Question

Gambler's ruin chain:
Suppose a person decides to participate in the following game: a fair coin is tossed and the person bets on the face that would fall, if you hit then you win the same amount as you bet and your bet is returned, otherwise you lose your bet. The player follows a strategy where they bet all their money in case of having $ 5 or less and Bet $ 1 otherwise, stopping playing when your capital is $ 10 or goes bankrupt. If the initial capital Player's is $ 2. What is the probability that the player will fold with $ 10?

Answer 1

The Gambler's Ruin Problem is a case of probability computation in a random walk. To understand the method of computation, the following steps are required:

• The transition rates are state-dependent, hence consider the following transition matrix (A):

Here, each entry aij gives the probability of moving from state i to state j in a single step, given that the current state is i. The probability of moving in one step from state 1 to state 0, for instance, is b + c while the probability of moving in one step from state 1 to state 2 is a; there is no probability of moving anywhere else in one step from state 1. By assumption, a + b + c = 1 since we must move somewhere.

• The corresponding Markov chain for state dependent transition states is shown below:

• We can solve for the respective probabilities pk by solving the following matrix system:

Here, pk = probability (eventual absorption at state N + 1 / initially at state k)

qk = probability (eventual ruin / initially at state k) = 1 - pk

• Here, pk can be obtained as a recursive function of p1, as follows:

• Upon solving the recurison polynomial, we obtain the following expression of pk, which is fairly simplified:

Here, N=10, k=9. Fixing the said variables, the required probability can be obtained.

The important takeaway from this question is the method of obtaining a general expression of the required probability for a random walk via a generic approach which can be utilised in other variations of the same problem as well. Thank you!