# Terminology for a game in which Black and White have the same "probability" to win

Consider a game between two players, Black and White. The game is sequential and ends after finitely many moves. White moves first. The game ends either in the victory of one of the players or in a draw.

Let us call "probability for White to win" the ratio between the number of games that end in a victory of White and the total number of games.

Is there a standard terminology for those games in which Black and White have the same probability to win, i.e. the number of games won by White equals the number of games won by Black, i.e. there is no advantage in playing White?

Moreover, which games (are known to) fall into this category and which not, among chess, draughts, Go and so on?

• Commented Jan 6, 2016 at 13:55
• Chess, draughts, and go are all perfect-information games: if both sides play perfectly, there is no probability involved. The game is either a win, a draw, or a loss for the first player (we do know that 8x8 draughts is a draw when played perfectly by both sides). And if you are asking whether such games are fair when played by imperfect primates like us, that is not a mathematical question. Commented Jan 6, 2016 at 14:03

The answer to this related question is excellent.

A fair game is probably closest to what you want. However, under optimal play, the game you describe, which is sequential and finite, must always end in either black winning, white winning, or in a draw. A game that always ends in a win for one of the players is sometimes called unfair. A game that always ends in a draw is sometimes called futile but might also be called fair since it gives neither player an advantage.

That idea that white and black have equal chance of winning implies there must be randomness in your game. Checkes, chess, go have no such randomness and are either unfair or futile. Checkers ends in a draw. Chess and go are unsolved.

It seems that you have two or three different concepts in your question, and I'm not sure which you intended to focus on.

there is no advantage in playing White

If a game has no hidden information and no randomness and has a designated starting player and ends in finitely many moves (like Go or Chess under certain rules barring infinite repetitions), then either white has a winning strategy (even if no one knows what it is), or black does, or both can force a draw. Since Go has no draws, Go can't be fair in this sense: either black or white has the advantage under perfect play. Chess has ties in the form of stalemate, so maybe Chess is "a tie under perfect play".

the number of games won by White equals the number of games won by Black

This is meaningful, but not a very interesting definition. Consider a game in which Black selects "B" or "W" and White selects "1" or "2", and the letter Black chose determines the winner. Certainly Black wins 2 of the 4 possible games, but this game is very unfair since black can win easily. If you seat to include games like this, I doubt there's a name for this.

probability for White to win

You can talk about this, but probability would be in the context of some sort of randomized play. For example, you could take the top 100 Chess players according to FIDE, and speak of the theoretical probability of white winning if two of those players were selected uniformly randomly to play, but that doesn't necessarily tell you what perfect play would be like. I think in Chess it's suspected that White has the advantage, but in the example of Go, different play styles even among the top 10 players makes it unclear which color might have the advantage under perfect play.

There is no single word, you could use:

• unbiased
• nonpartisan
• fair
• neutral
• objective
• even handed
• equitable
• impartial
• Nonpartisan might be confused for the technical terms "impartial" or "not partizan" in combinatorial game theory. Commented Jan 6, 2016 at 14:07
– JMP
Commented Jan 6, 2016 at 14:09
• of course, that's a fine definition of nonpartisan, and applies here. If this were English.SE, I wouldn't have said anything; I just wanted to warn people interested in the mathematical study of these games that that word in particular may lead to confusion. Commented Jan 6, 2016 at 14:25
• i wouldn't know
– JMP
Commented Jan 6, 2016 at 14:29

The other answers are pretty good, but I wanted to resurrect this 8-year-old post to point out that your usage of the term "probability" that white will win is too vague to be worked with, which is why the other answers waffle a bit.

Since this is a math exchange post, we can discuss that issue formally. The crux of the problem with your question is that your usage of the term "probability" implies a probability space. This is a triplet, say $$(\Omega, \mathcal{F}, P),$$ where $$\Omega$$ is the set of outcomes, $$\mathcal{F}$$ is a $$\sigma$$-algebra on $$\Omega$$ for which probabilities are defined, and $$P$$ is a probability measure assigning probabilities to elements of $$F$$. (You can learn more about these by studying Kolmogorov's axioms of probability.) Once we specify a probability space, we can answer the terminology question more precisely.

I'll enumerate what I see as possible probability spaces to frame your question. We'll assume a game with a finite game tree, like chess, so we can ignore $$\Omega$$ and $$\mathcal{F}$$ (since we'll take $$\Omega$$ to be the set of all possible game endings and $$\mathcal{F}$$ to be the power set, $$2^\Omega$$) and focus on defining $$P$$. That way, once we have $$P$$, we can compute the "probability that white wins" as $$\mathbb{E}_P[\mathbb{1}\{\text{white wins}\}].$$

We'll also assume that the game is deterministic (no element of chance like poker has), with no hidden information, like battleship has.

1. All games are equally likely--$$P(\omega) = \frac{1}{|\Omega|}, \forall \omega \in \Omega$$. This is the most basic interpretation of how to assign chess games a probability. It's more or less equivalent to saying, what are the odds white wins if both players play randomly?

For chess, this is not known, but it's also not very interesting, because we assume the players are not just playing randomly. If white wins 99% of games when they play randomly but black can force a win by playing intelligently, the practical chance white wins is 0 once black knows the winning strategy. This is what Mark S. is getting at in part of his answer:

Consider a game in which Black selects "B" or "W" and White selects "1" or "2", and the letter Black chose determines the winner. Certainly Black wins 2 of the 4 possible games, but this game is very unfair since black can win easily. If you seat to include games like this, I doubt there's a name for this.

So there is no terminology for a game where white wins in exactly 50% of the leaf nodes of a game tree.

1. If we assume players move at least somewhat intelligently, then we can define $$P$$ to allocate more probability mass to games that would actually happen between two humans. This distribution could only be estimated by looking at large databases of human games as an approximation for all possible human games.

In this case, we might colloquially call the game fair if the expectation above is 0.5, or, in other words, if white wins in half of all realistic games between two humans. If I have the sense that as an average person, I go into a game with a 50% chance to win against an average opponent, it feels fair to me.

But only to me. What if win ratios differ with skill levels? Or between different opponents? Or over time? In chess, we see a phenomenon where black and white are quite equal for average players, but white is increasingly better than black as ratings rise. In some sense, we feel that the better a player is at the game, the more qualified they are to say if it is fair or unfair as a whole, rather than to them personally. This leads to

1. A scenario where we weight games by how likely they are to occur between two perfect opponents. Since the games are fully determined and perfectly known by the players, there is a very limited set of games that could occur, and they will all share the same outcome. If white can force a win, they will never choose not to, and the same for black; but if draws are possible and the otherwise losing player can force one, they will do that every time. So, the "probability that white wins" is either 100% or 0%, depending on the game.

A game with a 100% chance of one player winning with perfect play is called unfair, while a game where players can force a draw is called futile or else fair. This dilemma between futility and unfairness is exactly what makes tic-tac-toe uninteresting. It would make chess, go, etc. uninteresting as well, except that the game trees are so large that perfect play is next to impossible, simulating an element of randomness where none exists--because, in the end, the only way a game can be fair and still have a winner is by incorporating an element of randomness. Ironically, games with randomness often prompt players to call the game unfair anyway, but in a different sense--the game is unfair to all players, rather than favoring one player over another.

TL;DR: If you specify your question more thoroughly, it can be answered more precisely. Anyway, thanks for an interesting question, math stack exchange rando from 2016 ;)