  In tug-of-war, two players compete by moving a counter along edges of a
graph, each winning the right to move at a given turn according to the flip of
a possibly biased coin. The game ends when the counter reaches the boundary, a
fixed subset of the vertices, at which point one player pays the other an
amount determined by the boundary vertex. Economists and mathematicians have
independently studied tug-of-war for many years, focussing respectively on
resource-allocation forms of the game, in which players iteratively spend
precious budgets in an effort to influence the bias of the coins that determine
the turn victors; and on PDE arising in fine mesh limits of the constant-bias
game in a Euclidean setting.
  In this article, we offer a mathematical treatment of a class of tug-of-war
games with allocated budgets: each player is initially given a fixed budget
which she draws on throughout the game to offer a stake at the start of each
turn, and her probability of winning the turn is the ratio of her stake and the
sum of the two stakes. We consider the game played on a tree, with boundary
being the set of leaves, and the payment function being the indicator of a
single distinguished leaf. We find the game value and the essentially unique
Nash equilibrium of a leisurely version of the game, in which the move at any
given turn is cancelled with constant probability after stakes have been
placed. We show that the ratio of the players' remaining budgets is maintained
at its initial value $\lambda$; game value is a biased infinity harmonic
function; and the proportion of remaining budget that players stake at a given
turn is given in terms of the spatial gradient and the $\lambda$-derivative of
game value. We also indicate examples in which the solution takes a different
form in the non-leisurely game.
