Accumulation games on graphs

Accumulation games on discrete locations were introduced by W. Ruckle and K. Kikuta. The Hider secretly distributes his total wealth h 1 over locations 1; 2; : : : ; n: The Searcher con…scates the material from any r of these locations. The Hider wins if the wealth remaining at the n r unsearched locations sums to at least 1; otherwise the Searcher wins. Their game models problems in which the Hider needs to have, after con…scation (or loss by natural causes), a su¢ cient amount of material (food, wealth, arms) to carry out some objective (survive the winter, buy a house, start an insurrection). This paper takes the hiding locations to be the nodes of a graph and restricts the node sets the Searcher can remove to be drawn from a given family: the edges, the connected r-sets, or some other given sets of nodes. This models the case where the pilferer, or storm, is known to act only on a set of close locations. Unlike the original game, our game requires mixed strategies.