315,777 research outputs found
Three-Player Entangled XOR Games are NP-Hard to Approximate
We show that for any Є > 0 the problem of finding a factor (2 - Є) approximation to the entangled value of a three-player XOR game is NP-hard. Equivalently, the problem of approximating the largest possible quantum violation of a tripartite Bell correlation inequality to within any multiplicative constant is NP-hard. These results are the first constant-factor hardness of approximation results for entangled games or quantum violations of Bell inequalities shown under the sole assumption that P≠NP. They can be thought of as an extension of Håstad's optimal hardness of approximation results for MAX-E3-LIN2 [J. ACM, 48 (2001), pp. 798--859] to the entangled-player setting. The key technical component of our work is a soundness analysis of a plane-vs-point low-degree test against entangled players. This extends and simplifies the analysis of the multilinearity test by Ito and Vidick [Proceedings of the 53rd FOCS, IEEE, Piscataway, NJ, 2012, pp. 243-252]. Our results demonstrate the possibility of efficient reductions between entangled-player games and our techniques may lead to further hardness of approximation results
Three-player entangled XOR games are NP-hard to approximate
We show that for any ε > 0 the problem of finding a factor (2 - ε) approximation to the entangled value of a three-player XOR game is NP-hard. Equivalently, the problem of approximating the largest possible quantum violation of a tripartite Bell correlation inequality to within any multiplicative constant is NP-hard. These results are the first constant-factor hardness of approximation results for entangled games or quantum violations of Bell inequalities shown under the sole assumption that P≠NP. They can be thought of as an extension of Hástad's optimal hardness of approximation results for MAX-E3-LIN2 (JACM'01) to the entangled-player setting. The key technical component of our work is a soundness analysis of a point-vs-plane low-degree test against entangled players. This extends and simplifies the analysis of the multilinearity test by Ito and Vidick (FOCS'12). Our results demonstrate the possibility for efficient reductions between entangled-player games and our techniques may lead to further hardness of approximation results
Computing Approximate Nash Equilibria in Polymatrix Games
In an -Nash equilibrium, a player can gain at most by
unilaterally changing his behaviour. For two-player (bimatrix) games with
payoffs in , the best-known achievable in polynomial time is
0.3393. In general, for -player games an -Nash equilibrium can be
computed in polynomial time for an that is an increasing function of
but does not depend on the number of strategies of the players. For
three-player and four-player games the corresponding values of are
0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general
-player games where a player's payoff is the sum of payoffs from a number of
bimatrix games. There exists a very small but constant such that
computing an -Nash equilibrium of a polymatrix game is \PPAD-hard.
Our main result is that a -Nash equilibrium of an -player
polymatrix game can be computed in time polynomial in the input size and
. Inspired by the algorithm of Tsaknakis and Spirakis, our
algorithm uses gradient descent on the maximum regret of the players. We also
show that this algorithm can be applied to efficiently find a
-Nash equilibrium in a two-player Bayesian game
Quantified Boolean Formula Games and Their Complexities
Consider QBF, the Quantified Boolean Formula problem, as a combinatorial game
ruleset. The problem is rephrased as determining the winner of the game where
two opposing players take turns assigning values to boolean variables. In this
paper, three common variations of games are applied to create seven new games:
whether each player is restricted to where they may play, which values they may
set variables to, or the condition they are shooting for at the end of the
game. The complexity for determining which player can win is analyzed for all
games. Of the seven, two are trivially in P and the other five are
PSPACE-complete. These varying properties are common for combinatorial games;
reductions from these five hard games can simplify the process for showing the
PSPACE-hardness of other games.Comment: 14 pages, 0 figures, for Integers 2013 Conference proceeding
Bejeweled, Candy Crush and other Match-Three Games are (NP-)Hard
The twentieth century has seen the rise of a new type of video games targeted
at a mass audience of "casual" gamers. Many of these games require the player
to swap items in order to form matches of three and are collectively known as
\emph{tile-matching match-three games}. Among these, the most influential one
is arguably \emph{Bejeweled} in which the matched items (gems) pop and the
above gems fall in their place. Bejeweled has been ported to many different
platforms and influenced an incredible number of similar games. Very recently
one of them, named \emph{Candy Crush Saga} enjoyed a huge popularity and
quickly went viral on social networks. We generalize this kind of games by only
parameterizing the size of the board, while all the other elements (such as the
rules or the number of gems) remain unchanged. Then, we prove that answering
many natural questions regarding such games is actually \NP-Hard. These
questions include determining if the player can reach a certain score, play for
a certain number of turns, and others. We also
\href{http://candycrush.isnphard.com}{provide} a playable web-based
implementation of our reduction.Comment: 21 pages, 12 figure
Three-Player Entangled XOR Games are NP-Hard to Approximate
We show that for any Є > 0 the problem of finding a factor (2 - Є) approximation to the entangled value of a three-player XOR game is NP-hard. Equivalently, the problem of approximating the largest possible quantum violation of a tripartite Bell correlation inequality to within any multiplicative constant is NP-hard. These results are the first constant-factor hardness of approximation results for entangled games or quantum violations of Bell inequalities shown under the sole assumption that P≠NP. They can be thought of as an extension of Håstad's optimal hardness of approximation results for MAX-E3-LIN2 [J. ACM, 48 (2001), pp. 798--859] to the entangled-player setting. The key technical component of our work is a soundness analysis of a plane-vs-point low-degree test against entangled players. This extends and simplifies the analysis of the multilinearity test by Ito and Vidick [Proceedings of the 53rd FOCS, IEEE, Piscataway, NJ, 2012, pp. 243-252]. Our results demonstrate the possibility of efficient reductions between entangled-player games and our techniques may lead to further hardness of approximation results
Comparing dynamitic difficulty adjustment and improvement in action game
A thesis submitted to the University of Bedfordshire in partial fulfilment of the requirements for the degree of Master ResearchDesigning a game difficulty is one of the key things as a game designer. Player will be feeling boring when the game designer makes the game too easy or too hard. In the past decades, most of single player games can allow players to choose the game difficulty either easy, normal or hard which define the overall game difficulty. In action game, these options are lack of flexibility and they are unsuitable to the player skill to meet the game difficulty. By using Dynamic Difficulty Adjustment (DDA), it can change the game difficulty in real time and it can match different player skills. In this paper, the final goal is the comparison of the three DDA systems in action game and apply an improved DDA. In order to apply a new improved DDA, this thesis will evaluate three chosen DDA systems with chosen action decision based AI for action game. A new DDA measurement formula is applied to the comparing section
Percolation games, probabilistic cellular automata, and the hard-core model
Let each site of the square lattice be independently assigned
one of three states: a \textit{trap} with probability , a \textit{target}
with probability , and \textit{open} with probability , where
. Consider the following game: a token starts at the origin, and two
players take turns to move, where a move consists of moving the token from its
current site to either or . A player who moves the token
to a trap loses the game immediately, while a player who moves the token to a
target wins the game immediately. Is there positive probability that the game
is \emph{drawn} with best play -- i.e.\ that neither player can force a win?
This is equivalent to the question of ergodicity of a certain family of
elementary one-dimensional probabilistic cellular automata (PCA). These
automata have been studied in the contexts of enumeration of directed lattice
animals, the golden-mean subshift, and the hard-core model, and their
ergodicity has been noted as an open problem by several authors. We prove that
these PCA are ergodic, and correspondingly that the game on has
no draws.
On the other hand, we prove that certain analogous games \emph{do} exhibit
draws for suitable parameter values on various directed graphs in higher
dimensions, including an oriented version of the even sublattice of
in all . This is proved via a dimension reduction to a
hard-core lattice gas in dimension . We show that draws occur whenever the
corresponding hard-core model has multiple Gibbs distributions. We conjecture
that draws occur also on the standard oriented lattice for
, but here our method encounters a fundamental obstacle.Comment: 35 page
Playing Dominoes Is Hard, Except by Yourself
Dominoes is a popular and well-known game possibly dating back three millennia. Players are given a set of domino tiles, each with two labeled square faces, and take turns connecting them into a growing chain of dominoes by matching identical faces. We show that single-player dominoes is in P, while multiplayer dominoes is hard: when players cooperate, the game is NP-complete, and when players compete, the game is PSPACE-complete. In addition, we show that these hardness results easily extend to games involving team play
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