Voronoi choice games

Meena Boppana, Rani Hod, Michael Mitzenmacher, Tom Morgan

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We study novel variations of Voronoi games and associated random processes that we call Voronoi choice games. These games provide a rich framework for studying questions regarding the power of small numbers of choices in multi-player, competitive scenarios, and they further lead to many interesting, non-trivial random processes that appear worthy of study. As an example of the type of problem we study, suppose a group of n miners (or players) are staking land claims through the following process: each miner has m associated points independently and uniformly distributed on an underlying space (such as the unit circle, the unit square, or the unit torus), so the kth miner will have associated points pk1, pk2, . . . , pkm. We generally here think of m as being a small constant, such as 2. Each miner chooses one of these points as the base point for their claim. Each miner obtains mining rights for the area of the square that is closest to their chosen base; that is, they obtain the Voronoi cell corresponding to their chosen point in the Voronoi diagram of the n chosen points. Each player's goal is simply to maximize the amount of land under their control. What can we say about the players' strategy and the equilibria of such games? In our main result, we derive bounds on the expected number of pure Nash equilibria for a variation of the 1-dimensional game on the circle where a player owns the arc starting from their point and moving clockwise to the next point. This result uses interesting properties of random arc lengths on circles, and demonstrates the challenges in analyzing these kinds of problems. We also provide several other related results. In particular, for the 1-dimensional game on the circle, we show that a pure Nash equilibrium always exists when each player owns the part of the circle nearest to their point, but it is NP-hard to determine whether a pure Nash equilibrium exists in the variant when each player owns the arc starting from their point clockwise to the next point. This last result, in part, motivates our examination of the random setting.

Original languageEnglish
Title of host publication43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016
EditorsYuval Rabani, Ioannis Chatzigiannakis, Davide Sangiorgi, Michael Mitzenmacher
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770132
DOIs
StatePublished - 1 Aug 2016
Externally publishedYes
Event43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 - Rome, Italy
Duration: 12 Jul 201615 Jul 2016

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume55
ISSN (Print)1868-8969

Conference

Conference43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016
Country/TerritoryItaly
CityRome
Period12/07/1615/07/16

Keywords

  • Correlated equilibria
  • Hotelling model
  • Power of two choices
  • Voronoi games

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