Maximizing the discounted survival probability in Vardi's casino

Robert Chen, Ilie Grigorescu, Larry Shepp

Research output: Contribution to journalArticlepeer-review


We are in a subfair casino, with fortune f 0 ∈ and we want to turn it into a fortune of size 1 in discounted time. We may stake any amount 0<s≤f if our fortune is f at any time, n≥0, and at any odds, r>0. We assume that every such gamble has a pay-off with fixed expected value cs, with -1<c<0. The problem is to determine the maximum expected discounted pay-off V(f 0;b,c) = max s,rE S,R f0B τX(f τ = 1), 0<b<1, where the maximum is taken over all choices of stakes and odds, S are stakes, and R are odds, and we get a positive pay-off only if our final fortune is 1. We determine the explicit recurrence satisfied by the unique optimal strategy and discounted pay-off. The optimal function is piecewise smooth in intervals [φ n-1n], n≥1, φ 0 = 0 between critical fortunes φ n ↑ 1.

Original languageEnglish (US)
Pages (from-to)623-638
Number of pages16
Issue number4-6
StatePublished - Aug 2011


  • Vardi casino
  • bold play
  • optimal strategy

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation


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