3 research outputs found
Bidding combinatorial games
Combinatorial Game Theory is a branch of mathematics and theoretical computer
science that studies sequential 2-player games with perfect information. Normal
play is the convention where a player who cannot move loses. Here, we
generalize the classical alternating normal play to infinitely many game
families, by means of discrete Richman auctions (Develin et al. 2010, Larsson
et al. 2021, Lazarus et al. 1996). We generalize the notion of a perfect play
outcome, and find an exact characterization of outcome feasibility. As a main
result, we prove existence of a game form for each such outcome class; then we
describe their lattice structures. By imposing restrictions to the general
families, such as impartial and {\em symmetric termination}, we find surprising
analogies with alternating play.Comment: 5 figure
Constructive comparison in bidding combinatorial games
A class of discrete Bidding Combinatorial Games that generalize alternating
normal play was introduced by Kant, Larsson, Rai, and Upasany (2022). The major
questions concerning optimal outcomes were resolved. By generalizing standard
game comparison techniques from alternating normal play, we propose an
algorithmic play-solution to the problem of game comparison for bidding games.
We demonstrate some consequences of this result that generalize classical
results in alternating play (from Winning Ways 1982 and On Numbers and Games
1976). In particular, integers, dyadics and numbers have many nice properties,
such as group structures, but on the other hand the game * is non-invertible.
We state a couple of thrilling conjectures and open problems for readers to
dive into this promising path of bidding combinatorial games.Comment: 23 pages, 1 figur
Bidding Combinatorial Games
Combinatorial Game Theory is a branch of mathematics and theoretical computer science that studies sequential 2-player games with perfect information. Normal play is the convention where a player who cannot move loses. Here, we generalize the classical alternating normal play to infinitely many game families, by means of discrete Richman auctions (Develin et al. 2010, Larsson et al. 2021, Lazarus et al. 1996). We generalize the notion of a perfect play outcome, and find an exact characterization of outcome feasibility. As a main result, we prove existence of a game form for each such outcome class; then we describe their lattice structures. By imposing restrictions to the general families, such as impartial and symmetric termination, we find surprising analogies with alternating play.</jats:p