  Bidding chess is a chess variant where instead of alternating play, players
bid for the opportunity to move. Generalizing a known result on so-called
Richman games, we show that for a natural class of games including bidding
chess, each position can be assigned rational upper and lower values
corresponding to the limit proportion of money that Black (say) needs in order
to force a win and to avoid losing, respectively.
  We have computed these values for all three-piece endgames, and in all cases,
the upper and lower values coincide. Already with three pieces, the game is
quite complex, and the values have denominators of up to 138 digits.
