Impartial coloring games

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of $k > 0$ colors. Each different ruleset specifies that game's coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, and sequential coloring. For each, we study the outcome classes for special cases and general computational complexity. In some cases we pay special attention to the Grundy function

Quantum-like models cannot account for the conjunction fallacy

Human agents happen to judge that a conjunction of two terms is more probable than one of the terms, in contradiction with the rules of classical probabilities—this is the conjunction fallacy. One of the most discussed accounts of this fallacy is currently the quantum-like explanation, which relies on models exploiting the mathematics of quantum mechanics. The aim of this paper is to investigate the empirical adequacy of major quantum-like models which represent beliefs with quantum states. We first argue that they can be tested in three different ways, in a question order effect configuration which is different from the traditional conjunction fallacy experiment. We then carry out our proposed experiment, with varied methodologies from experimental economics. The experimental results we get are at odds with the predictions of the quantum-like models. This strongly suggests that this quantum-like account of the conjunction fallacy fails. Future possible research paths are discussed

Building Nim

The game of nim, with its simple rules, its elegant solution and its historical importance is the quintessence of a combinatorial game, which is why it led to so many generalizations and modifications. We present a modification with a new spin: building nim. With given finite numbers of tokens and stacks, this two-player game is played in two stages (thus belonging to the same family of games as e.g. nine-men's morris): first building, where players alternate to put one token on one of the, initially empty, stacks until all tokens have been used. Then, the players play nim. Of course, because the solution for the game of nim is known, the goal of the player who starts nim play is a placement of the tokens so that the Nim-sum of the stack heights at the end of building is different from 0. This game is trivial if the total number of tokens is odd as the Nim-sum could never be 0, or if both the number of tokens and the number of stacks are even, since a simple mimicking strategy results in a Nim-sum of 0 after each of the second player's moves. We present the solution for this game for some non-trivial cases and state a general conjecture

The Maker-Maker domination game in forests

We study the Maker-Maker version of the domination game introduced in 2018 by Duch\^ene et al. Given a graph, two players alternately claim vertices. The first player to claim a dominating set of the graph wins. As the Maker-Breaker version, this game is PSPACE-complete on split and bipartite graphs. Our main result is a linear time algorithm to solve this game in forests. We also give a characterization of the cycles where the first player has a winning strategy

Partizan Subtraction Games

Partizan subtraction games are combinatorial games where two players, say Left and Right, alternately remove a number n of tokens from a heap of tokens, with $n \in S_L$ (resp. $n \in S_R$) when it is Left's (resp. Right's) turn. The first player unable to move loses. These games were introduced by Fraenkel and Kotzig in 1987, where they introduced the notion of dominance, i.e. an asymptotic behavior of the outcome sequence where Left always wins if the heap is sufficiently large. In the current paper, we investigate the other kinds of behaviors for the outcome sequence. In addition to dominance, three other disjoint behaviors are defined, namely weak dominance, fairness and ultimate impartiality. We consider the problem of computing this behavior with respect to $S_L$ and $S_R$, which is connected to the well-known Frobenius coin problem. General results are given, together with arithmetic and geometric characterizations when the sets $S_L$ and $S_R$ have size at most 2