Today I learnt about an interesting gambling problem which has a lot usages in life. Indeed, our lives has so many things to gambling on, so many thing to trade of. To maximize your chance of winning you can take a look at this Theory and it might save you a fortune!
Problem Statement
The Gambler’s Ruin Problem in its most basic form consists of two gamblers A and B who are playing a probabilistic game multiple times against each other. Every time the game is played, there is a probability p (0 < p < 1) that gambler A will win against gambler B. Likewise, using basic probability axioms, the probability that gambler B will win is 1 — p. Each gambler also has an initial wealth that limits how much they can bet. The total combined wealth is denoted by k and gambler A has an initial wealth denoted by i, which implies that gambler B has an initial wealth of k — i. Wealth is required to be positive. The last condition we apply to this problem is that both gamblers will play indefinitely until one of them has lost all their initial wealth and thus cannot play anymore.
Imagine that gambler A’s initial wealth i is an integer dollar amount and that each game is played for one dollar. That means that gambler A will have to play at least i games for their wealth to drop to zero. The probability that they win one dollar in each game is p, which will be equal to 1/2 if the game is fair for both gamblers. If p > 1/2, then gambler A has a systematic advantage and if p < 1/2 then gambler A has a systematic disadvantage. The series of games can only end in two outcomes: gambler A has a wealth of k dollars (gambler B lost all their money), or gambler A has a wealth of 0 dollars (gambler B has all the wealth). The main focus of the analysis is to determine the probability that gambler A will end up with a wealth of k dollars instead of 0 dollars. Regardless of the outcome, one of the gamblers will end up in financial ruin, hence the name Gambler’s Ruin.
Problem Solution
Continuing with the same structure outlined above, we now want to determine the probability aᵢ that gambler A will end up with k dollars given that they started out with i dollars. For the sake of simplicity, we will add an additional assumption here: all games are identical and independent. Every time the gamblers play a new game, it can be interpreted as a new iteration of the Gambler’s Ruin problem where the initial wealth of each gambler are different, depending on the outcome of the most recent game. Mathematically, we can think of each sequence of games that ends in gambler A having j dollars where j = 0, … , k. The probability gambler A wins given that a specific sequence occurs is aⱼ. Any sequence that ends in gambler A having k dollars means that they won, so aₖ=1. Likewise, any sequence than ends in gambler A having 0 dollars means that they are in ruin so a₀=0. We are interested in determining the probabilities for all values of i=1, … , k — 1.
Using event notation, we can denote A₁ as the event that gambler A wins game 1. Similarly, B₁ is the event that gambler B wins game 1. The event W occurs when gambler A ends up with k dollars before they end up with 0 dollars. The probability that this event occurs can be derived using properties of conditional probability.
Given that gambler A starts with i dollars, the probability that they win is P(W)=aᵢ. If gambler A wins one dollar in the first game, then their wealth becomes i+1. If they lost one dollar in the first game, their fortune becomes i — 1. The probability that they win the entire sequence will then depend on whether they won the first game. When we apply this logic to the previous equation, we know have an expression of the probability of winning the entire sequence of games that depends on the probability of winning one dollar each game and the conditional probabilities of winning the sequence given the gamblers wealth.
The wealth of gambler A at any given point in time varies between the total initial wealth of both gamblers and zero. In other words, given i=1, … , k — 1, we can plug in all possible values of i into the equation above to obtain k — 1 equations that determine the probability of winning based on adjacent values of i. We can use elementary algebra to aggregate these equations into a standardized format that can be simplified into a single formula. This formula specifies a fundamental relation between the probability of winning each game p, the total initial wealth of both gamblers k, and the probability of winning given a wealth of one dollar a₁. Once we have determined a₁, we can iteratively loop through all k — 1 to derive the probability aᵢ for all possible values of i.
We will now consider two possibilities: a fair game and an unfair game, which depends on the value of p that we plug into the equation above. In a fair game, where p=1/2, the base of the exponent in the right side of the equation can be simplified to (1 — p)/p=1. We can then simplify the entire equation as follows: 1 — a₁=(k — 1)a₁, which can be rearranged to a₁=1/k. If we iterate over all previous equations that determine the probabilities of winning for different values of i, we arrive at a general solution for fair games.
The equation above is a remarkable result: given that the game is fair, the probability that gambler A will end up with k dollars before they end up with zero dollars is equal to their initial wealth i divided by the total wealth of both gamblers k.
If p is not equal to 1/2 the game is unfair, as one of the two gamblers has a systematic advantage. We can similarly derive a general solution which depends on the value of p and the wealth parameters.
