  In this paper we introduce and study {\em all-pay bidding games}, a class of
two player, zero-sum games on graphs. The game proceeds as follows. We place a
token on some vertex in the graph and assign budgets to the two players. Each
turn, each player submits a sealed legal bid (non-negative and below their
remaining budget), which is deducted from their budget and the highest bidder
moves the token onto an adjacent vertex. The game ends once a sink is reached,
and \PO pays \PT the outcome that is associated with the sink. The players
attempt to maximize their expected outcome. Our games model settings where
effort (of no inherent value) needs to be invested in an ongoing and stateful
manner. On the negative side, we show that even in simple games on DAGs,
optimal strategies may require a distribution over bids with infinite support.
A central quantity in bidding games is the {\em ratio} of the players budgets.
On the positive side, we show a simple FPTAS for DAGs, that, for each budget
ratio, outputs an approximation for the optimal strategy for that ratio. We
also implement it, show that it performs well, and suggests interesting
properties of these games. Then, given an outcome $c$, we show an algorithm for
finding the necessary and sufficient initial ratio for guaranteeing outcome $c$
with probability~$1$ and a strategy ensuring such. Finally, while the general
case has not previously been studied, solving the specific game in which \PO
wins iff he wins the first two auctions, has been long stated as an open
question, which we solve.
