1. Bidding Chess

In chess, positions with only three pieces (the two kings and one more piece) are perfectly understood. The only such endgame requiring some finesse is king and pawn versus king, but even that endgame is played flawlessly by amateur players. In this article we investigate a chess variant where already positions with three pieces exhibit a complexity far beyond what can be embraced by a human.

Bidding Chess is a chess variant where instead of the two players alternating turns, the move order is determined by a bidding process. Each player has a stack of chips and at every turn, the players bid for the right to make the next move. The highest bidding player then pays what they bid to the opponent, and makes a move. The goal is to capture the opponent's king, and therefore there are no concepts of checkmate or stalemate.

As the total number of chips tends to infinity, there is in each position a limit proportion of chips that a player needs in order to force a win. We have computed these limits for all positions with three pieces, and the results (see for example Figure 7) show that already with such limited material, the game displays a remarkable intricacy.

Similar bidding games were introduced by David Richman in the 1980s, and presented in [4; 5] only after his tragic death. Bidding chess has been discussed in [1; 2; 3].

There are various reasonable protocols for making bids and handling situations of equal bids [3].