  In two-player games on graphs, the players move a token through a graph to
produce an infinite path, which determines the winner or payoff of the game. We
study {\em bidding games} in which the players bid for the right to move the
token. Several bidding rules were studied previously. In {\em Richman} bidding,
in each round, the players simultaneously submit bids, and the higher bidder
moves the token and pays the other player. {\em Poorman} bidding is similar
except that the winner of the bidding pays the "bank" rather than the other
player. {\em Taxman} bidding spans the spectrum between Richman and poorman
bidding. They are parameterized by a constant $\tau \in [0,1]$: portion $\tau$
of the winning bid is paid to the other player, and portion $1-\tau$ to the
bank. We present, for the first time, results on {\em infinite-duration} taxman
games. Our most interesting results concern quantitative taxman games, namely
{\em mean-payoff} games, where poorman and Richman bidding differ. A central
quantity in these games is the {\em ratio} between the two players' initial
budgets. While in poorman mean-payoff games, the optimal payoff a player can
guarantee depends on the initial ratio, in Richman bidding, the payoff depends
only on the structure of the game. In both games the optimal payoffs can be
found using (different) probabilistic connections with {\em random-turn based}
games in which in each turn, a coin is tossed to determine which player moves.
The payoff with Richman bidding equals the payoff of a random-turn based game
with an un-biased coin, and with poorman bidding, the coin is biased according
to the initial budget ratio. We give a complete classification of mean-payoff
taxman games using a probabilistic connection. Our results show that Richman
bidding is the exception; namely, for every $\tau <1$, the value of the game
depends on the initial ratio.
