Abstract
In twoplayer 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 tau in [0,1]: portion tau of the winning bid is paid to the other player, and portion 1tau to the bank. While finiteduration (reachability) taxman games have been studied before, we present, for the first time, results on infiniteduration taxman games. It was previously shown that both Richman and poorman infiniteduration games with qualitative objectives reduce to reachability games, and we show a similar result here. Our most interesting results concern quantitative taxman games, namely meanpayoff 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 meanpayoff 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 randomturn 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 randomturn game with an unbiased coin, with poorman bidding, the bias in the coin is the initial ratio of the budgets. We give a complete classification of meanpayoff taxman games that is based on a probabilistic connection: the value of a taxman bidding game with parameter tau and initial ratio r, equals the value of a randomturn game that uses a coin with bias F(tau, r) = (r+tau * (1r))/(1+tau). Thus, we show that Richman bidding is the exception; namely, for every tau <1, the value of the game depends on the initial ratio. Our proof technique simplifies and unifies the previous proof techniques for both Richman and poorman bidding.
BibTeX  Entry
@InProceedings{avni_et_al:LIPIcs:2019:10955,
author = {Guy Avni and Thomas A. Henzinger and Dorde Zikelic},
title = {{Bidding Mechanisms in Graph Games}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {11:111:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771177},
ISSN = {18688969},
year = {2019},
volume = {138},
editor = {Peter Rossmanith and Pinar Heggernes and JoostPieter Katoen},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10955},
URN = {urn:nbn:de:0030drops109553},
doi = {10.4230/LIPIcs.MFCS.2019.11},
annote = {Keywords: Bidding games, Richman bidding, poorman bidding, taxman bidding, meanpayoff games, randomturn games}
}
Keywords: 

Bidding games, Richman bidding, poorman bidding, taxman bidding, meanpayoff games, randomturn games 
Seminar: 

44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019) 
Issue Date: 

2019 
Date of publication: 

23.08.2019 