Fair Game

In statistics, a fair game is a scenario where neither player has any advantage over the other.

Simply put, in a fair game, the amount each player bets should be proportional to their chance of winning.

$$ S(A):p(A)=S(B):p(B) $$

Here, S(A) and S(B) represent the amounts bet by players A and B, respectively, while p(A) and p(B) are their corresponding probabilities of winning.

This means that, given the probability of each outcome and the associated bet, no player holds a mathematical advantage.

A practical example

Let’s take the example of rolling a six-sided die.

The die is fair, meaning each side has an equal chance of appearing, which is 1/6​ for each face.

Player A wins if a 6 is rolled, while player B wins if any other number (1, 2, 3, 4, or 5) comes up.

To make this game fair, the amount bet by each player must be proportional to their respective chances of winning.

$$ S(A):p(A)=S(B):p(B) $$

Or,

$$ \frac{S(A)}{p(A)} = \frac{S(B)}{p(B)} $$

The probability of player A winning is p(A)=1/6 because there is a 1 in 6 chance that the die will show a 6.

On the other hand, player B’s chance of winning is p(B)=5/6 since any other number (1, 2, 3, 4, or 5) would result in a win for them.

 

$$ \frac{S(A)}{ \frac{1}{6} } = \frac{S(B)}{ \frac{5}{6} } $$

$$ 6 \cdot S(A) = \frac{6}{5} \cdot S(B) $$

$$ 6 \cdot S(A) \cdot \frac{5}{6} = \frac{6}{5} \cdot S(B) \cdot \frac{5}{6} $$

$$ 5 \cdot S(A) = S(B) $$

This shows that player B’s bet needs to be five times the amount of player A’s bet.

So, if player A bets 10 with a 1/6 chance of winning, player B should bet 50 with a 5/6 chance of winning to ensure the game is fair.

 

And so on.