ENSPM201 | 15 julho | Persi Diaconis

Gambler's Ruin and the ICM

Abstract: Picture three gambler’s with A, B and C respectively.Eachtime,apairofgamblersarechosenatrandom,afaircoinisflippedand1 is transferred as the coin comes up heads or tails. Eventually, one of the gamblers is eliminated and two play the game until one winds up with A+B+C. In joint work with Stew Ethier, we study ‘what happens?” how long until the first gamber is eliminated? When this happens, how is the amount left distributed? And, how do the answers depend on A, B, C? These questions are interesting in tournament poker where the chances depend on the initial stack sizes, A,B,C but the winner and runner up get fixed prizes. then the chances of the different elimination orders (eg the chance that say 3 is eliminated first then 1 leaving 2 with all the cash) become important. If you want a puzzle, suppose A=B=1 and C=998 what is the chance that C is eliminated first?
The mathematical tools used are adapted from classical geometric analysis: Whitney covers, John and inner regular domains, moderate growth, Nash-Sobolev estimates. Of course with a good dose of probability theory mixed in. Actual computation involves a sojourn into modern numerical analysis. I will try to explain things to a general audience. Throughout, results are illustrated with data from the final tournaments in the World Series of Poker.