Yukihiko Funaki

Coincidence of and collinearity between game theoretic solutions

SummaryThe first part is the study of several conditions which are sufficient for the coincidence of the prenucleolus concept and the egalitarian nonseparable contribution (ENSC-) method. The main sufficient condition for the coincidence involved requires that the maximal excesses at the ENSC-solution are determined by the (n-1)-person coalitions in then-person game. The second part is the study of both a new type of games, the so-calledk-coalitionaln-person games, and the interrelationship between solutions on the class of those games. The main results state that the Shapley value of ak-coalitionaln-person game can be written as a convex or affine combination of the ENSC-solution and the centre of the imputation set.ZusammenfassungIm ersten Teil der Arbeit werden verschiedene hinreichende Bedingungen für die Koinzidenz des Prenukleolus-Lösungskonzepts und der ENSC-Regel vorgestellt. Es wird dabei gezeigt, daß der Prenukleolus mit der ENSC-Lösung zusammenfällt, falls die maximalen Exzesse der ENSC-Lösung durch die (n-1)-Personen Koalitionen desn-Personenspiels bestimmt werden. Im zweiten Teil der Arbeit untersuchen wir eine Klasse von Spielen, die sogenanntenk-Koalitions-n-Personenspiele und untersuchen die Zusammenhänge zwischen den Lösungskonzepten für diesen speziellen Typ von Spielen. Es stellt sich heraus, daß der Shapleywert einesk-Koalitions-n-Personenspiels beschrieben werden kann als eine Linearkombination der ENSC-Lösung und des Schwerpunktes der Auszahlungsmenge.

The Core of an Economy with a Common Pool Resource: A Partition Function Form Approach

Abstract. In this paper we consider a model of an economy with a common pool resource. Under decreasing returns to scale, it is well-known that no Nash equilibrium attains Pareto efficiency. We examine whether it is possible to achieve Pareto efficiency and avoid the tragedy of the commons through cooperation among players. For that purpose, we use the notion of a game in partition function form. Whether or not the core exists depends crucially on the expectations of each coalition regarding the coalition formation of the outsiders. If each coalition has pessimistic expectations, then the core always exists, while if it has optimistic expectations, the core may be empty.

Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values

One of the main issues in economic allocation problems is the trade-off between marginalism and egalitarianism. In the context of cooperative games this trade-off can be framed as one of choosing to allocate according to the Shapley value or the equal division solution. In this paper we provide three different characterizations of egalitarian Shapley values being convex combinations of the Shapley value and the equal division solution. First, from the perspective of a variable player set, we show that all these solutions satisfy the same reduced game consistency. Second, on a fixed player set, we characterize this class of solutions using monotonicity properties. Finally, towards a strategic foundation, we provide a non-cooperative implementation for these solutions which only differ in the probability of breakdown at a certain stage of the game. These characterizations discover fundamental differences as well as intriguing connections between marginalism and egalitarianism.

Theoretical and experimental investigations of the performance of keyword auction mechanisms

Two keyword auction mechanisms, the Generalized Second-Price auction (GSP) and the Vickrey-Clarke-Groves mechanism (VCG), were compared theoretically and experimentally. The former is widely used in practice; the latter is not, but it has a dominant strategy equilibrium where all participants bid their true values. In the theoretical investigation, by applying the �locally envy-free Nash equilibrium� to the VCG, we found that the allocations are efficient and that upper and lower bounds of the auctioneers revenue coincide in the two mechanisms. A laboratory experiment, in which the revenues and efficiencies were similar in both mechanisms, supported this result

A coalition formation value for games in partition function form

The coalition formation problem in an economy with externalities can be adequately modeled by using games in partition function form (PFF games), proposed by Thrall and Lucas. If we suppose that forming the grand coalition generates the largest total surplus, a central question is how to allocate the worth of the grand coalition to each player, i.e., how to find an adequate solution concept, taking into account the whole process of coalition formation. We propose in this paper the original concepts of scenario-value, process-value and coalition formation value, which represent the average contribution of players in a scenario (a particular sequence of coalitions within a given coalition formation process), in a process (a sequence of partitions of the society), and in the whole (all processes being taken into account), respectively. We give also two axiomatizations of our coalition formation value.

Sustaining cooperation in social dilemmas: Comparison of centralized punishment institutions

This study investigates two centralized punishment institutions for a linear public goods game. These institutions require a certain contribution level and sanction under-contributing players. The two differ in who, among those who do not meet this requirement, receive sanctions. In one institution, all violators are sanctioned, and in the other, only the worst violator(s) is sanctioned. Theoretically, the public goods game of the latter institution yields contributions equal to or greater than that of the former institution with the same requirement and sanction level. The results of an experiment support this theoretical prediction. However, there is a discrepancy between the theory and laboratory observations in that the institution with the theoretically optimal requirement did not yield the highest profit.

THE CORE AND CONSISTENCY PROPERTIES: A GENERAL CHARACTERISATION

In this paper, we unify various axiomatisations of the core of TU games by means of consistency with respect to different definitions of reduced games. First, we introduce a general definition of reduced games including the reduced games due to Davis and Maschler (1965), Moulin (1985), and Funaki (1995) as special cases. Then, we provide a general theorem from which the characterisations due to Peleg (1986), Tadenuma (1992), and Funaki (1995) can be obtained. Our general theorem clarifies how the three characterisations of the core differ and are related.

Reduced Game Properties of Egalitarian Division Rules for TU-Games

The egalitarian non-individual contribution (ENIC-)value represents the equal division of the surplus of the total profits, given that each player is already allocated some kind of a yet unspecified individual contribution. Four particular versions, the CIS-, ENSC-, ENPAC-, ENBC-values, are also considered by choosing the individual worth, the separable contribution, the pairwise-averaged contribution and the Banzhaf contribution as the notion of individual contribution. Axiomatic characterizations of the ENIC-value in general and the four particular ENIC values are provided on the class of cooperative games with a fixed player set as well as a variable player set. The latter axiomatization involves a consistency axiom in terms of reduced games.

Collinearity between the Shapley value and the egalitarian division rules for cooperative games

For each cooperativen-person gamev and eachh∈{1, 2, ⋯,n}, letvh be the average worth of coalitions of sizeh andvhi the average worth of coalitions of sizeh which do not contain playeri∈N. The paper introduces the notion of a proportional average worth game (or PAW-game), i.e., the zero-normalized gamev for which there exist numbersch∈ℝ such thatvh−vhi=ch (vn−1−vn−1/i) for allh∈{2, 3, ⋯,n−1}, andi∈N. The notion of average worth is used to prove a formula for the Shapley value of a PAW-game. It is shown that the Shapley value, the value representing the center of the imputation set, the egalitarian non-separable contribution value and the egalitarian non-average contribution value of a PAW-game are collinear. The class of PAW-games contains strictly the class ofk-coalitional games possessing the collinearity property discussed by Driessen and Funaki (1991). Finally, it is illustrated that the unanimity games and the landlord games are PAW-games.ZusammenfassungSeiv ein kooperativesn-Personenspiel und seih∈{1, 2, ⋯,n}. Mitvh bezeichnen wir die mittlere Auszahlung aller Koalitionen der Größeh und mitvhi die mittlere Auszahlung aller Koalitionen der Größeh, die den Spieleri∈N nicht enthalten. In dieser Arbeit, führen wir den Begriff des Spieles mit proportionaler mittlerer Auszahlung (oder PMA-Spiel) ein. Diese sind null-reduzierte Spielev, für die Zahlench∈ℝ existieren, sodaß die Beziehungvh−vhi=ch (vn−1−vn−1/i) für jedesh∈{2, 3, ⋯,n−1 undi∈N gilt. Der Begriff der mittleren Auszahlung wird dann benutzt, um eine Formel für den Shapley-Wert der PMA-Spiele abzuleiten. Wir zeigen, daß der Shapley-Wert, und die durch das Zentrum der Imputationsmenge, die gleichmäßigen nicht-separablen Beiträge, bzw. gleichmäßigen nicht-gemittelten Beiträge definierten Werte der PMA-Spiele kollinear sind. Die Klasse aller PMA-Spiele enthält im strengen Sinne die Klasse allerk-Koalitionsspiele, die die Kollinearitätseigenschaft haben (Driessen und Funaki, 1991). Schließlich zeigen wir, daß die Einstimmigkeitsspiele und die Grundbesitzerspiele auch PMA-Spiele sind.

The impact of the Tohoku earthquake and tsunami on social capital in Japan: Trust before and after the disaster

We report empirical results related to trust and trustworthiness based on a representative web survey carried out in March 2011 in Japan. Although it initially was intended as a pilot, our survey is unique and unrepeatable because the massive Tohoku earthquake that hit Japan in spring 2011 occurred during the data-collection process and created a natural experiment. Apart from exploring changes originated by the disaster, the novelty of our approach lies in using a multipurpose questionnaire assembled by researchers with diverse interests from different academic areas that allows for exploring political and other social correlates of the economic concepts of trust and trustworthiness as measured by the game-theoretical trust game.