We consider the problem of payoff division in indivisible coalitional games, where the value of the grand coalition is a natural number. This number represents a certain quantity of indivisible objects, such as parliamentary seats, kidney exchanges, or top features contributing to the outcome of a machine learning model. The goal of this paper is to propose a fair method for dividing these objects among players. To achieve this, we define the indivisible Shapley value and study its properties. We demonstrate our proposed technique using three case studies, in particular, we use it to identify key regions of an image in the context of an image classification task.

Reinforcement learning agents can achieve super-human performance in complex decision-making tasks, but their behaviour is often difficult to understand and explain. This lack of explanation limits deployment, especially in safety-critical settings where understanding and trust are essential. We identify three core explanatory targets that together provide a comprehensive view of reinforcement learning agents: behaviour, outcomes, and predictions. We develop a unified theoretical framework for explaining these three elements of reinforcement learning agents through the influence of individual features that the agent observes in its environment. We derive feature influences by using Shapley values, which collectively and uniquely satisfy a set of well-motivated axioms for fair and consistent credit assignment. The proposed approach, Shapley Values for Explaining Reinforcement Learning (SVERL), provides a single theoretical framework to comprehensively and meaningfully explain reinforcement learning agents. It yields explanations with precise semantics that are not only interpretable but also mathematically justified, enabling us to identify and correct conceptual issues in prior explanations. Through illustrative examples, we show how SVERL produces useful, intuitive explanations of agent behaviour, outcomes, and predictions, which are not apparent from observing agent behaviour alone.

Weaknesses: I believe the results proposed in this paper are related to existing work. The techniques used are close to existing methods - at the very least a detailed comparison is in order. The paper fails to acknowledge lots of literature on representing coalitional games in a restricted manner. In fact, many techniques have been proposed for concisely representing coalitional games, and approximately solving them. This issue is covered in depth in (e.g): Chalkiadakis, Georgios, Edith Elkind, and Michael Wooldridge.

While Shapley's analysis was originally thought to quantify the worth of human agents in a team, Research in the field of Multi-Agent Systems (MAS) suggests its application is straightforward to every other possible transferable viable pathways to solve complex tasks [1]. In a MAS utility coalitional game that respects the needed mathematical environment, every agent is, in principle, an individual independent properties. of one another with its own characteristics and skills. The field of possible applications of Shapley and Myerson The main idea is that by assigning to each agent a specific subtask analyses or their generalizations is broad. Shapley analysis or according to its perks and hence exploiting a delocalized its suitable generalizations can be applied for instance to estimate control, it is possible to solve a problem more efficiently. The the contributions of basketball players in a match using the human society itself is an example of a MAS since groups of recorded match data and statistics [3]. If the practitioner possesses individuals usually train according to their nature to exercise some information about the connectivity of interactions, specific professions that require different expertise: medical or, e.g., spatial rules of the game that restrict the interaction personnel, firefighters, engineers, etc. When analyzing the behavior among agents, Shapley and Myerson analyses can be used to of agents in a MAS a question arises immediately: according assess the importance of vertices, i.e., agents, in graphs. Recent to a common goal to be reached, which agent is contributing works investigated the Shapley and Myerson analyses of the most, and which are its most important individual transportation networks [4] and bus-holding strategies [5].

Feature attributions based on the Shapley value are popular for explaining machine learning models; however, their estimation is complex from both a theoretical and computational standpoint. We disentangle this complexity into two factors: (1)~the approach to removing feature information, and (2)~the tractable estimation strategy. These two factors provide a natural lens through which we can better understand and compare 24 distinct algorithms. Based on the various feature removal approaches, we describe the multiple types of Shapley value feature attributions and methods to calculate each one. Then, based on the tractable estimation strategies, we characterize two distinct families of approaches: model-agnostic and model-specific approximations. For the model-agnostic approximations, we benchmark a wide class of estimation approaches and tie them to alternative yet equivalent characterizations of the Shapley value. For the model-specific approximations, we clarify the assumptions crucial to each method's tractability for linear, tree, and deep models. Finally, we identify gaps in the literature and promising future research directions.

In constructing an econometric or statistical model, we pick relevant features or variables from many candidates. A coalitional game is set up to study the selection problem where the players are the candidates and the payoff function is a performance measurement in all possible modeling scenarios. Thus, in theory, an irrelevant feature is equivalent to a dummy player in the game, which contributes nothing to all modeling situations. The hypothesis test of zero mean contribution is the rule to decide a feature is irrelevant or not. In our mechanism design, the end goal perfectly matches the expected model performance with the expected sum of individual marginal effects. Within a class of noninformative likelihood among all modeling opportunities, the matching equation results in a specific valuation for each feature. After estimating the valuation and its standard deviation, we drop any candidate feature if its valuation is not significantly different from zero. In the simulation studies, our new approach significantly outperforms several popular methods used in practice, and its accuracy is robust to the choice of the payoff function.

A variety of recent papers discuss the application of Shapley values, a concept for explaining coalitional games, for feature attribution in machine learning. However, the correct way to connect a machine learning model to a coalitional game has been a source of controversy. The two main approaches that have been proposed differ in the way that they condition on known features, using either (1) an interventional or (2) an observational conditional expectation. While previous work has argued that one of the two approaches is preferable in general, we argue that the choice is application dependent. Furthermore, we argue that the choice comes down to whether it is desirable to be true to the model or true to the data. We use linear models to investigate this choice. After deriving an efficient method for calculating observational conditional expectation Shapley values for linear models, we investigate how correlation in simulated data impacts the convergence of observational conditional expectation Shapley values. Finally, we present two real data examples that we consider to be representative of possible use cases for feature attribution -- (1) credit risk modeling and (2) biological discovery. We show how a different choice of value function performs better in each scenario, and how possible attributions are impacted by modeling choices.

The research on coalitional games has focused on how to share the reward among a coalition such that players are in-centivised to collaborate together. It assumes that the (deterministic or stochastic) characteristic function is known in advance. This paper studies a new setting (a task allocation problem) where the characteristic function is not known and it is controlled by some private information from the players. Hence, the challenge here is twofold: (i) incentivize players to reveal their private information truthfully, (ii) incentivize them to collaborate together. We show that existing reward distribution mechanisms or auctions cannot solve the challenge. Hence, we propose the very first mechanism for the problem from the perspective of both mechanism design and coalitional games.

It's Not Whom You Know, It's What You (or Your Friends) Can Do: Succint Coalitional Frameworks for Network Centralities. September 25, 2019 Abstract We investigate the representation of game-theoretic measures of network centrality using a framework that blends a social network representation with the succint formalism of cooperative skill games. We discuss the expressiveness of the new framework and highlight some of its advantages, including a fixed-parameter tractability result for computing centrality measures under such representations. As an application we introduce new network centrality measures that capture the extent to which neighbors of a certain node can help it complete relevant tasks. 1 Introduction Measures of network centrality have a long and rich history in the social sciences [1] and Artificial Intelligence. Such measures have proved useful for a variety of tasks, such as identifying spreading nodes [2] and gatekeepers for information dissemination [3], advertising ...

Artificial Neural Networks have shown impressive success in very different application cases. Choosing a proper network architecture is a critical decision for a network's success, usually done in a manual manner. As a straightforward strategy, large, mostly fully connected architectures are selected, thereby relying on a good optimization strategy to find proper weights while at the same time avoiding overfitting. However, large parts of the final network are redundant. In the best case, large parts of the network become simply irrelevant for later inferencing. In the worst case, highly parameterized architectures hinder proper optimization and allow the easy creation of adverserial examples fooling the network. A first step in removing irrelevant architectural parts lies in identifying those parts, which requires measuring the contribution of individual components such as neurons. In previous work, heuristics based on using the weight distribution of a neuron as contribution measure have shown some success, but do not provide a proper theoretical understanding. Therefore, in our work we investigate game theoretic measures, namely the Shapley value (SV), in order to separate relevant from irrelevant parts of an artificial neural network. We begin by designing a coalitional game for an artificial neural network, where neurons form coalitions and the average contributions of neurons to coalitions yield to the Shapley value. In order to measure how well the Shapley value measures the contribution of individual neurons, we remove low-contributing neurons and measure its impact on the network performance. In our experiments we show that the Shapley value outperforms other heuristics for measuring the contribution of neurons.