This is an entry to the 17th fortnightly topic challenge.
Chess is an interesting game, but is so unrealistic. In real life, you couldn't tell order your knight to attack an enemy bishop and expect it win with 100% certainty. There would be a struggle, and the stronger fighter would be more likely to win, but could still get unlucky and lose.
Monte Carlo Chess is a variant which accounts for the messiness of real life by throwing probability into the mix.
Rules of Monte Carlo Chess
- All pieces move the same way as in regular chess, including en passant and castling.
- Winning: The goal is capture your entire opponent's army.
- Check: You are not required to protect your king when it is in check.
- Power Levels: Every piece has a certain power level. Initially, these are
Note that the power level of a pawn will not change if it is promoted.
Capturing moves only succeed with a certain probability proportional to the attacker's strength. If the attacking piece has a power level of $a$, and the target a level of $b$, then the attacker wins with probability $$\frac{a}{a+b}.$$ If the attacker wins, the move proceeds as normal. If not, then the attacking piece is removed from the board, while the target remains in its original position.
Leveling up: Whenever a piece wins a battle, its power level increases by the amount of the loser. So, if a piece with power level 7 attacks one with level 15, the winner will have a power level of 22. Pencil and paper can be used to keep track of the power levels of the remaining pieces.
50 moves rule and stalemate: If 50 turns happen without any pawn moves, castling, or captures, the game ends, and the winner is decided by chance. This also happens if a player cannot make any legal move when it is their turn. Specifically, white wins with probability $$\frac{W}{W+B},$$ where $W$ is the total of the power levels of all white pieces, and $B$ is the black total.
Your task, dear puzzlers, is to solve the game of Monte Carlo chess. That is, when black and white both play optimally, what is the probability of each player winning? And what are their optimal strategies?
Source: This is heavily inspired by the puzzle "Gladiators, Version 1" from Peter Winkler's Mathematical Puzzles: A Connoisseur's Collection.