# The expected outcome of a random game of chess?

Imagine a game of chess where both players generate a list of legal moves and pick one uniformly at random.

Q: What is the expected outcome for white?

• 1 point for black checkmated, 0.5 for a draw, 0 for white checkmated. So the expected outcome is given by
• Neither player resigns, nor are there any draw offers or claims.

As a chess player, I’m curious if white (who plays first) has some advantage here.

I’m not expecting an exact answer to be possible. Partial results (e.g. that the expectation is >0.5) and experimental results are welcome. (The expectation is not 0 or 1, since there are possible games where white does not win and where black does not win.)

I’m guessing this has been looked at before, so I decided to ask first (rather than implement a chess engine that makes random moves and hope to find something other than “draw, draw, draw, draw, …”). Searching for “random game of chess” lists Chess960 and other randomized variants, which is not what I want.

Technicalities:

• En passant capturing, castling, pawn promotion, etc. all apply as usual.

• The FIDE Laws of Chess will be updated 1 July 2014 with the following:

9.6 If one or both of the following occur(s) then the game is drawn:

• a. the same position has appeared, as in 9.2b, for at least five
consecutive alternate moves by each player.

• b. any consecutive
series of 75 moves have been completed by each player without the
movement of any pawn and without any capture. If the last move
resulted in checkmate, that shall take precedence.

This means that games of chess must be finite, and thus there is a finite number of possible games of chess.

I found a bug in the code given in Hooked’s answer (which means that my original reanalysis was also flawed): one also have to check for insufficient material when assessing a draw, i.e.

int(board.is_stalemate())


should be replaced with

int(board.is_insufficient_material() or board.is_stalemate())


This changes things quite a bit. The probabillity of a draw goes up quite a bit. So far with $$n=5⋅105n = 5\cdot 10^5$$ samples I find

$$E[Black]≈0.5002E[\text{Black}] \approx 0.5002$$
$$E[White]≈0.4998E[\text{White}] \approx 0.4998$$
$$P[Draw]≈84.4%P[\text{Draw}] \approx 84.4\%$$

A simple hypotesis test shows that with $$P(white)=P(black)=0.078, P(draw)=0.844P(\text{white})=P(\text{black})=0.078,~P(\text{draw})=0.844$$ and $$N=5⋅105N=5\cdot 10^5$$ samples the probabillity to get $$|E[Black]−0.5|>0.002|E[\text{Black}] - 0.5| > 0.002$$ is $$25%25\%$$ so our results are perfectly consistent with $$E[White]=E[Black]=0.5E[\text{White}]=E[\text{Black}]=0.5$$. The “hump” remains, but is now easily explained: it is due to black or white winning. Either they win early or the game goes to a draw.

(source: folk.uio.no)

Here is one of the shortest game I found, stupid black getting matted in four moves:

(source: folk.uio.no)