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Robin Hood Reachability Bidding Games

Shaull Almagor, Guy Avni, Neta Dafni

TL;DR

This work introduces Robin Hood bidding games, a wealth-regulated variant of bidding games with redistribution parameter $\lambda\in[0,\tfrac{1}{2})$, and studies reachability thresholds. It proves the existence of a threshold function $\mathsf{Th}$ and provides a MILP-based method to compute it, while showing that, unlike standard bidding games, the outcome may be undetermined exactly at the threshold. The analysis hinges on the Average Property and a reduction from general graphs to DAGs, and reveals a rich, piecewise dependence on $\lambda$, including a discontinuity at $\lambda=\tfrac{1}{4}$. The results yield both algorithmic tools and theoretical insights for regulated bidding dynamics and suggest directions for extending to infinite-duration settings and alternative wealth-regulation mechanisms.

Abstract

Two-player graph games are a fundamental model for reasoning about the interaction of agents. These games are played between two players who move a token along a graph. In bidding games, the players have some monetary budget, and at each step they bid for the privilege of moving the token. Typically, the winner of the bid either pays the loser or the bank, or a combination thereof. We introduce Robin Hood bidding games, where at the beginning of every step the richer player pays the poorer a fixed fraction of the difference of their wealth. After the bid, the winner pays the loser. Intuitively, this captures the setting where a regulating entity prevents the accumulation of wealth to some degree. We show that the central property of bidding games, namely the existence of a threshold function, is retained in Robin Hood bidding games. We show that finding the threshold can be formulated as a Mixed-Integer Linear Program. Surprisingly, we show that the games are not always determined exactly at the threshold, unlike their standard counterpart.

Robin Hood Reachability Bidding Games

TL;DR

This work introduces Robin Hood bidding games, a wealth-regulated variant of bidding games with redistribution parameter , and studies reachability thresholds. It proves the existence of a threshold function and provides a MILP-based method to compute it, while showing that, unlike standard bidding games, the outcome may be undetermined exactly at the threshold. The analysis hinges on the Average Property and a reduction from general graphs to DAGs, and reveals a rich, piecewise dependence on , including a discontinuity at . The results yield both algorithmic tools and theoretical insights for regulated bidding dynamics and suggest directions for extending to infinite-duration settings and alternative wealth-regulation mechanisms.

Abstract

Two-player graph games are a fundamental model for reasoning about the interaction of agents. These games are played between two players who move a token along a graph. In bidding games, the players have some monetary budget, and at each step they bid for the privilege of moving the token. Typically, the winner of the bid either pays the loser or the bank, or a combination thereof. We introduce Robin Hood bidding games, where at the beginning of every step the richer player pays the poorer a fixed fraction of the difference of their wealth. After the bid, the winner pays the loser. Intuitively, this captures the setting where a regulating entity prevents the accumulation of wealth to some degree. We show that the central property of bidding games, namely the existence of a threshold function, is retained in Robin Hood bidding games. We show that finding the threshold can be formulated as a Mixed-Integer Linear Program. Surprisingly, we show that the games are not always determined exactly at the threshold, unlike their standard counterpart.

Paper Structure

This paper contains 18 sections, 11 theorems, 9 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An Enlightening Example
  4. Alternative Tie Breaking
  5. The case $\frac{1}{4}\leq\lambda<\frac{1}{2}$
  6. The case $0<\lambda<\frac{1}{4}$
  7. Correct Tie Breaking
  8. The case $\lambda=\frac{1}{4}$
  9. The case $\frac{1}{4}<\lambda<\frac{1}{2}$
  10. The case $0\leq\lambda<\frac{1}{4}$
  11. Existence of Thresholds
  12. The Average Property
  13. Games Played on Directed Acyclic Graphs
  14. Games Played on General Graphs
  15. Characterization of Thresholds in Terms of the Average Property
  16. ...and 3 more sections

Key Result

corollary 1

There exists a game $\mathcal{G}$ and vertex $v$ such that the threshold function of $v$ is discontinuous as a function of $\lambda$.

Figures (7)

  • Figure 1: A Robin Hood game. The target for is $v_1$.
  • Figure 2: Threshold of $v_{\textrm{left}}$ in \ref{['fig: first example']} as a function of $\lambda$.
  • Figure 3: A losing play for .
  • Figure 4: A DAG (in this case, a tree) and the unique values satisfying the average property, for $n=2$ and $\lambda=\frac{1}{6}$.
  • Figure 5: Infinitely many functions satisfying the average property, for $\lambda=\frac{1}{4}$. The numbers are the thresholds.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Remark 3.1
  • corollary 1
  • Remark 3.2
  • Remark 3.3
  • definition 1
  • lemma 1
  • corollary 2
  • lemma 2
  • proposition 1
  • corollary 3
  • ...and 5 more