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Trapping the Ultimate Success

Alexander Gnedin, Zakaria Derbazi

Abstract

We introduce a betting game, where the gambler aims to guess the last success epoch from past observed data. The player may bet on the event that no further successes occur, or choose a `trap' which is any span of future times. In the latter case winning is achieved if the last success turns out to be the only one falling in the trap. The game is closely related to the sequential decision problem of maximising the probability of stopping on the last success in a finite sequence of trials. We use this connection to analyse the problem of stopping at the last record for trials paced by a Polya-Lundberg process with log-series distribution of the total number of trials.

Trapping the Ultimate Success

Abstract

We introduce a betting game, where the gambler aims to guess the last success epoch from past observed data. The player may bet on the event that no further successes occur, or choose a `trap' which is any span of future times. In the latter case winning is achieved if the last success turns out to be the only one falling in the trap. The game is closely related to the sequential decision problem of maximising the probability of stopping on the last success in a finite sequence of trials. We use this connection to analyse the problem of stopping at the last record for trials paced by a Polya-Lundberg process with log-series distribution of the total number of trials.

Paper Structure

This paper contains 19 sections, 9 theorems, 77 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions
  3. The probability model
  4. The trapping game and stopping problem
  5. The fixed-$n$ game
  6. Discrete time
  7. Monotonicity in $n$.
  8. Fixed $n$, trapping in continuous time
  9. Examples
  10. The best choice problem.
  11. The Karamata-Stirling profile.
  12. Random number of trials
  13. Tests for the monotone case of optimal stopping
  14. Unimodality and concavity
  15. The best choice problem under the log-series prior
  16. ...and 4 more sections

Key Result

Theorem 1

The optimal trapping strategy on $n$ trials is a $z$-strategy, where $z$ is the unique mode of $S_1( k,n;\cdot)$. The mode is $0$ in the case (trivial), and otherwise $z\in(0,1)$.

Figures (5)

  • Figure 1: Bernstein polynomials for $p_k=1/k$.
  • Figure 2: Bernstein polynomials for $p_k=\theta/(\theta+k-1)$.
  • Figure 3: Concavity (\ref{['concave']}) holds for the region between parabolas.
  • Figure 4: Bounds on the optimal strategy $I_k(x)$
  • Figure 5: Stop and Continuation Values

Theorems & Definitions (17)

  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 7 more