratic time.

Abstract

Bob cuts a pizza into slices of not necessarily equal size and shares it with Alice by alternately taking turns. One slice is taken in each turn. The first turn is Alice's. She may choose any of the slices. In all other turns only those slices can be chosen that have a neighbor slice already eaten. We prove a conjecture of Peter Winkler by showing that Alice has a strategy for obtaining 4/9 of the pizza. This is best possible, that is, there is a cutting and a strategy for Bob to get 5/9 of the pizza. We also give a characterization of Alice's best possible gain depending on the number of slices. For a given cutting of the pizza, we describe a linear time algorithm that computes Alice's strategy gaining at least 4/9 of the pizza and another algorithm that computes the optimal strategy for both players in any possible position of the game in quadratic time. We distinguish two types of turns, shifts and jumps. We prove that Alice can gain 4/9, 7/16 and 1/3 of the pizza if she is allowed to make at most two jumps, at most one jump and no jump, respectively, and the three constants are the best possible.