Stake-governed tug-of-war and the biased infinity Laplacian
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We introduce a two-person zero-sum game that we call stake-governed tug-of-war. The game develops the classic tug-of-war random-turn game from <cit.>. In tug-of-war, two players compete by moving a counter along adjacent edges of a graph, each winning the right to move at a given turn according to the outcome of the flip of a fair coin; a payment is made from one player to the other when the counter reaches a boundary set on which the terminal payment value is specified. The player Mina who makes the payment seeks to minimize its mean; her opponent Maxine seeks to maximize it. The game's value is the infinity harmonic extension of the payment boundary data. In the stake-governed version, both players first receive a limited budget. At the start of each turn, each stakes an amount drawn from her present budget, and the right to move at the turn is won randomly by a player with probability equal to the ratio of her stake and the combined stake just offered. For certain graphs, we present the solution of a leisurely version of the game, in which, after stakes are bid at a turn, the upcoming move is cancelled with probability 1 - ϵ∈ (0,1). With the parameter ϵ small enough, and for finite trees whose leaves are the boundary set and whose payment function is the indicator on a given leaf, we determine the value of the game and the set of Nash equilibria. When the ratio of the initial fortunes of Maxine and Mina is λ, Maxine wins each turn with a probability λ1+λ under optimal play, and game value is a biased infinity harmonic function h(λ,v); each player stakes a shared non-random proportion of her present fortune, a formula for which we give in terms of the spatial gradient and λ-derivative of h(λ,v). We also show with some examples how the solution can differ when ϵ is one.
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