Publication Type

Conference Proceeding Article

Version

publishedVersion

Publication Date

8-2019

Abstract

In two-player games on graphs, the players move a token through a graph to produce a finite or infinite path, which determines the qualitative winner or quantitative payoff of the game. We study bidding games in which the players bid for the right to move the token. Several bidding rules were studied previously. In Richman bidding, in each round, the players simultaneously submit bids, and the higher bidder moves the token and pays the other player. Poorman bidding is similar except that the winner of the bidding pays the “bank” rather than the other player. Taxman bidding spans the spectrum between Richman and poorman bidding. They are parameterized by a constant τ ∈ [0, 1]: portion τ of the winning bid is paid to the other player, and portion 1 − τ to the bank. While finite-duration (reachability) taxman games have been studied before, we present, for the first time, results on infinite-duration taxman games. It was previously shown that both Richman and poorman infinite-duration games with qualitative objectives reduce to reachability games, and we show a similar result here. Our most interesting results concern quantitative taxman games, namely mean-payoff games, where poorman and Richman bidding differ significantly. A central quantity in these games is the ratio between the two players’ initial budgets. While in poorman mean-payoff games, the optimal payoff of a player depends on the initial ratio, in Richman bidding, the payoff depends only on the structure of the game. In both games the optimal payoffs can be found using (different) probabilistic connections with random-turn games in which in each turn, instead of bidding, a coin is tossed to determine which player moves. While the value with Richman bidding equals the value of a random-turn game with an un-biased coin, with poorman bidding, the bias in the coin is the initial ratio of the budgets. We give a complete classification of mean-payoff taxman games that is based on a probabilistic connection: the value of a taxman bidding game with parameter τ and initial ratio r, equals the value of a random-turn game that uses a coin with bias F(τ, r) = r+τ·(1−r) 1+τ . Thus, we show that Richman bidding is the exception; namely, for every τ

Keywords

Bidding games, Richman bidding, poorman bidding, taxman bidding, meanpayoff games, random-turn games

Discipline

Graphics and Human Computer Interfaces | Theory and Algorithms

Research Areas

Intelligent Systems and Optimization

Areas of Excellence

Digital transformation

Publication

Proceedings of the 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019), Germany, August 26-30

First Page

1

Last Page

13

Identifier

10.4230/LIPICS.MFCS.2019.11

Publisher

Schloss Dagstuhl

City or Country

Germany

Copyright Owner and License

Authors

Additional URL

https://doi.org/10.4230/LIPICS.MFCS.2019.11

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