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Peeking inside the “black box”: post-hoc interpretability
Published in Brandon M. Greenwell, Tree-Based Methods for Statistical Learning in R, 2022
The Shapley value [Shapley, 2016] is an idea from coalitional/cooperative game theory. In a coalitional game, assume there are p players that form a grand coalition (S) worth a certain payout (ΔS). Suppose it is also known how much any smaller coalition (Q⊆S) (i.e., any subset of p players) is worth (ΔQ). The goal is to distribute the total payout ΔS to the individual p players in a “fair” way; that is, so that each player receives their “fair” share. The Shapley value is one such solution and the only one that uniquely satisfies a particular set of “fairness properties.”
Multi-objective optimization with genetic algorithm, fuzzy logic and game theory
Published in Franklin Y. Cheng, Kevin Z. Truman, Structural Optimization, 2017
Franklin Y. Cheng, Kevin Z. Truman
Game theory was developed for both cooperative and non-cooperative games. Cooperative games are those in which participants have the opportunity to communicate with one another and to form binding and enforceable agreements. In non-cooperative games, each player acts independently in an effort to maximize his/her own pay-off, which produces an outcome that may be favourable for one player but unfavourable for another. The concept of player cooperation therefore becomes important when considering compromise game outcomes. The measurement of success of cooperative play is embodied in the concept of the Pareto optimum; a Pareto optimum has the property that if any other solution is used, at least one player does worse or they all do the same. A cooperative game theory consists of ways to analyze conflicts existing in objectives or interest groups (players), to provide a neutral forum for discussion and negotiations among players, and then to suggest a compromise solution acceptable to all players.
Cross-Layer Cooperative Communication in Wireless Networks
Published in Jerry D. Gibson, Mobile Communications Handbook, 2017
Matthew Nokleby, Gareth Middleton, Behnaam Aazhang
To improve upon the single-stage NE, we turn to the Nash bargaining solution (NBS) from cooperative game theory [7]. Formally, a two-player bargaining game is defined by a set of feasible utilities Uand a disagreement point δ∈U. The set U contains utility vectors u = (u1,u2), denoting the payoff to both players, for all possible strategies the players may implement. The disagreement point δ represents the “status quo” prior to bargaining or utility guaranteed to each player should bargaining fail. In bargaining games, players cooperatively choose a compromise point. That is, rather than individually focusing on payoff maximization, players jointly choose a mutually agreeable utility vector and agree to enact the strategies associated therewith. The Nash bargain is an axiomatic solution, meaning that the solution point is defined by a set of (ostensibly) reasonable axioms rather than a concrete bargaining process. Rather than give a full description of the axioms, we briefly note that the NBS is characterized as the point on the Pareto boundary that satisfies affine invariance, symmetry between players, and independence to irrelevant alternatives. Nash showed that the unique* point satisfying the specified axioms is
Robustness with respect to class imbalance in artificial intelligence classification algorithms
Published in Journal of Quality Technology, 2021
Jiayi Lian, Laura Freeman, Yili Hong, Xinwei Deng
Moreover, based on the estimated model, it is useful to quantify the impact of the predictor variables to the response. Here we adopt the SHAP (SHapley Additive exPlanations) approach (Lundberg and Lee 2017) to quantitatively assess the impact of predictor variables to the response. The Shapley value, based on the cooperative game theory (Shapley 1953), can be applied in a wide variety of models and is not affected by the unit of measurement. The SHAP method assigns each predictor variable a Shapley value of importance for the predictive model. For the linear model in (1), the SHAP has an explicit form. The detailed explanations of the Shapely formula under the linear model can be found in the appendix. Denote as the importance of the label proportion variable xj to the model output for individual observation i. Similarly, we can define For the ith observation in our proposed model, the importance of predictor variables xj’s, ’s, ’s, and have the following forms:
Multi-objective optimization and cost-based output pricing of a standalone hybrid energy system integrated with desalination
Published in The Engineering Economist, 2020
Xi Luo, Yanfeng Liu, Xiaojun Liu
The Shapley value, developed by Shapley in (1953), is a solution concept in cooperative game theory. It ensures that the benefit to each player is equal to the average marginal contribution of the player in the coalition. For each cooperative game, this value assigns a unique distribution of the total surplus generated by the coalition of all players (Luo & Liu, 2016). The Shapley value is computed as follows: where is a set of all players in the game. Any subset is called a coalition and refers to the coalitions formed by players in the set based on their interests. For each coalition, is the corresponding characteristic function, which is the profit of each coalition in this study.
Moral hazard problem and collaborative coordination in supply chain with capacity reservation contract
Published in International Journal of Production Research, 2019
Yasuhiko Takemoto, Ikuo Arizono
In the cooperative game theory, the Nash bargaining solution (Nash 1950) has been proposed in the case that two players bargain about a deal. In the situation of this paper, the Nash bargaining solution is obtained from the following Nash product: where is particularly called the disagreement point of bargaining. The disagreement point of bargaining means the profits in the case that the conclusion of the contract would be not realised. In this paper, the disagreement point of bargaining is the combination of the profits in the supplier and manufacturer in the ordinary transaction without the reservation option. Note that the respective profits in the ordinary transaction without the reservation option are given as Equations (1) and (4) for .