NESTClass: What is martingale?

NEST is a project based on game theory. In game theory, there is an interesting concept called martingale.

What is martingale?

The name martingale comes from the French town of Martique, where the inhabitants are so careful with their money that they estimate how much they will spend this week based on how much they will spend next week. The small amount of money they will spend next week is most likely to equal the amount they spend today. A French mathematician was so impressed by the hardworking spirit of the people of the town that he simply named the theory he had developed after Martingale.

So, in the eyes of mathematicians, the money spent by the town’s inhabitants on one day next week will be expected to equal the money spent today. See what I mean? We have introduced the concept of “expectation”. This concept is also often used in gambling. Why is expectation important?

Here is a perception-bending example for you:

Consider a gambling game that starts with a bet of 1. If you win, you gain the bet and the bet for the next game is reset to 1. If you lose, you lose the bet and the bet for the next game is doubled to 2. Assuming equal probability of winning and losing, ask whether there is a good strategy for this one gambling problem. To be clearer, let me give you an example, let’s say that 4 games are played and game 1 is lost, then $1 will be lost. If you lose again in game 2, then you will lose $2, for a total loss of $3. If you win the 3rd game, you would get $4, making yourself in possession of the bet for $1. Another win in game 4 would win another $1, so a total of $2 would be won.

The first and more important observation is that whenever a person wins then they must net $1, so a natural sense is that the game is in the player’s favour because whenever a player wins once they get a positive return of $1. But is this really the case?

Let’s assume we show 6 games played, first we can analyse the expectation of a gain. The maximum gain would be 6, with a probability of 1 in 64 at this point, or of course it could be a gain of 5 with a probability of 5 in 64, etc. etc. There are many scenarios, so I won’t list them here. Anyway, if you are interested, you can calculate the returns and the corresponding probabilities. If you multiply the payoffs and probabilities to get the expected value, we find that the expectation happens to be 0. So, the first thing you can see is that the game is not very profitable in the long run.

So, it may seem that this game design is beneficial to the players, but in fact, it is fair to both sides of the game. Martingale has a strict mathematical definition, but in layman’s terms, martingale is the key to measuring a fair game.

When dealing with uncertain returns, we must trade using a martingale information flow to achieve a fair result.

Martingale brings us fairness.