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This paper reviews the recent literature on the “social ranking problem”, that is, the problem of converting group rankings into individual rankings. We introduce and categorize existing social ranking methods and we briefly explain their attributes. Three main categories of social ranking methods are identified: lexicographic social rankings, methods based on voting mechanisms, and those inspired by the theory of coalitional games. An open-source R package called socialranking for computing the majority of the existing social rankings is also presented and discussed.
Computing Shapley values for large cooperative games is an NP-hard problem. For practical applications, stochastic approximation via permutation sampling is widely used. In the context of machine learning applications of the Shapley value, the concept of antithetic sampling has become popular. The idea is to employ the reverse permutation of a sample in order to reduce variance and accelerate convergence of the algorithm. We study this approach for the Shapley and Banzhaf values, as well as for the Owen value which is a solution concept for games with precoalitions. We combine antithetic samples with established stratified sampling algorithms. Finally, we evaluate the performance of these algorithms on four different types of cooperative games.
The Asymmetric Public Goods Game (APGG) C++ framework offers an easy to use environment to study game theoretical questions. Specifically, it is designed to address questions in the domain of asymmetric public goods games.
We hereby publish version 1.1.2 of our software APGG (Release for DOI via Zenodo for paper in JOSS, the Journal of Open Source Software).
The Asymmetric Public Goods Game (APGG) C++ framework offers an easy to use environment to study game theoretical questions. Specifically, it is designed to address questions in the domain of asymmetric public goods games. The modular architecture allows for a vast amount of scenarios and setups for experimenting with different public goods games, using easy to change parameters. Users can experiment with well mixed and structured populations as well as with symmetric and asymmetric payoffs. APGG also features group level payoffs and individual payoffs, and different evolutionary selection mechanisms (Miller et al., 1995) and replication schemes. Results are automatically saved in semantic and descriptive structures and can be easily visualized with the included Python scripts. This paper aims to explain the functionality and the structure of the framework, to show the workflow that APGG follows, to present the different modules that are available, and to show how APGG can be used to run experiments with public goods games on example scenarios.
The article studies the efficient computation of the Public Good index defined by Manfred Holler in 1982 (and also known as the Holler index or as the Holler–Packel index) as well as variations of that power index defined in scientific works by Manfred Holler allowing for precoalitions among subsets of players. Starting from the state-of-the-art algorithm for computing the Public Good index for weighted voting games the paper presents a framework for fast algorithms for six variants of the Public Good index with precoalitions. The study discusses implementations of the Public Good indices with precoalitions in C++, reviews computing times, and points out that the new algorithms are applicable for large numbers of players.
The purpose of this paper is to introduce new methods to measure the indirect control power of firms in complex corporate shareholding structures using the concept of power indices from cooperative game theory. The proposed measures vary in desirable properties satisfied, as well as in the bargaining models of power indices used to construct them. Hence, they can be used to produce different pictures of the coalitional strength of firms in control of other firms in mutual shareholding networks with the presence of cycles. Precisely, in the framework of Karos and Peters from 2015, ten power indices substitute the original Shapley and Shubik power index in a modular fashion. In this way, we obtain a set of new measures called aggregated indices. The float shareholders typically hold less than 5 percent of the outstanding shares, which is an uncertain element of indirect control in complex shareholding structures. The fuzzy number seems appropriate to model these shareholders’ behavior. The novelty is that we model the behavior of float using Z-fuzzy numbers. The new methods are tested in an example.
The article belongs to the Special Issue "Decision Optimization in Information Theory and Game Theory" of the journal "Entropy".
Package ‘rSRD’
(2023)
We provide an implementation for Sum of Ranking Differences (SRD),
a novel statistical test introduced by Héberger (2010)
<doi:10.1016/j.trac.2009.09.009>. The test allows the comparison of
different solutions through a reference by first performing a rank
transformation on the input, then calculating and comparing the distances
between the solutions and the reference - the latter is measured in the
L1 norm. The reference can be an external benchmark (e.g. an established
gold standard) or can be aggregated from the data. The calculated distances,
called SRD scores, are validated in two ways, see Héberger and Kollár-Hunek
(2011) <doi:10.1002/cem.1320>. A randomization test (also called permutation
test) compares the SRD scores of the solutions to the SRD scores of randomly
generated rankings. The second validation option is cross-validation that
checks whether the rankings generated from the solutions come from the same
distribution or not. For a detailed analysis about the cross-validation
process see Sziklai, Baranyi and Héberger (2021) <arXiv:2105.11939>. The
package offers a wide array of features related to SRD including the computation
of the SRD scores, validation options, input preprocessing and plotting tools.
This paper discusses algorithms for measuring indirect control in complex corporate shareholding networks and investigates the importance of mutual connections in the network in the sense of shareholdings of one firm in another. Our algorithms rely on the concept of power indices from cooperative game theory. We focus on a variant of the implicit power index by Stach and Mercik based on the absolute Banzhaf index. We extend this algorithm by determining the number of regressions in an adaptive network-dependent manner taking into account the maximal length of a path to each controlled company in the network and by a model for the float, i.e., the set of unidentified small shareholders. We compare our method with existing algorithms and discuss the importance of linkages by investigating divestment of shares for a theoretical network with 21 players.
This document gives a few use cases for the EvolutionaryGames package. EvolutionaryGames provides basic concepts of evolutionary game theory, like e.g. finding evolutionary stable strategies and computing and drawing evolutionarily stable sets as well as phase diagrams for various evolutionary dynamics for single-population games with two, three and four different phenotypes.
We study the efficient computation of power indices for weighted voting games with precoalitions amongst subsets of players (reflecting, e.g., ideological proximity) using the paradigm of dynamic programming. Starting from the state-of-the-art algorithms for computing the Banzhaf and Shapley–Shubik indices for weighted voting games, we present a framework for fast algorithms for the three most common power indices with precoalitions, i.e., the Owen index, the Banzhaf–Owen index and the symmetric coalitional Banzhaf index, and point out why our new algorithms are applicable for large numbers of players. We discuss implementations of our algorithms for the three power indices with precoalitions in C++ and review computing times, as well as storage requirements.
This article deals with measuring indirect control in complex corporate shareholding networks using the concept of power indices from cooperative game theory. We focus on the approaches by Mercik-Łobos and Stach-Mercik which measure the control power of all firms involved in shareholding networks with algorithms based on the raw Johnston index. We point out how these approaches can be generalized replacing the raw Johnston index by various other power indices in a modular fashion. We further extend the algorithmic framework by investigating more than one regression and present requirements for software and modelling. Finally, we test the new framework of generalized implicit power indices for a network with 21 players and discuss how properties of the underlying power index like efficiency or null player removability influence the measurements of indirect control.
The aim of the article is to propose a new method of valuation of a company, considering its ownership relations with other companies. For this purpose, the concept of the Shapley value from cooperative game theory is used as the basis for assessing such dependent companies. The paper presents proposals for Shapley value calculation algorithms for our model. We expand our model by discussing personal relations in addition to ownership relations and point out how intuitionistic fuzzy sets may be helpful in this context. As a result, we propose two new expanded models. In the first probabilistic model, we apply Pearson’s correlation coefficient, in the second, we use a correlation coefficient between intuitionistic fuzzy sets to determine the personal relationships. Finally, we present and interpret results for a real-world economic network with 17 companies.
We study the efficient computation of power indices for weighted voting games using the paradigm of dynamic programming. We survey the state-of-the-art algorithms for computing the Banzhaf and Shapley–Shubik indices and point out how these approaches carry over to related power indices. Within a unified framework, we present new efficient algorithms for the Public Good index and a recently proposed power index based on minimal winning coalitions of the smallest size, as well as a very first method for computing the Johnston indices for weighted voting games efficiently. We introduce a software package providing fast C++ implementations of all the power indices mentioned in this article, discuss computing times, as well as storage requirements.
Package ‘EvolutionaryGames’
(2017)
Package ‘CoopGame’
(2019)
The theory of cooperative games with transferable utility offers useful insights into the way parties can share gains from cooperation and secure sustainable agreements, see e.g. one of the books by Chakravarty,Mitra and Sarkar (2015, ISBN:978-1107058798) or by Driessen (1988,ISBN:978-9027727299) for more details. A comprehensive set of tools for cooperative game theory with transferable utility is provided. Users can create special families of cooperative games, like e.g. bankruptcy games,cost sharing games and weighted voting games. There are functions to check various game properties and to compute five different set-valued solution concepts for cooperative games. A large number of point-valued solution concepts is available reflecting the diverse application areas of cooperative game theory. Some of these point-valued solution concepts can be used to analyze weighted voting games and measure the influence of individual voters within a voting body. There are routines for visualizing both set-valued and point-valued solutions in the case of three or four players.
This document gives a brief and concise overview of the various functionalities of the package CoopGame and presents a few use cases. In particular, we introduce the capabilities of CoopGame to create special families of cooperative games, to check game properties and to compute set-valued and point-valued solutions. We also introduce the usage of CoopGame for visualizing set-valued and point-valued solutions in the caseof three or four players. We end with a brief outlook to future developments. This vignette accompanies version 0.2.1 of the package CoopGame.
Using EvolutionaryGames
(2017)
This html-vignette gives a few use cases for the R package EvolutionaryGames. EvolutionaryGames provides basic concepts of evolutionary game theory, like e.g. finding evolutionary stable strategies and computing and drawing evolutionarily stable sets as well as phase diagrams for various evolutionary dynamics for single-population games with two, three and four different phenotypes.
We are studying the Gately point, an established solution concept for cooperative games. We point out that there are superadditive games for which the Gately point is not unique, i.e. in general the concept is rather set-valued than an actual point. We derive conditions under which the Gately point is guaranteed to be a unique imputation and provide a geometric interpretation. The Gately point can be understood as the intersection of a line defined by two points with the set of imputations. Our uniqueness conditions guarantee that these two points do not coincide. We provide demonstrative interpretations for negative propensities to disrupt. We briefly show that our uniqueness conditions for the Gately point include quasibalanced games and discuss the relation of the Gately point to the τ-value in this context. Finally, we point out relations to cost games and the ACA method and end upon a few remarks on the implementation of the Gately point and an upcoming software package for cooperative game theory.
The public goods game is a famous example illustrating the tragedy of the commons (Hardin in Science 162:1243–1248, 1968). In this game cooperating individuals contribute to a pool, which in turn is distributed to all members of the group, including defectors who reap the same rewards as cooperators without having made a contribution before. The question is now, how to incentivize group members to all cooperate as it maximizes the common good. While costly punishment (Helbing et al. in New J Phys 12:083005, 2010) presents one such method, the cost of punishment still reduces the common good. The selfishness of the group members favors defectors. Here we show that including other members of the groups and sharing rewards with them can be another incentive for cooperation, avoiding the cost required for punishment. Further, we show how punishment and this form of inclusiveness interact. This work suggests that a redistribution similar to a basic income that is coupled to the economic success of the entire group could overcome the tragedy of the commons.