Chance of winning a game of hearts with four players

Question

I play hearts with a computer game program. The game is set up so that four people are playing the game. The question is: What are the mistakes, if any, with assuming that the probability of winning a game of hearts is 1 in 4 or 25%? This solution need not be restricted to include just playing with a computer. It needs to be answered as well with the possibility that people could be playing the game with each other without the aid of a computer. I would appreciate any help with answering my question. George

Answer 1

Suppose you fix a (possibly randomized) strategy for the four players $A,B,C,D$ and that the first player is chosen randomly. Suppose also that we choose some tie breaking rule so that each game has exactly one winner. Then both of the following are true:

1. If players $A,B,C,D$ all have the same strategy, then each of them wins with probability $1/4$.
2. Let $P$ be a random player chosen among $A,B,C,D$. Then $P$ wins with probability $1/4$.

What is not true in general is that the probability that the different players win is the same.

Here is a simpler example to illustrate this situation. Consider the following game, involving two players $A,B$. Each of the players chooses a number, either $1$ or $2$. The winner is the player choosing the largest number. In the case of a tie, the winner is chosen randomly. In this case, it is easy to check that if $A,B$ choose the same strategy, then they both win with probability $1/2$ (here a (randomized) strategy specifies the probability that the player chooses $1$); this follows from the symmetry of the situation. However, if $A$ chooses $1$ always and $B$ chooses $2$ always, $B$ will always win. Even in this case, however, the probability that a random player $P$ chosen among $A,B$ wins is exactly $1/2$, since we choose the loser $A$ with probability $1/2$ and the winner $B$ with probability $1/2$.