Interesting. So either closed teams or unrestricted access to the game, and in both cases each player can bet each turn on a move, but they only bet before the game on black or white.
I guess one could test this game with a friend, to see how much one would spend trying to sabotage for the opponent instead of focusing on ones own moves.
The "auction" would have to be blind, move revealed after all bets? Closed teams can at any point brute-force a move, so it becomes a game of not being "one move away from losing."
First two moves would see 0 bets. Then it would be a matter of all-in when game is 2 moves from check-mate. The only question is how much do you bet on the first, and how much on the second move.
Say black is check mate if black moves X and white moves Y. And let us say both teams know this. black would need to have more funds that white when it is white's turn if black moves X. So either they go all in, sure to avoid black to X. Or they save ALL their cash to make white move some other way. Or they have to gamble, ...