16,918 research outputs found

    The structure of pairing strategies for k-in-a-row type games

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    In Maker-Breaker positional games two players, Maker and Breaker, play on a finite or infinite board with the goal of claiming or preventing the opponent from getting a finite winning set, respectively. For different games there are several winning strategies for Maker or Breaker. One class of winning strategies is the so-called pairing (paving) strategies. Here, we describe all possible pairing strategies for the 9-in-a-row game. Furthermore, we define a graph of the pairings, containing 194,543 vertices and 532,107 edges, in order to give them a structure. A complete characterization of the graph is also given

    Characterising and recognising game-perfect graphs

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    Consider a vertex colouring game played on a simple graph with kk permissible colours. Two players, a maker and a breaker, take turns to colour an uncoloured vertex such that adjacent vertices receive different colours. The game ends once the graph is fully coloured, in which case the maker wins, or the graph can no longer be fully coloured, in which case the breaker wins. In the game gBg_B, the breaker makes the first move. Our main focus is on the class of gBg_B-perfect graphs: graphs such that for every induced subgraph HH, the game gBg_B played on HH admits a winning strategy for the maker with only ω(H)\omega(H) colours, where ω(H)\omega(H) denotes the clique number of HH. Complementing analogous results for other variations of the game, we characterise gBg_B-perfect graphs in two ways, by forbidden induced subgraphs and by explicit structural descriptions. We also present a clique module decomposition, which may be of independent interest, that allows us to efficiently recognise gBg_B-perfect graphs.Comment: 39 pages, 8 figures. An extended abstract was accepted at the International Colloquium on Graph Theory (ICGT) 201

    Biased Weak Polyform Achievement Games

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    In a biased weak (a,b)(a,b) polyform achievement game, the maker and the breaker alternately mark a,ba,b previously unmarked cells on an infinite board, respectively. The maker's goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the (a,b)(a,b) game can be built from winning strategies for games involving fewer marks for the maker and the breaker. A new type of breaker strategy called the priority strategy is introduced. The winners are determined for all (a,b)(a,b) pairs for polyiamonds and polyominoes up to size four

    Stated versus inferred beliefs: A methodological inquiry and experimental test

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    If asking subjects their beliefs during repeated game play changes the way those subjects play, using those stated beliefs to evaluate and compare theories of strategic behavior is problematic. We experimentally verify that belief elicitation can alter paths of play in a repeated asymmetric matching pennies game. In this setting, belief elicitation improves the goodness of fit of structural models of belief learning, and the prior beliefs implied by such structural models are both stronger and more realistic when beliefs are elicited than when they are not. These effects are, however, confined to the player type who sees a strong asymmetry between payoff possibilities for her two strategies in the game. We also find that “inferred beliefs” (beliefs estimated from past observed actions of opponents) can be better predictors of observed actions than the “stated beliefs” resulting from belief elicitation.beliefs; stated beliefs; belief elicitation; inferred beliefs; estimated beliefs; belief updating; repeated games; experimental methods

    Achieving snaky

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    We prove that the polyomino generally known as snaky is a three-dimensional winner, that it loses on an 8 × 8 board, and that its handicap number is at most one

    Bargaining Outcomes as the Result of Coordinated Expectations: An Experimental Study of Sequential Bargaining

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    Experimental studies of two-person sequential bargaining demonstrate that the concept of subgame perfection is not a reliable point predictor of actual behavior. Alternative explanations argue that 1) fairness influences outcomes and 2) that bargainer expectations matter and are likely not to be coordinated at the outset. This paper examines the process by which bargainers in two-person dyads coordinate their expectations on a bargaining convention and how this convention is supported by the seemingly empty threat of rejecting positive but small subgame perfect offers. To organize the data from this experiment, we develop a Markov model of adaptive expectations and bounded rationality. The model predicts actual behavior quite closely.Sequential Bargaining, Experiment, Convention, Fairness, Finite Markov Chain, Bounded Rationality

    A Simple Test of Learning Theory?

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    We report experiments designed to test the theoretical possibility, first discovered by Shapley (1964), that in some games learning fails to converge to any equilibrium, either in terms of marginal frequencies or of average play. Subjects played repeatedly in fixed pairings one of two 3 × 3 games, each having a unique Nash equilibrium in mixed strategies. The equilibrium of one game is predicted to be stable under learning, the other unstable, provided payoffs are sufficiently high. We ran each game in high and low payoff treatments. We find that, in all treatments, average play is close to equilibrium even though there are strong cycles present in the data.: Games, Learning, Experiments, Stochastic Fictitious Play, Mixed Strategy Equilibria.
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