83,871 research outputs found
The -operator and Invariant Subtraction Games
We study 2-player impartial games, so called \emph{invariant subtraction
games}, of the type, given a set of allowed moves the players take turn in
moving one single piece on a large Chess board towards the position
. Here, invariance means that each allowed move is available
inside the whole board. Then we define a new game, of the old game, by
taking the -positions, except , as moves in the new game. One
such game is \W^\star= (Wythoff Nim), where the moves are defined by
complementary Beatty sequences with irrational moduli. Here we give a
polynomial time algorithm for infinitely many -positions of \W^\star. A
repeated application of turns out to give especially nice properties
for a certain subfamily of the invariant subtraction games, the
\emph{permutation games}, which we introduce here. We also introduce the family
of \emph{ornament games}, whose -positions define complementary Beatty
sequences with rational moduli---hence related to A. S. Fraenkel's `variant'
Rat- and Mouse games---and give closed forms for the moves of such games. We
also prove that (-pile Nim) = -pile Nim.Comment: 30 pages, 5 figure
Move ordering and communities in complex networks describing the game of go
We analyze the game of go from the point of view of complex networks. We
construct three different directed networks of increasing complexity, defining
nodes as local patterns on plaquettes of increasing sizes, and links as actual
successions of these patterns in databases of real games. We discuss the
peculiarities of these networks compared to other types of networks. We explore
the ranking vectors and community structure of the networks and show that this
approach enables to extract groups of moves with common strategic properties.
We also investigate different networks built from games with players of
different levels or from different phases of the game. We discuss how the study
of the community structure of these networks may help to improve the computer
simulations of the game. More generally, we believe such studies may help to
improve the understanding of human decision process.Comment: 14 pages, 21 figure
The game of go as a complex network
We study the game of go from a complex network perspective. We construct a
directed network using a suitable definition of tactical moves including local
patterns, and study this network for different datasets of professional
tournaments and amateur games. The move distribution follows Zipf's law and the
network is scale free, with statistical peculiarities different from other real
directed networks, such as e. g. the World Wide Web. These specificities
reflect in the outcome of ranking algorithms applied to it. The fine study of
the eigenvalues and eigenvectors of matrices used by the ranking algorithms
singles out certain strategic situations. Our results should pave the way to a
better modelization of board games and other types of human strategic scheming.Comment: 6 pages, 9 figures, final versio
Remarks on the k-error linear complexity of p(n)-periodic sequences
Recently the first author presented exact formulas for the number of 2âżn-periodic binary sequences with given 1-error linear complexity, and an exact formula for the expected 1-error linear complexity and upper and lower bounds for the expected k-error linear complexity, k >2, of a random 2âżn-periodic binary sequence. A crucial role for the analysis played the Chan-Games algorithm. We use a more sophisticated generalization of the Chan-Games algorithm by Ding et al. to obtain exact formulas for the counting function and the expected value for the 1-error linear complexity for pâżn-periodic sequences over Fp, p prime. Additionally we discuss the calculation of lower and upper bounds on the k-error linear complexity of pâżn-periodic sequences over Fp
Infinite sequential Nash equilibrium
In game theory, the concept of Nash equilibrium reflects the collective
stability of some individual strategies chosen by selfish agents. The concept
pertains to different classes of games, e.g. the sequential games, where the
agents play in turn. Two existing results are relevant here: first, all finite
such games have a Nash equilibrium (w.r.t. some given preferences) iff all the
given preferences are acyclic; second, all infinite such games have a Nash
equilibrium, if they involve two agents who compete for victory and if the
actual plays making a given agent win (and the opponent lose) form a
quasi-Borel set. This article generalises these two results via a single
result. More generally, under the axiomatic of Zermelo-Fraenkel plus the axiom
of dependent choice (ZF+DC), it proves a transfer theorem for infinite
sequential games: if all two-agent win-lose games that are built using a
well-behaved class of sets have a Nash equilibrium, then all multi-agent
multi-outcome games that are built using the same well-behaved class of sets
have a Nash equilibrium, provided that the inverse relations of the agents'
preferences are strictly well-founded.Comment: 14 pages, will be published in LMCS-2011-65
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