Simultaneous Combinatorial Game Theory

Combinatorial game theory (CGT), as introduced by Berlekamp, Conway and Guy, involves two players who move alternately in a perfect information, zero-sum game, and there are no chance devices. Also the games have the finite descent property (every game terminates in a finite number of moves). The two players are usually called Left and Right. The games often break up into components and the players must choose one of the components in which to play. One main aim of CGT is to analyze the components individually (rather than analyzing the sum as a whole) then use this information to analyze the sum. In this paper, the players move simultaneously in a combinatorial game. Three sums are considered which are defined by the termination rules: (i) one component does not have a simultaneous move; (ii) no component has a simultaneous move; (iii) one player has no move in any component. These are combined with a winning convention which is either: (i) based on which player has moves remaining; or (ii) the greatest score. In each combination, we show that equality of games induces an equivalence relation and the equivalence classes are partially ordered. Also, where possible, given games $A$ and $B$, we give checks to determine if Left prefers to replace $A$ by $B$ in a sum

Games and Complexes I: Transformation via Ideals

Placement games are a subclass of combinatorial games which are played on graphs. We will demonstrate that one can construct simplicial complexes corresponding to a placement game, and this game could be considered as a game played on these simplicial complexes. These complexes are constructed using square-free monomials

When waiting moves you in scoring combinatorial games

Combinatorial Scoring games, with the property `extra pass moves for a player does no harm', are characterized. The characterization involves an order embedding of Conway's Normal-play games. Also, we give a theorem for comparing games with scores (numbers) which extends Ettinger's work on dicot Scoring games.Comment: 19 pages, 5 figure

Mis\`ere-play Hackenbush Sprigs

A Hackenbush Sprig is a Hackenbush String with the ground edge colored green and the remaining edges either red or blue. We show that in canonical form a Sprig is a star-based number (the ordinal sum of star and a dyadic rational) in mis\`ere-play, as well as in normal-play. We find the outcome of a disjunctive sum of Sprigs in mis\`ere-play and show that it is the same as the outcome of that sum plus star in normal-play. Along the way it is shown that the sum of a Sprig and its negative is equivalent to 0 in the universe of mis\`ere-play dicotic games, answering a question of Allen.Comment: 13 pages, 2 figures (1 in color

Absolute Combinatorial Game Theory

This is the study of game comparison in \emph{combinatorial game spaces}, and their sub-spaces, called \emph{universes} of games. The concept of a combinatorial game space allows for a general framework, which includes many standard classes of terminating games. In an \emph{absolute} universe, it is shown that the comparison of two games need only include followers of the two and atomic games instead of all games. This leads to a constructive game comparison for studied absolute universes (e.g. Milnor, Conway, Ettinger, Siegel, Milley and Renault). On the way, we introduce the dicot \emph{kernel} of a game space, and show how to construct infinitely many absolute universes, by extending specifically the mis\`ere kernel.Comment: 24 pages, 1 figur

Finding Golden Nuggets by Reduction

We introduce a class of normal play partizan games, called Complementary Subtraction. Let $A$ denote your favorite set of positive integers. This is Left's subtraction set, whereas Right subtracts numbers not in $A$. The Golden Nugget Subtraction Game has the $A$ and $B$ 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

Atomic weights and the combinatorial game of Bipass

We define an all-small ruleset, Bipass, within the framework of normal-play combinatorial games. A game is played on finite strips of black and white stones. Stones of different colors are swapped provided they do not bypass one of their own kind. We find a surjective function from the strips to integer atomic weights (Berlekamp, Conway and Guy 1982) that measures the number of units in all-small games. This result provides explicit winning strategies for many games, and in cases where it does not, it gives narrow bounds for the canonical form game values. We prove that the game value *2 does not appear as a disjunctive sum of Bipass. Moreover, we find game values for some parametrized families of games, including an infinite number of strips of value *.Comment: 22 pages, 6 figure

Guaranteed Scoring Games

The class of Guaranteed Scoring Games (GS) are two-player combinatorial games with the property that Normal-play games (Conway et. al.) are ordered embedded into GS. They include, as subclasses, the scoring games considered by Milnor (1953), Ettinger (1996) and Johnson (2014). We present the structure of GS and the techniques needed to analyze a sum of guaranteed games. Firstly, GS form a partially ordered monoid, via defined Right- and Left-stops over the reals, and with disjunctive sum as the operation. In fact, the structure is a quotient monoid with partially ordered congruence classes. We show that there are four reductions that when applied, in any order, give a unique representative for each congruence class. The monoid is not a group, but in this paper we prove that if a game has an inverse it is obtained by `switching the players'. The order relation between two games is defined by comparing their stops in \textit{any} disjunctive sum. Here, we demonstrate how to compare the games via a finite algorithm instead, extending ideas of Ettinger, and also Siegel (2013).Comment: 29 page

Simultaneous Moves with Cops and an Insightful Robber

The 'Cheating Robot' is a simultaneous play, yet deterministic, version of Cops and Robbers played on a finite, simple, connected graph. The robot knows where the cops will move on the next round and has the time to react. The cops also know that the robot has inside information. More cops are required to capture a robot than to capture a robber. Indeed, the minimum degree is a lower bound on the number of cops required to capture a robot. Only on a tree is one cop guaranteed to capture a robot, although two cops are sufficient to capture both a robber and a robot on outerplanar graphs. In graphs where retracts are involved, we show how cop strategies against a robber can be modified to capture a robot. This approach gives exact numbers for hypercubes, and $k$-dimensional grids in general.Comment: 15 pages, 2 figure

A Deterministic Model for Simultaneous Play Games: The Cheating Robot and Insider Information

Combinatorial games are two-player games of pure strategy where the players, usually called Left and Right, move alternately. In this paper, we introduce simultaneous-play combinatorial games except that Right has extra information: he knows what move Left is about to play and can react in time to modify his move. Right is `cheating' and we assume that Left is aware of this. This knowledge makes for deterministic, not probabilistic, strategies. The basic theory and properties are developed, including showing that there is an equivalence relation and partial order on the games. Whilst there are no inverses in the class of all games, we show that there is a sub-class, simple hot games, in which the `integers' have inverses. In this sub-class, the optimal strategies are obtained by the solutions to a minimum-weight matching problem on a graph whose number of vertices equals the number of summands in the disjunctive sum. This is further refined, in a version of simultaneous toppling dominoes, by reducing the number of edges in the underlying graph to be linear in the number of vertices.Comment: 24 pages, 6 figure