282 research outputs found
Impartial coloring games
Coloring games are combinatorial games where the players alternate painting
uncolored vertices of a graph one of 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 (resp. ) 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 and , which is connected to the well-known Frobenius coin
problem. General results are given, together with arithmetic and geometric
characterizations when the sets and have size at most 2
- …