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A Stopping Game on Zero-Sum Sequences

Adrian Dumitrescu, Arsenii Sagdeev

Abstract

We introduce and analyze a natural game formulated as follows. In this one-person game, the player is given a random permutation $A=(a_1,\dots, a_n)$ of a multiset $M$ of $n$ reals that sum up to $0$, where each of the $n!$ permutation sequences is equally likely. The player only knows the value of $n$ beforehand. The elements of the sequence are revealed one by one and the player can stop the game at any time. Once the process stops, say, after the $i$th element is revealed, the player collects the amount $\sum_{j=i+1}^{n} a_j$ as his/her payoff and the game is over (the payoff corresponds to the unrevealed part of the sequence). Three online algorithms are given for maximizing the expected payoff in the binary case when $M$ contains only $1$'s and $-1$'s. $\texttt{Algorithm 1}$ is slightly suboptimal, but is easier to analyze. Moreover, it can also be used when $n$ is only known with some approximation. $\texttt{Algorithm 2}$ is exactly optimal but not so easy to analyze on its own. $\texttt{Algorithm 3}$ is the simplest of all three. It turns out that the expected payoffs of the player are $Θ(\sqrt{n})$ for all three algorithms. In the end, we address the general problem and deal with an arbitrary zero-sum multiset, for which we show that our $\texttt{Algorithm 3}$ returns a payoff proportional to $\sqrt{n}$, which is worst case-optimal.

A Stopping Game on Zero-Sum Sequences

Abstract

We introduce and analyze a natural game formulated as follows. In this one-person game, the player is given a random permutation of a multiset of reals that sum up to , where each of the permutation sequences is equally likely. The player only knows the value of beforehand. The elements of the sequence are revealed one by one and the player can stop the game at any time. Once the process stops, say, after the th element is revealed, the player collects the amount as his/her payoff and the game is over (the payoff corresponds to the unrevealed part of the sequence). Three online algorithms are given for maximizing the expected payoff in the binary case when contains only 's and 's. is slightly suboptimal, but is easier to analyze. Moreover, it can also be used when is only known with some approximation. is exactly optimal but not so easy to analyze on its own. is the simplest of all three. It turns out that the expected payoffs of the player are for all three algorithms. In the end, we address the general problem and deal with an arbitrary zero-sum multiset, for which we show that our returns a payoff proportional to , which is worst case-optimal.

Paper Structure

This paper contains 19 sections, 4 theorems, 16 equations, 2 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Example.
  3. Comment.
  4. Our results.
  5. Related work.
  6. Preliminaries
  7. A simple threshold algorithm for the binary case
  8. Algorithm outline.
  9. Algorithm analysis.
  10. Remarks.
  11. An optimal algorithm for the binary case
  12. Algorithm analysis.
  13. Example.
  14. A 'Stop in the Middle' algorithm for the binary case
  15. Algorithm analysis.
  16. ...and 4 more sections

Key Result

Theorem 1

Let $M=\{a_1,a_2,\ldots,a_n\}$ be an arbitrary zero-sum multiset of reals. There is an online algorithm that reads a random permutation of $M$ and stops with an expected payoff of $\Omega(\mu \sqrt{n})$, where $\mu =\sum_{i=1}^n |a_i|/n$ is the average absolute value for the input sequence. This bou

Figures (2)

  • Figure 1: Left: a lattice path from $(x,y)=(0,0)$ to $(6,6)$ that reaches the red line $y=x-2$ (left), and one that does not (right). Here $n=6$. The line $y=x$ is in blue.
  • Figure 2: Left: an optimal strategy for $n=8$: $T[0,0]=1$, $T[1,0]=47/35$, $T[2,0]=2$, $\ldots$, $T[4,4]=0$. Right: an optimal strategy for $n=20$; to avoid the clutter, only the reachable stopping states are shown. Note that all stopping states are below the main diagonal (with one exception, the zero entry for the automatic stop in the end).

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4