Grabbing Olives on Linear Pizzas and Pissaladières

Abstract

In this paper we revisit the problem entitled Sharing a Pizza stated by P. Winkler by considering a new puzzle called Sharing a Pissaladiere. The game is played by two polite coatis Alice and Bob who share a pissaladière (a p×q grid) which is divided into rectangular slices. Alice starts in a corner and then the coatis alternate removing a remaining slice adjacent to at most two other slices. On some slices there are precious olives of Nice and the aim of each coati is to grab the maximum number of olives. We first study the particular case of 1×n grid (i.e. a path) where the game is a graph grabbing game known as Sharing a linear pizza. In that case each player can take only an end vertex of the remaining path. These problems are particular cases of a new class of games called d-degenerate games played on a graph with non negative weights assigned to the vertices with the rule that coatis alternatively take a vertex of degree at most d. 
Our main results are the following. We give optimal strategies for paths (linear pizzas) with no two adjacent weighty vertices. We also give a recurrence formula to compute the gains which depend only on the parity of n and of the respective parities of weighty vertices with a complexity in O(h²) where h denotes the number of parity changes in the weighty vertices. When the weights are only {0,1} we reduce the computation of the average number of olives collected by each player to a word counting problem. We solve Sharing a pissaladière with {0,1} weights, when there is one olive or 2 olives. In that case Alice (resp. Bob) grabs almost all the olives if the number of vertices of the grid n = p×q is odd (resp. even). We prove that for a 2×q grid with a fixed number k of olives Bob grabs at least ⌈(k-1)/3⌉ olives and almost always grabs all the k olives.