The Wikipedia article is here
I’m going to assume at this point that you have no knowledge of game theory. In what follows, I’m going to provide a somewhat simplistic explanation, not because I want to patronise you but because this may all be quite new, and the Wikipedia article is far from simple.
Take the very simplest game: a coin toss. One player tosses a coin, the other calls heads or tails. If the caller is right, she wins. If the caller is wrong, the one who tosses the coin wins. If the coin doesn’t come down or ends up on its edge, the game doesn’t count.
In probability theory, the total probability that every possible allowed outcome of an event happens is equal to 1.
In this case there are 2 events: heads or tails. If the coin is perfectly fair, therefore, the probability of heads is 0.5, and so is the probability of tails. 0.5 + 0.5 = 1.
Even if the coin is not perfectly fair, the probability of heads plus the probability of tails is 1. If in 100 coin tosses heads comes up 40 times, then tails has come up 60 times.
This is called a zero sum game. If one player wins, the other player loses.
In fact a zero sum game is a bit more complicated than that. For the game to be zero sum, the players need to bet on the result. If they each start with £5, and the loser pays the winner 10p, the game can continue until one player runs out of money. The total amount of money involved, however, remains at £10 no matter what.
There are other sorts of game which are not zero sum; for instance a poker tournament where, in addition to their own bets, players compete to win prizes. We won’t get on to these for quite a while.
If a zero sum game is a game of pure chance, like coin tossing, it isn’t very interesting. There are only 2 possible results. What makes even coin tossing more interesting is that human beings are not rational about games.
Let’s start with a completely fair coin and throw it three times. Each time it comes up heads. What are the chances of this happening?
The first time the chance is 0.5. The second time it is also 0.5, and again for the third. To work out the overall chance we multiply together those probabilities – 0.5 x 0.5 * 0.5 = 0.125 or 1/8. One time in 8, we expect to get three heads in a row.
So, after three heads, what is the probability of a fourth head? For our fair coin, the answer is again 0.5. The coin has no memory. Each time it starts from scratch.
But human beings often don’t reason like this. They either think that heads has come up too often and it is now time for the coin to come up tails, or they think that something is biasing the coin to come up heads. The former is the commonest “explanation”.
In the case of coin tossing it doesn’t really matter. As the chance of heads or tails is identical each time, the player has exactly the same chance of winning whichever is chosen. Guessing heads or tails after a run of heads does not affect the chance of winning.
But with games in which two players each make a choice, this is not the case. As we will see, in such games the player who understands the mathematics can have an advantage.
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