# fair and unfair games

so, as far as i understand there are two types of mathematical games: fair and unfair.

fair games are games where both (all) players have exactly the same chance of winning (outcome of the game is not affected by the order of players taking turns). i'd say, if there is pure luck involved - it's most likely to be a fair game. good examples could be backgammon and russian roulette.

unfair game are those where there is a distinction between who moves first which affects who wins (outcome of the game is affected by the order of players taking turns). like, for example, tic-tac-toe: second player, if the game is played perfectly, can never win, he can force a draw at the most. The same applies to abacus game (sorry, i don't know the real name for it. the rules are the following: players take turns at selecting one line on the abacus and picking any amount of stones on it. the player, that picks the last stone - loses.): second player, if played perfectly, always loses.

so, basically, the rule i read somewhere is "in unfair game, a player with less starting advantage always loses or, at best, forces a draw".

is that right?

second part of my question is this: how about more complex games, like checkers, go, reversi and chess?

as far as my research went - checkers is an unfair game: when played perfectly, it always ends in a draw. ok. the most questionable is chess - as i read, it is considered fair, but mostly just because it is to complex (like centuries at our calculation capabilities) to calculate a perfect match. is it so? because it sounds kinda artificial. Also, statistics show, that whites (player 1) have a little bit more chance to win.

PS. of course, all psychological aspects should be left aside. question is purely mathematical: i think about it as "if two computers would play it as a perfect game, which one would win - player 1, or player 2"

• In a on-player game such as Russian roulette (?!?!?!), all players have trivially the same chance to win (or survive). What you call the "abacus game" is known as Nim. – Michael Greinecker Mar 20 '12 at 8:55
• There is nothing unfair about tic-tac-toe. With best play, every game is a draw. – Gerry Myerson Mar 20 '12 at 12:21

Fairness in a game is usually defined as: both (all $n$) players have the same theoretical chance of winning. That is, if both players play with perfect strategy - both players are perfectly skilled.

In reality, fairness doesn't matter, because players may play imperfectly (i.e. make a mistake)

Draws do not really count towards fairness, because neither player wins.

I've tried to analyse each game that you suggest.

Backgammon: Not known, but likely unfair.

It's very likely that one of the players has a slight advantage. It could be that the first player has first-move advantage, or it could be that the second player has an advantage from having more information about his opponents move. These advantages are unlikely to balance perfectly.

This is difficult to analyse, however, because of the large number of random dice rolls that are possible during a game.

The randomness of Backgammon makes this different than the other games - which are 'deterministic'.

In Backgammon, even if the first player has an advantage, they might still lose through bad luck. In the deterministic games below, if the first player has an advantage, they won't give it up unless they make a mistake.

Russian Roulette: Fair (with 2, 3 or 6 players, but only theoretically).

Both/all players have an equal chance of death. However, the consequences of Russian Roulette are so dire that players are highly motivated to cheat. There may be out-of-game consequences that could be asymmetrical for the surviving players, making it difficult to say that the game is fair in reality.

Tic-tac-toe: Fair

Tic-tac-toe, played perfectly, will always end in a draw. So, both players have a 0% chance of winning (the same chance!)

Abacus game/Nim: Unfair

There are no draws in Nim, and no randomness, so one player will always win. If both players play perfectly, either the first player will always win, or the second player will always win, depending on the exact setup.

Checkers: Fair

Checkers has been analysed enough that we know perfect play will always end in a draw - same situation as Tic-Tac-Toe.

Go: not known, but almost certainly unfair

Go has a system of compensating for a perceived level of unfairness to try and make the game "fair". One player receives a chosen number of komi stones at the start of the game. It would be very unlikely that a whole number of komi stones will ever make the game fair. But, Go is hard to analyse so we don't know for sure.

They also might add stones to compensate for difference in your level - it might seem odd to say this, but this is designed to deliberately make the game "unfair", because it gives the weaker player a theoretically better chance of winning! It's "unfair" because the weaker player should win if they play perfectly. But, of course, the weaker player won't play perfectly - the unfairness compensates for the player's weakness!

Reversi: not known, but believed to be fair

Analysis so far suggests that perfect-play games will end in a draw. This would put it in the same category as Tic-Tac-Toe. But, this is not fully analysed yet.

Chess: not known

As you say, in reality, it seems that White wins more often, so it's commonly said that first player (White) has an advantage. However, these games are not played perfectly, so that actually doesn't help us to decide if this game is theoretically "fair".

If the first player really did have a theoretical advantage, then White should be able to win every time, not only some of the time! Because, there's no randomness - only a mistake will change the result.

How to make an unfair game fair

In each case, I have considered whether or not the first player is in a fair position against the second player.

So, we can make each game theoretically "fair" by flipping a (fair) coin to decide who goes first! Then, even if the first player has an advantage, you have a fair chance of playing first :)

• This is an excellent answer. – user12998 Apr 2 '12 at 14:19
• A very interesting case is Hex: Unfair, with the advantage for player one. But we do not know what optimal play looks like in general. – Michael Greinecker Apr 2 '12 at 15:22
• Chess is known to be drawn by perfect play. – Joshua Nov 20 '15 at 23:32
• @Joshua Any reference for this claim??? – J.-E. Pin Jan 6 '16 at 15:43

It seems you read some rather unusual definitions of these terms. Under the usual definition, a two-player zero-sum game is fair if the expected payoff assuming optimal play is zero. Though I doubt that the expected payoff of Backgammon with optimal play is known, it's unlikely to be zero. Most of the games you mention are either known to be unfair or likely to be unfair; regarding Russian roulette, see In Russian roulette, is it best to go first?. [Correction:] I previously claimed that chess and checkers can't be fair; Ilmari rightly points out that they can both end in a draw and that in fact optimal play in checkers has been shown to lead to a draw; so you actually had it the wrong way around; checkers is the only one of the games you mentioned known to be fair.

• Actually, both chess and checkers can end in a draw, so they may be fair if that's the outcome of mutually optimal play (and, indeed, it appears that this is true for checkers). – Ilmari Karonen Mar 20 '12 at 12:45
• @Ilmari: You're right, of course, and thanks for the link; I hadn't heard that checkers was solved. I've corrected the answer. – joriki Mar 20 '12 at 13:23
• I would understand a different definition of 'fair' in this context - that both players have the same expected outcome. This would mean that Russian Roulette, played strictly, is theoretically fair - both players are equally likely to win/lose. The answers suggesting it is not fair are (correctly) assuming your opponent will cheat, or we may also believe there are out-of-game consequences of an untimely death that may be asymmetric. – Ronald Apr 2 '12 at 13:43