Comparative accuracy of value solutions in non-sidepayment games with empty core

This article reports a test of the predictive accuracy of solution concepts in cooperative non-sidepayment n-person games with empty core. Six solutions were tested. Three of these were value solutions (i.e., -transfer value, -transfer nucleolus, and -transfer disruption value) and three were equilibrium solutions (deterrence set, stable set, and imputation set). The test was based on a laboratory experiment utilizing 5-person, 2-choice normal form games with empty core; other related data sets were also analyzed. Goodness-of-fit results based on discrepancy scores show that the three value solutions are about equally accurate in predicting outcomes, and that all three are substantially more accurate than the other solutions tested.     

A test of the characteristic function and the Harsanyi function in N-person normal form sidepayment games

This paper reports the results of a laboratory experiment investigating sidepayment games represented in normal form. Attempts to predict payoff allocations via the application of solution concepts (such as the Shapley value or the nucleolus) encounter a problem in games of this form, because the game must first be transformed into some other form. Commonly, this other form is a set function defined over coalitions, such as the von Neumann-Morgenstern characteristic function. Because there are numerous possible transformations, the question arises as to which one provides the most accurate basis for prediction of payoffs.The laboratory experiment tested three such transformations - the mixed strategy characteristic function, the pure strategy characteristic function, and the Harsanyi threat function. Payoff predictions from two solution concepts (Shapley value, nucleolus) were computed on the basis of each of these transformations, making a total of six theories under test.Results of the study show, in general, that payoff predictions based on the Harsanyi threat function and on the mixed strategy characteristic function were more accurate than those based on the pure strategy characteristic function. The most accurate theories were the Shapley value computed from the Harsanyi function, the nucleolus computed from the Harsanyi function, and the Shapley value computed from the mixed strategy characteristic function. Less accurate were the nucleolus computed from the mixed strategy characteristic function and both the nucleolus and the Shapley value computed from the pure strategy characteristic function.This research was supported by grants SOC-7726932 and SES-8319322 from the National Science Foundation. The authors express appreciation to Yat-Tuck See, Jyh-Jen Horng Shiau, and Raymond Wong for assistance in computer programming, and to Jennifer Brandt, Young C. Choi, David C. Dettman, Laurel Dettman, Stephen B. Geisheker, Irving J. Ginsberg, Mike P. Griffin, Kimberly Ihm, Todd Isaacson, Christy Kinney, Mary Kohl, Sue Pope, Tammy Schmieden, Jill Schwarze, Susan Winter, and Kenneth Yuen for assistance in data collection and analysis.     

Domination structures and multicriteria problems in n-person games

Multiple criteria decision problems with one decision maker have been recognized and discussed in the recent literature in optimization theory, operations research and management science. The corresponding concept with n-decision makers, namely multicriteria n-person games, has not yet been extensively explored.In this paper we first demonstrate that existing solution concepts for single criterion n-person games in both normal form and characteristic function form induce domination structures (similar to those defined and studied by Yu [39] for multicriteria single decision maker problems) in various spaces, including the payoff space, the imputation space and the coalition space. This discussion provides an understanding of some underlying assumptions of the solution concepts and provides a basis for generalizing and generating new solution concepts not yet defined. Also we illustrate that domination structures may be regarded as a measure of power held by the players.We then illustrate that a multicriteria problem can naturally arise in decision situations involving (partial) conflict among n-persons. Using our discussion of solution concepts for single criterion games as a basis, various approaches for resolving both normal form and characteristic function form multicriteria n-person games are proposed. For multicriteria games in characteristic function form, we define a multicriteria core and show that there exists a single game point whose core is equal to the multicriteria core. If we reduce a multicriteria game to a single criterion game, domination structures which are more general than classical ones must be considered, otherwise some crucial information in the game may be lost. Finally, we discuss a parametrization process which, for a given multicriteria game, associates a single criterion game to each point in a parametric space. This parametrization provides a basis for the discussion of solution concepts in multicriteria n-person games.     

Procedural and optimization implementation of the weighted ENSC value

The main purpose of this article is to introduce the weighted ENSC value for cooperative transferable utility games which takes into account players’ selfishness about the payoff allocations. Similar to Shapley’s idea of a one-by-one formation of the grand coalition [Shapley (1953)], we first provide a procedural implementation of the weighted ENSC value depending on players’ selfishness as well as their marginal contributions to the grand coalition. Second, in the spirit of the nucleolus [Schmeidler (1969)], we prove that the weighted ENSC value is obtained by lexicographically minimizing a complaint vector associated with a new complaint criterion relying on players’ selfishness.

     

A test of the Core,Bargaining Set,Kernel and Shapley models in n-person quota games with one weak player

n-person (n – 1)-quota-games where the quotas are positive for certain n – 1 (strong) players and negative for the remaining (weak) player, are discussed. Normative solutions predicted by the Core,the Kernel, the Bargaining Set, the Competitive Bargaining Set, and by the Shapley Value are presented and exemplified.Each of twelve groups of subjects participated in a four-person and a five-person (n – 1)-quota games with one weak player. The weak player was always excluded from the ratified coalition. The division of payoffs among the strong players was more egalitarian than the Kernel solution but less egalitarian than the Shapley value. The Core and the Bargaining Sets were fully supported for the two strongest players, but less supported for the other players. Analyses of the bargaining process confirmed a dynamic interpretation of the Bargaining Set Theory.This research was performed while the author was at the University of North Carolina. The research was partially supported by a PHS Research Grant No. MH-10006 from the National Institute of Mental Health. The author thanks Professor Amnon Rapoport for helpful advice in the design of this study.     

Population monotonic path schemes for simple games

A path scheme for a game is composed of a path, i.e., a sequence of coalitions that is formed during the coalition formation process and a scheme, i.e., a payoff vector for each coalition in the path. A path scheme is called population monotonic if a player’s payoff does not decrease as the path coalition grows. In this study, we focus on Shapley path schemes of simple games in which for every path coalition the Shapley value of the associated subgame provides the allocation at hand. Obviously, each Shapley path scheme of a game is population monotonic if and only if the Shapley allocation scheme of the game is population monotonic in the sense of Sprumont (Games Econ Behav 2:378–394, 1990). We prove that a simple game allows for population monotonic Shapley path schemes if and only if the game is balanced. Moreover, the Shapley path scheme of a specific path is population monotonic if and only if the first winning coalition that is formed along the path contains every minimal winning coalition. We also show that each Shapley path scheme of a simple game is population monotonic if and only if the set of veto players of the game is a winning coalition. Extensions of these results to other efficient probabilistic values are discussed.     

Uncertainty with Partial Information on the Possibility of the Events

The Choquet expected utility model deals with nonadditive probabilities (or capacities). Their dependence on the information the decision-maker has about the possibility of the events is taken into account. Two kinds of information are examined: interval information (for instance, the percentage of white balls in an urn is between 60% and 100%) and comparative information (for instance, the information that there are more white balls than black ones). Some implications are shown with regard to the core of the capacity and to two additive measures which can be derived from capacities: the Shapley value and the nucleolus. Interval information bounds prove to be satisfied by all probabilities in the core, but they are not necessarily satisfied by the nucleolus (when the core is empty) and the Shapley value. We must introduce the constrained nucleolus in order for these bounds to be satisfied, while the Shapley value does not seem to be adjustable. On the contrary, comparative information inequalities prove to be not necessarily satisfied by all probabilities in the core and we must introduce the constrained core in order for these inequalities be satisfied. However, both the nucleolus and the Shapley value satisfy the comparative information inequalities, and the Shapley value does it more strictly than the nucleolus. This revised version was published online in June 2006 with corrections to the Cover Date.     

An Empirical Comparison of Probabilistic Coalition Structure Theories in 3-Person Sidepayment Games

This article reports a comparative test of the central-union theory vis-à-vis several other game-theoretic solution concepts in 3-person sidepayment games. Based on a laboratory experiment, this comparison utilizes nine games in characteristic function form. The solution concepts under test include the equal excess model, the Myerson–Shapley solution, the kernel, and two variants of the central-union theory (CU-1 and CU-2). With regard to the player's payoffs, results show that the CU-1, CU-2, kernel, and equal excess theories have essentially equal predictive accuracy and that all of these are more accurate than Myerson–Shapley. When the solution concepts are extended and coalition structure probability predictions are incorporated in the test, one version of the central-union theory (CU-2) is overall more accurate than the other solutions.     

Game-theoretic analyses of coalition behavior

Coalitions are frequently more visible than payoffs. The theory of n-person games seeks primarily to identify stable allocations of valued resources; consequently, it gives inadequate attention to predicting which coalitions form. This paper explores a way of correcting this deficiency of game-theoretic reasoning by extending the theory of two-person cooperative games to predict both coalitions and payoffs in a three-person game of status in which each player seeks to maximize the rank of his total score. To accomplish this, we analyze the negotiations within each potential two-person coalition from the perspective of Nash's procedure for arbitrating two-person bargaining games, then assume that players expect to achieve the arbitrated outcome selected by this procedure and use these expectations to predict achieved ranks and to identify players' preferences between alternative coalition partners in order to predict the probability that each coalition forms. We test these payoff and coalition predictions with data from three laboratory studies, and compare the results with those attained in the same data by von Neumann and Morgenstern's solution of two-person cooperative games, Aumann and Maschler's bargaining set solution for cooperative n-person games, and an alternative model of coalition behavior in three-person sequential games of status.

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Three-valued simple games

In this paper we study three-valued simple games as a natural extension of simple games. We analyze to which extent well-known results on the core and the Shapley value for simple games can be extended to this new setting. To describe the core of a three-valued simple game we introduce (primary and secondary) vital players, in analogy to veto players for simple games. Moreover, it is seen that the transfer property of Dubey (1975) can still be used to characterize the Shapley value for three-valued simple games. We illustrate three-valued simple games and the corresponding Shapley value in a parliamentary bicameral system.     

Hyperplane games,prize games and NTU values

The Shapley value is a well-known solution concept for TU games. The Maschler–Owen value and the NTU Shapley value are two well-known extensions of the Shapley value to NTU games. A hyperplane game is an NTU game in which the feasible set for each coalition is a hyperplane. On the domain of monotonic hyperplane games, the Maschler–Owen value is axiomatized (Hart Essays in game theory. Springer, 1994). Although the domain of hyperplane game is a very interesting class of games to study, unfortunately, on this domain, the NTU Shapley value is not well-defined, namely, it assigns an empty set to some hyperplane games. A prize game (Hart Essays in game theory. Springer, 1994) is an NTU game that can be obtained by “truncating” a hyperplane game. As such, a prize game describes essentially the same situation as the corresponding hyperplane game. It turns out that, on the domain of monotonic prize games, the NTU Shapley value is well-defined. Thus, one can define a value which is well-defined on the domain of monotonic hyperplane games as follows: given a monotonic hyperplane game, first, transform it into a prize game, and then apply the NTU Shapley value to it. We refer to the resulting value as the “generalized Shapley value” and compare the axiomatic properties of it with those of the Maschler–Owen value on the union of the class of monotonic hyperplane games and that of monotonic prize games. We also provide axiomatizations of the Maschler–Owen value and the generalized Shapley value on that domain.

     

Predictive superiority of the beta-characteristic function in cooperative non-sidepayment N-person games

This article reports an experimental study of decision-making outcomes in cooperative non-sidepayment games. The objective of this test was to determine which characteristic function, V (S) or V (S), provides the most accurate basis for payoff predictions from solution concepts. The experiment tested three solution concepts (core, stable set, imputation set) in the context of 5-person, 2-strategy non-sidepayment games. Predictions from each of the three solution concepts were computed on the basis of both V (S) and V (S), making a total of six predictive theories under test. Consistent with earlier studies (Michener et al., 1984a; Michener et al., 1985), two basic findings emerged. First, the data show that for each of the solutions tested, the prediction from any solution concept computed from V(S) was more accurate than the prediction from the same solution concept computed from V (S). Second, the -core was the most accurate of the six theories tested. Overall, these results support the view that V (S) is superior to V (S) as a basis for payoff predictions in cooperative non-sidepayment games.     

A Banzhaf share function for cooperative games in coalition structure

A cooperative game with transferable utility–or simply a TU-game– describes a situation in which players can obtain certain payoffs by cooperation. A value function for these games assigns to every TU-game a distribution of payoffs over the players. Well-known solutions for TU-games are the Shapley and the Banzhaf value. An alternative type of solution is the concept of share function, which assigns to every player in a TU-game its share in the worth of the grand coalition. In this paper we consider TU-games in which the players are organized into a coalition structure being a finite partition of the set of players. The Shapley value has been generalized by Owen to TU-games in coalition structure. We redefine this value function as a share function and show that this solution satisfies the multiplication property that the share of a player in some coalition is equal to the product of the Shapley share of the coalition in a game between the coalitions and the Shapley share of the player in a game between the players within the coalition. Analogously we introduce a Banzhaf coalition structure share function. Application of these share functions to simple majority games show some appealing properties.     

An axiomatic analysis of the Nash equilibrium concept

The purpose of this paper is to analyze axiomatically the Nash equilibrium concept. The class of games under study is a (relatively large) subclass of n-person normal form games. Solutions are correspondences which associate to each game a non empty set of strategy vectors of this game. It is shown that if a solution satisfies the axioms Independence of irrelevant alternatives (IIA) and Individual rationality (IR), then all the strategy vectors in this solution are Nash equilibria. This result holds good also if IR is replaced by Strong individual monotonicity (SIM) or Weak principle of fair compromise (WPFC).     

The stability of bargains behind the veil of ignorance

Although analyses about what representative individuals would choose behind the veil of ignorance have been regarded as n-person non-zero-sum cooperative games, none of the apparatus of game theory beyond 2-person non-zero-sum noncooperative games has actually been used. The grand coalition of all representative individuals emerges from behind the veil of ignorance to form a society unanimously. This paper investigates the consequences of extending the original position to allow three persons the possibility of forming binding coalitions behind the veil of ignorance. Just enough information and structure is added to the traditional analysis, to make bargaining feasible. The result is that whether or not representative individuals know the payoff structure for forming a society, a stable unanimous agreement may not emerge. The analysis shows yet another way in which original position arguments are sensitive to assumptions about information and criteria of rational decision behind the veil of ignorance.     

Bargaining strength in three-person characteristic-function games with v(i)> 0 a reanalysis of Kahan and Rapoport (1977)

Kahan and Rapoport (1977) investigated the effects of guaranteed payoffs on bargaining in three-person cooperative games by systematically varying different sources of power: the power arising from the 1-person values, the power emerging from the pair coalitions as reflected by the quotas of the non-normalized game, and the grand coalition value. In the present paper it is suggested that one additionally take into account the assumption of strategic equivalence and that one analyze games with v(i)>0 in terms of the quotas of the zero-normalized game. Ostmann's (1984) rather sophisticated game theoretic framework, permitting the standardization of all three-person games, is introduced. A reanalysis of Kahan and Rapoport's data employing this perspective yields results which can be interpreted more easily than those of the original study. Moreover, they are consistent with the findings of almost all studies on 3-person characteristic-function games. It is argued that one could use the introduced analytical framework to investigate the range of empirical validity of the mathematical assumption of invariance under strategic equivalence.     

Games with a local permission structure: separation of authority and value generation

It is known that peer group games are a special class of games with a permission structure. However, peer group games are also a special class of (weighted) digraph games. To be specific, they are digraph games in which the digraph is the transitive closure of a rooted tree. In this paper we first argue that some known results on solutions for peer group games hold more general for digraph games. Second, we generalize both digraph games as well as games with a permission structure into a model called games with a local permission structure, where every player needs permission from its predecessors only to generate worth, but does not need its predecessors to give permission to its own successors. We introduce and axiomatize a Shapley value-type solution for these games, generalizing the conjunctive permission value for games with a permission structure and the $\beta $ -measure for weighted digraphs.     

Cooperative provision of indivisible public goods

A community faces the obligation of providing an indivisible public good that each of its members is able to provide at a certain cost. The solution is to rely on the member who can provide the public good at the lowest cost, with a due compensation from the other members. This problem has been studied in a non-cooperative setting by Kleindorfer and Sertel (J Econ Theory 64:20–34, 1994). They propose an auction mechanism that results in an interval of possible individual contributions whose lower bound is the equal division. Here, instead we take a cooperative stand point by modelling this problem as a cost sharing game that turns out to be a ‘reverse’ airport game whose core is shown to have a regular structure. This enables an easy calculation of the nucleolus that happens to define the upper bound of the Kleindorfer–Sertel interval. The Shapley value instead is not an appropriate solution in this context because it may imply compensations to non-providers.     

Punishing greediness in divide-the-dollar games

Brams and Taylor 1994 presented a version of the divide-the-dollar game (DD), which they call DD1. DD1 suffers from the following drawback: when each player demands approximately the entire dollar, then if the least greedy player is unique, then this player obtains approximately the entire dollar even if he is only slightly less greedy than the other players. I introduce a parametrized family of 2-person DD games, whose “endpoints” (the games that correspond to the extreme points of the parameter space) are (1) a variant of DD1, and (2) a game that completely overcomes the greediness-related problem. I also study an n-person generalization of this family. Finally, I show that the modeling choice between discrete and continuous bids may have far-reaching implications in DD games.