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## Über dieses Buch

This contributed volume contains fourteen papers based on selected presentations from the European Conference on Game Theory SING11-GTM 2015, held at Saint Petersburg State University in July 2015, and the Networking Games and Management workshop, held at the Karelian Research Centre of the Russian Academy of Sciences in Petrozvavodsk, Russia, also in July 2015. These papers cover a wide range of topics in game theory, including recent advances in areas with high potential for future work, as well as new developments on classical results. Some of these includeA new approach to journal ranking using methods from social choice theory;
A differential game of a duopoly in which two firms are competing for market share in an industry with network externalities;
The impact of information propagation in the model of tax audits;
A voting model in which the results of previous votes can affect the process of coalition formation in a decision-making body;
The Selten-Szidarovsky technique for the analysis of Nash equilibria of games with an aggregative structure;
Generalized nucleoli and generalized bargaining sets for games with restricted cooperation;
Bayesian networks and games of deterrence; and
A new look at the study of solutions for games in partition function form.

The maturity and vitality of modern-day game theory are reflected in the new ideas, novel applications, and contributions of young researchers represented in this collection. It will be of interest to anyone doing theoretical research in game theory or working on one its numerous applications.

## Inhaltsverzeichnis

### Ranking Journals in Sociology, Education, and Public Administration by Social Choice Theory Methods

An analysis of journals’ rankings based on five commonly used bibliometric indicators (impact factor, article influence score, SNIP, SJR, and H-index) has been conducted. It is shown that despite the high correlation, these single-indicator-based rankings are not identical. Therefore, new approach to ranking academic journals is proposed based on the aggregation of single bibliometric indicators using several ordinal aggregation procedures. In particular, we use the threshold procedure, which allows to reduce opportunities for manipulations.

Fuad T. Aleskerov, Anna M. Boriskova, Vladimir V. Pislyakov, Vyacheslav I. Yakuba

### On the Position Value for Special Classes of Networks

This paper deals with a particular class of TU-games, whose cooperation is restricted by a network structure. We consider a communication situation (or graph game) in which a network is produced by subsequent formation of links among players and at each step of the network formation process, the surplus generated by a link is shared between the players involved, according to some rule. As a consequence, we obtain a family of solution concepts that we investigate on particular network structures. This approach provides a different interpretation of the position value, introduced by Borm et al. (SIAM J Discret Math 5(3):305–320, 1992), since it turns out that a specific symmetric rule leads to this solution concept. Moreover, we investigate the problem of computing the position value on particular classes of networks.

Giulia Cesari, Margherita Maria Ferrari

### A Differential Game of a Duopoly with Network Externalities

In this work, we develop a differential game of a duopoly where two firms compete for market share in an industry with Network Externalities. Here the evolution of the market share is modeled in such a way that the effects of advertising efforts that both firms make are a function of the share itself. This means that the efficacy of marketing efforts are diminished with low market share and enhanced when it is higher. We show that Network Externalities can influence the decision a firm makes about marketing expenditures. Particularly, when a firm is large enough, the creation of a monopoly is easier when this market structure is present. For this, we obtain the optimal strategies for the firms and test them on a simulation, where we compare the market with and without this kind of externalities. We find that the value of the market share in proportion with the cost of obtaining it by advertising efforts is the key to know the long term equilibrium market share.

Mario Alberto García-Meza, José Daniel López-Barrientos

### The Shapley Value as a Sustainable Cooperative Solution in Differential Games of Three Players

The contribution of the paper is twofold: first, it has been shown that the Yeung’s conditions can be used to construct a strongly time-consistent core. In this core there is a supporting imputation which has the property that a single deviation from this imputation in favor of any other imputation from the core still leads to the payment from the core. The obtained results were formulated for the Shapley value taken as the supporting imputation. Second, a particular class of differential games was considered. For this class of games the δ-characteristic function turns out to be superadditive and the Yeung’s conditions are satisfied without any additional restrictions on the parameters of the model. All results are presented in the analytic form.

Ekaterina Gromova

### Impact of Propagation Information in the Model of Tax Audit

An effective tax system is an important part of economic and social interactions in human society. The key element of the tax system is tax control which provides the main functions of taxation and allows for increasing tax revenue and fees to the state budget. However, total tax audits of a population of taxpayers is economically unreasonable, and even selective tax audits are not always profitable. In this case the propagation of information can be viewed as an “infection of the mind,” and its spread shows an interesting resemblance to that of epidemics. We thus use a modification of the classical Susceptible-Infected-Recovery model to describe the process. We assume that information propagates through the population by pairwise contacts between spreaders and others in the population and Informed agents disseminate information through their network of contacts or social networks. We study a model of spreading information in a large population of taxpayers and describe the dynamics of this process in complex social networks. We formulate an optimal control problem of tax auditing and analyze the behavior of agents in different subgroups depending on received information.

Elena Gubar, Suriya Kumacheva, Ekaterina Zhitkova, Olga Porokhnyavaya

### An Infinite Horizon Differential Game of Optimal CLV-Based Strategies with Non-atomic Firms

We study the structure of optimal customer acquisition and customer retention strategies as a differential game over an infinite horizon in an industry with a large number of non-atomic firms. The optimal retention effort is constant over time and the optimal acquisition effort is proportional to the size of potential customer base. Greater customer profitability leads to higher per- capita acquisition and retention efforts, larger size of firms, and lower churn rate. A greater discount rate leads to lower per-capita acquisition and retention efforts, smaller firm size, and a greater churn rate. Tougher competition lowers the firms’ acquisition and retention expenditures and it does not affect per-capita values. Both the churn rate and the share of acquisition expenditures in the total marketing budget decrease as firms grow over time. We revisit the concepts of the customer lifetime value (CLV) and the value of the firm in the dynamic equilibrium of an industry with a large number of players and demonstrate the equivalence between maximization of the value of the firm and maximization of a firm’s individual CLV.

Gerasimos Lianos, Igor Sloev

### A Dynamic Model of a Decision Making Body Where the Power of Veto Can Be Invoked

Classical analysis of the power of individuals or groups in decision making bodies tends to consider processes of coalition formation in isolation from each other. However, the results of attempts to form a winning coalition will affect the dynamics of coalition formation in the future, particularly when certain players are endowed with the power of veto. When a voter invokes their power of veto to block a generally popular motion, this is likely to provoke retaliation from the supporters of the motion. For this reason, even if a player is not in favor of a bill and can block it, it may be favorable for that party to abstain rather than veto in return for support regarding issues to be considered in the future. Hence, players should only use their power of veto if they are very strongly against a bill. In this paper, we present a model of voting in which the results of previous votes can affect the process of coalition formation. We present a model of such a dynamic voting game and present the form of an equilibrium in such a game. This theory is illustrated using an example based on the voting procedures used in the United Nations Security Council.

Jacek Mercik, David M. Ramsey

### The Selten–Szidarovszky Technique: The Transformation Part

A technique due to Selten and Szidarovszky for the analysis of Nash equilibria of games with an aggregative structure is reconsidered. Among other things it is shown that the transformation part of this technique can be extended to abstract games with co-strategy mappings and allows for a purely algebraic setting.

Pierre von Mouche

### Generalized Nucleoli and Generalized Bargaining Sets for Games with Restricted Cooperation

A generalization of the theory of the bargaining set, the kernel, and the nucleolus for cooperative TU-games, where objections and counter-objections are permitted only between the members of a collection of coalitions ?? $$\mathcal{A}$$ and can use only the members of a collection of coalitions ℬ⊃?? $$\mathcal{B}\supset \mathcal{A}$$ , is considered. Four versions of generalized bargaining set are possible. Three versions of generalized kernel and two versions of generalized nucleolus are defined. One generalized kernel, one generalized nucleolus, and the corresponding generalized bargaining sets were examined in Naumova (Contributions to Game Theory and Management, vol. 5, pp. 230–242. Graduate School of Management, St. Petersburg University, St. Petersburg, 2012; Contributions to Game Theory and Management GTM2014, vol. 8, pp. 231–242. St. Petersburg State University, St. Petersburg, 2015). Conditions on ?? $$\mathcal{A}$$ and ℬ $$\mathcal{B}$$ that ensure existence of the second generalized kernel are obtained. Weakly mixed collections of coalitions are defined. For such collections of coalitions, the second generalized nucleolus is contained in the second generalized kernel and in two generalized bargaining sets. If ?? $$\mathcal{A}$$ does not contain singletons such inclusion is valid for all games only if ?? $$\mathcal{A}$$ is a weakly mixed collection of coalitions. For weakly mixed collection of coalitions ?? $$\mathcal{A}$$ an iterative procedure that converges to a point in the second generalized kernel is described.

Natalia Naumova

### Occurrence of Deception Under the Oversight of a Regulator Having Reputation Concerns

This paper studies deceptions conducted by agents in the presence of a regulator. The regulator is supposed to detect deviations from the “rightful” behavior through costly monitoring; thus she may not choose to be diligent in her job because of the associated costs. The goal is to understand the occurrence of deceptions when the interaction of the parties is not contractible, their behavior is not observable and the regulator has reputation concern for being perceived as diligent in a repeated incomplete-information setting. It is found that when the regulator faces a sequence of myopic agents, her payoff at any Nash equilibrium converges to the maximum payoff as the discount factor approaches to one for any prior belief on the regulator’s type. This suggests that, contrary to the well-known disappearance of reputation results in the literature, the reputation of the regulator for being diligent persists in the long-run in any equilibrium. These findings imply that socially undesirable behavior of the agents could be prevented through reputation concerns in this repeated setting.

Ayça Özdog̃an

### Bayesian Networks and Games of Deterrence

The present paper analyzes possible bridges between two types of graphical models: on the one hand, Bayesian Networks, and on the other hand, Graphs of Deterrence associated with a particular category of qualitative games called matrix Games of Deterrence, in which players do not look for optimal outcomes but for acceptable ones. Three related-types of relations are scrutinized: implications and rebuttals; priors and hidden parts of the graph; probability and playability.

Michel Rudnianski, Utsav Sadana, Hélène Bestougeff

### A New Look at the Study of Solutions for Games in Partition Function Form

This chapter studies the structure of games in partition function and according to an axiomatic point of view, we provide a global description of linear symmetric solutions by means of a decomposition of the set of such games (as well as of a decomposition of the space of payoff vectors). The exhibition of relevant subspaces in such decomposition and based on the idea that every permutation of the set of players may be thought of as a linear map, allow for a new look at linear symmetric solutions.

Joss Sánchez-Pérez

### A Model of Tacit Collusion: Nash-2 Equilibrium Concept

We examine an equilibrium concept for 2-person non-cooperative games with boundedly rational agents which we call Nash-2 equilibrium. It is weaker than Nash equilibrium and equilibrium in secure strategies: a player takes into account not only current strategies but also all profitable next-stage responses of the partners to her deviation from the current profile that reduces her relevant choice set. We provide a condition for Nash-2 existence in finite games and complete characterization of Nash-2 equilibrium in strictly competitive games. Nash-2 equilibria in Hotelling price-setting game are found and interpreted in terms of tacit collusion.

Marina Sandomirskaia

### Strong Coalitional Structure in an Open Vehicle Routing Game

In the chapter it is investigated a special case of one-product open vehicle routing game, in which there is a central warehouse or wholesaler, several customers, who are considered to be players. Each player is placed in a node of the transportation network and is characterized by demand and distance to the warehouse. For such a problem a coalitional transportation game (CTG) is formalized. In such a game each customer (player) should rent a track to deliver goods from the central warehouse. It is assumed that all tracks have the same capacity. The players tend to minimize their transportation costs and totally supply their demands. A player may rent a vehicle alone, or chose a coalition of players to cooperate. In cooperation the players of coalitions find the shortest path form the central depot to all the player of coalition. Transportation costs are allocated between players according to the Nash arbitration scheme. Strong equilibrium which is stable against deviations of any coalition of players is found in a CTG. A computation procedure for strong equilibrium construction is proposed. Implementation of procedure is illustrated with a numerical example.

Nikolay Zenkevich, Andrey Zyatchin
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