# A Reluctant Random Walk

A Reluctant Random Walk

There are cards labeled 1, 2, ..., . Initially, the cards are randomly divided between two players, A and B. For example, if is even, each player could get half of the cards. At each move of the game, one of the numbers from 1 to is chosen at random and the player with this card must give it to the other player. The game continues until one of the player has all the cards. This Demonstration shows games for one of the players, with expected duration and probability of ruin.

n

n

n

n

This process is a discrete-time Markov chain with two absorbing barriers. The corresponding walk is said to be reluctant because the process hesitates before stopping. Indeed, the more the walker approaches one of the absorbing barriers, the lower the probability becomes that the next move is toward this barrier. The result is that the game lasts a long time on average.