31 research outputs found

    The per capita Shapley support levels value

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    The per capita Shapley support levels value extends the Shapley value to cooperative games with a level structure. This value prevents symmetrical groups of players of different sizes from being treated equally. We use efficiency, additivity, the null player property, and two new properties to give an axiomatic characterization. The first property, called joint productivity, is a fairness property within components and makes the difference to the Shapley levels value. If all players of two components are only jointly productive, they should receive the same payoff. Our second axiom, called neutral collusions, is a fairness axiom for players outside a component. Regardless of how players of a component organize their power, as long as the power of the coalitions that include all players of the component remains the same, the payoff to players outside the component does not change

    Disjointly and jointly productive players and the Shapley value

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    Central to this study is the concept of disjointly productive players where no cooperation gain occurs when one of two such players joins a coalition containing the other. Our first new axiom states that the payoff to a player does not change when another player, disjointly productive to that player, leaves the game. The second axiom implies that the payoff to a third player does not change if we merge two disjointly productive players into a new player. These two axioms, along with efficiency, characterize the Shapley value and may be advantageous sometimes to improve the runtime for computing the Shapley value. Further axiomatizations are provided, using, for example, a modification of behavior property where the payoff for two players in two new games in which their behavior changes once to total dislike and once to total affection is equal to the payoff in the original game

    Mutual information-based group explainers with coalition structure for machine learning model explanations

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    In this article, we propose and investigate ML group explainers in a general game-theoretic setting with the focus on coalitional game values and games based on the conditional and marginal expectation of an ML model. The conditional game takes into account the joint distribution of the predictors, while the marginal game depends on the structure of the model. The objective of the article is to unify the two points of view under predictor dependencies and to reduce the complexity of group explanations. To achieve this, we propose a feature grouping technique that employs an information-theoretic measure of dependence and design appropriate groups explainers. Furthermore, in the context of coalitional game values with a two-step formulation, we introduce a theoretical scheme that generates recursive coalitional game values under a partition tree structure and investigate the properties of the corresponding group explainers.Comment: 46 pages, 69 figure

    Disjointly productive players and the Shapley value

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    Central to this study is the concept of disjointly productive players. Two players are disjointly productive if there is no synergy effect if one of these players joins a coalition containing the other. Our first new axiom states that the payoff to a player does not change when another player, disjointly productive with that player, is removed from the game. The second new axiom means that if we merge two disjointly productive players into a new player, the payoff to a third player in a game with the merged player does not change. These two axioms, along with efficiency, characterize the Shapley value and can lead to improved run times for computing the Shapley value in games with some disjointly productive players

    Harsanyi support levels solutions

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    We introduce a new class of values with transferable utility for level structures. In these hierarchical structures, each level corresponds to a partition of the player set, which becomes increasingly coarse from the trivial partition containing only singletons to the partition containing only the grand coalition. The new values, called Harsanyi support levels solutions, extend the Harsanyi solutions to level structures. As an important subset of these values, we present the class of weighted Shapley support levels values as a further result. The values from this class extend the weighted Shapley values to level structures and contain the Shapley levels value as a special case. Axiomatizations of the studied classes are provided

    The Harsanyi paradox and the “right to talk” in bargaining among coalitions

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    We describe a coalitional value from a non-cooperative point of view, assuming coalitions are formed for the purpose of bargaining. The idea is that all the players have the same chances to make proposals. This means that players maintain their own "right to talk" when joining a coalition. The resulting value coincides with the weighted Shapley value in the game between coalitions, with weights given by the size of the coalitions. Moreover, the Harsanyi paradox (forming a coalition may be disadvantageous) disappears for convex games

    The weighted Shapley support levels values

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    This paper presents a new class of weighted values for level structures. The new values, called weighted Shapley support levels values, extend the weighted Shapley values to level structures and contain the Shapley levels value (Winter, 1989) as a special case. Since a level structure with only two levels coincides with a coalition structure we obtain, as a side effect, also new axiomatizations of weighted coalition structure values, presented in Levy and McLean (1989)

    The weighted Shapley support levels values

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    This paper presents a new class of weighted values for level structures. The new values, called weighted Shapley support levels values, extend the weighted Shapley values to level structures and contain the Shapley levels value (Winter, 1989) as a special case. Since a level structure with only two levels coincides with a coalition structure we obtain, as a side effect, also new axiomatizations of weighted coalition structure values, presented in Levy and McLean (1989)

    Harsanyi support levels payoffs and weighted Shapley support levels values

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    This paper introduces a new class of values for level structures. The new values, called Harsanyi support levels payoffs, extend the Harsanyi payoffs from the Harsanyi set to level structures and contain the Shapley levels value (Winter, 1989) as a special case. We also look at extensions of the weighted Shapley values to level structures. These values, we call them weighted Shapley support levels values, constitute a subset of the class of Harsanyi support levels payoffs and coincide on a level structure with only two levels with a class of weighted coalition structure values, already mentioned in Levy and McLean (1989) and discussed in McLean (1991). Axiomatizations of the studied classes are provided for both exogenously and endogenously given weights
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