305 research outputs found
An Extension of the Normal Play Convention to -player Combinatorial Games
We examine short combinatorial games for three or more players under a new
play convention in which a player who cannot move on their turn is the unique
loser. We show that many theorems of impartial and partizan two-player games
under normal play have natural analogues in this setting. For impartial games
with three players, we investigate the possible outcomes of a sum in detail,
and determine the outcomes and structure of three-player Nim.Comment: 45 pages; Presented at Integers Conference 2018, submitted to
Proceeding
Partizan Kayles and Misere Invertibility
The impartial combinatorial game Kayles is played on a row of pins, with
players taking turns removing either a single pin or two adjacent pins. A
natural partizan variation is to allow one player to remove only a single pin
and the other only a pair of pins. This paper develops a complete solution for
"Partizan Kayles" under misere play, including the misere monoid all possible
sums of positions, and discusses its significance in the context of misere
invertibility: the universe of Partizan Kayles contains a position whose
additive inverse is not its negative, and, moreover, this position is an
example of a right-win game whose inverse is previous-win
The Combinatorial Game Theory of Well-Tempered Scoring Games
We consider the class of "well-tempered" integer-valued scoring games, which
have the property that the parity of the length of the game is independent of
the line of play. We consider disjunctive sums of these games, and develop a
theory for them analogous to the standard theory of disjunctive sums of
normal-play partizan games. We show that the monoid of well-tempered scoring
games modulo indistinguishability is cancellative but not a group, and we
describe its structure in terms of the group of normal-play partizan games. We
also classify Boolean-valued well-tempered scoring games, showing that there
are exactly seventy, up to equivalence.Comment: 60 pages, 21 figure
Restricted developments in partizan misere game theory
Much progress has been made in misere game theory using the technique of
restricted misere play, where games can be considered equivalent inside a
restricted set of games without being equal in general. This paper provides a
survey of recent results in this area, including two particularly interesting
properties of restricted misere games: (1) a game can have an additive inverse
that is not its negative, and (2) a position can be reversible through an end
(a game with Left but not Right options, or vice versa). These properties are
not possible in normal play and general misere play, respectively. Related open
problems are discussed
Mis\`ere canonical forms of partizan games
We show that partizan games admit canonical forms in mis\`ere play. The proof
is a synthesis of the canonical form theorems for normal-play partizan games
and mis\`ere-play impartial games. It is fully constructive, and algorithms
readily emerge for comparing mis\`ere games and calculating their canonical
forms.
We use these techniques to show that there are precisely 256 games born by
day 2, and to obtain a bound on the number of games born by day 3.Comment: 12 page
Infinite cyclic impartial games
We define the family of {\it locally path-bounded} digraphs, which is a class
of infinite digraphs, and show that on this class it is relatively easy to
compute an optimal strategy (winning or nonlosing); and realize a win, when
possible, in a finite number of moves. This is done by proving that the
Generalized Sprague-Grundy function exists uniquely and has finite values on
this class.Comment: To appear in Proc. Computer Games 199
Cumulative Games: Who is the current player?
Combinatorial Game Theory (CGT) is a branch of game theory that has developed
almost independently from Economic Game Theory (EGT), and is concerned with
deep mathematical properties of 2-player 0-sum games that are defined over
various combinatorial structures. The aim of this work is to lay foundations to
bridging the conceptual and technical gaps between CGT and EGT, here
interpreted as so-called Extensive Form Games, so they can be treated within a
unified framework. More specifically, we introduce a class of -player,
general-sum games, called Cumulative Games, that can be analyzed by both CGT
and EGT tools. We show how two of the most fundamental definitions of CGT---the
outcome function, and the disjunctive sum operator---naturally extend to the
class of Cumulative Games. The outcome function allows for an efficient
equilibrium computation under certain restrictions, and the disjunctive sum
operator lets us define a partial order over games, according to the advantage
that a certain player has. Finally, we show that any Extensive Form Game can be
written as a Cumulative Game.Comment: 54 pages, 4 figure
The strange algebra of combinatorial games
We present an algebraic framework for the analysis of combinatorial games.
This framework embraces the classical theory of partizan games as well as a
number of misere games, comply-constrain games, and card games that have been
studied more recently. It focuses on the construction of the quotient monoid of
a game, an idea that has been successively applied to several classes of games.Comment: 21 page
Finding Golden Nuggets by Reduction
We introduce a class of normal play partizan games, called Complementary
Subtraction. Let denote your favorite set of positive integers. This is
Left's subtraction set, whereas Right subtracts numbers not in . The Golden
Nugget Subtraction Game has the and sequences, from Wythoff's game, as
the two complementary subtraction sets. As a function of the heap size, the
maximum size of the canonical forms grows quickly. However, the value of the
heap is either a number or, in reduced canonical form, a switch. We find the
switches by using properties of the Fibonacci word and standard Fibonacci
representations of integers. Moreover, these switches are invariant under
shifts by certain Fibonacci numbers. The values that are numbers, however, are
distinct, and we find a polynomial time bit characterization for them, via the
ternary Fibonacci representation.Comment: 28 page
Equality Classes of Nim Positions under Mis\`ere Play
We determine the mis\`{e}re equivalence classes of Nim positions under two
equivalence relations: one based on playing disjunctive sums with other
impartial games, and one allowing sums with partizan games. In the impartial
context, the only identifications we can make are those stemming from the known
fact about adding a heap of size 1. In the partizan context, distinct Nim
positions are inequivalent.Comment: 10 pages, LaTeX; fixed typos, improved formatting consistency, added
section on transfinite Ni
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