Axiomatizations of the normalized Banzhaf value and the Shapley value

A cooperative game with transferable utilities– or simply a TU-game – describes a situation in which players can obtain certain payoffs by cooperation. A solution concept for these games is a function which assigns to every such a game a distribution of payoffs over the players in the game. Famous solution concepts for TU-games are the Shapley value and the Banzhaf value. Both solution concepts have been axiomatized in various ways. An important difference between these two solution concepts is the fact that the Shapley value always distributes the payoff that can be obtained by the `grand coalition' consisting of all players cooperating together while the Banzhaf value does not satisfy this property, i.e., the Banzhaf value is not efficient. In this paper we consider the normalized Banzhaf value which distributes the payoff that can be obtained by the `grand coalition' proportional to the Banzhaf values of the players. This value does not satisfy certain axioms underlying the Banzhaf value. In this paper we introduce some new axioms that characterize the normalized Banzhaf value. We also provide an axiomatization of the Shapley value using similar axioms. Received: 10 April 1996 / Accepted: 2 June 1997     

Income inequality games

The paper explores different applications of the Shapley value for either inequality or poverty measures. We first investigate the problem of source decomposition of inequality measures, the so-called additive income sources inequality games, based on the Shapley value, introduced by Chantreuil and Trannoy (1999) and Shorrocks (1999). We show that multiplicative inequality games provide dual results compared with Chantreuil and Trannoy’s ones. We also investigate the case of multiplicative poverty games for which indices are non additively decomposable in order to capture contributions of sub-indices, which are multiplicatively connected with, as in the Sen-Shorrocks-Thon poverty index. We finally show, in the case of additive poverty indices, that the Shapley value may be equivalent to traditional methods of decomposition such as subgroup consistency and additive decomposition.     

The value of a player in n-person games

. The article decomposes the Shapley value into a value matrix which gives the value of every player to every other player in n-person games. Element Φ_ij(v) in the value matrix is positive, zero, or negative, dependent on whether row player i is beneficial, has no impact, or is not beneficial for column player j. The elements in each row and in each column of the value matrix sum up to the Shapley value of the respective player. The value matrix is illustrated by the voting procedure in the European Council of Ministers 1981–1995. Received: 9 September 1998/Accepted: 11 February 2000     

CONTRACT DESIGN AND SELF‐CONTROL WITH ASYMMETRIC INFORMATION

We study optimal contracting by a monopolistic seller of investment goods to a time‐inconsistent consumer and, in doing so, introduce asymmetric information to the model of DellaVigna and Malmendier (2004) . We find (1) the below‐marginal‐cost‐pricing rule may fail for a low‐value consumer; (2) the firm's profit is no longer unaffected by the consumer's short‐run impatience, as the latter is sophisticated. We find that there is an important threshold value of short‐run patience. When the consumer's short‐term patience is below this level, then, as the patience increases, the firm suffers. When the consumer's short‐run patience is above this threshold, then, as it increases, the firm benefits. Finally, we show that unlike monopoly, perfect competition with asymmetric information achieves the first‐best outcome. (JEL D03, D82, D91)     

PRODUCT BUNDLING AND INCENTIVES FOR MERGERS AND STRATEGIC ALLIANCES

This paper analyzes firms' choice of a merger or a strategic alliance in bundling their products with other complementary products. Tying two products of unequal value makes them equally valuable as they become inseparable for purchase. Consequently, firms can charge a higher price for the bundled products than before. If foreclosure is not the main purpose of bundling, firms would prefer strategic alliances to mergers because mergers only intensify competition by internalizing the complementarities of two products. In equilibrium, bundling occurs only through strategic alliances. (JEL L4, L11, L13, L23)     

The consensus value: a new solution concept for cooperative games

To generalize the standard solution for 2-person TU games into n-person cases, this paper introduces a recursive two-sided negotiation process to establish cooperation between all players. This leads to a new solution concept for cooperative games: the consensus value. An explicit comparison with the Shapley value is provided, also at the axiomatic level. Moreover, a class of possible generalizations of the consensus value is introduced and axiomatized with the Shapley value at one end and the equal surplus solution at the other. Finally, we discuss a non-cooperative mechanism which implements the consensus value.     

THE COMBINED EFFECT OF SALARY RESTRICTIONS AND REVENUE SHARING IN SPORTS LEAGUES

Many major sports leagues are characterized by a combination of cross‐subsidization mechanisms like revenue‐sharing arrangements and payroll restrictions. Up to now, the effects of these policy tools have only been analyzed separately. This article provides a theoretical model of a team sports league and analyzes the combined effect of salary restrictions (caps and floors) and revenue sharing. It shows that the effect on club profits, player salaries, and competitive balance crucially depends on the mix of these policy tools. Moreover, the invariance proposition does not hold even under Walrasian‐conjectures if revenue sharing is combined with a salary cap or floor. (JEL L83, C72, L11)     

FISCAL DECENTRALIZATION AND ECONOMIC GROWTH: SPENDING VERSUS REVENUE DECENTRALIZATION

This article examines whether the efficiency gains accompanying fiscal decentralization generate higher growth in more decentralized economies, applying pooled‐mean group techniques to a panel dataset of 23 Organization for Economic Co‐operation and Development (OECD) countries, 1972–2005. We find that spending decentralization has tended to be associated with lower economic growth while revenue decentralization has been associated with higher growth. Since OECD countries are substantially more spending than revenue decentralized, this is consistent with Oates' (1972) hypothesis that maximum efficiency gains require a close match between spending and revenue decentralization. It suggests reducing expenditure decentralization, and simultaneously increasing the fraction financed locally, would be growth‐enhancing. (JEL E62, H71, H72)     

MONOPOLY® PRICING

There is a large and rich literature in economics on monopoly pricing. However, a close examination of this literature reveals a surprising gap: there have been virtually no studies of Monopoly^® pricing. We fill that void with an empirical examination of the pricing of the board game Monopoly^®. In particular, we examine market‐level, quarterly prices from 1990 to 2002 and find substantial evidence of the phenomenon of customary pricing. (JEL L83, L1, L12, Z11, D42)     

AUCTIONS WITH RESALE OPPORTUNITIES: AN EXPERIMENTAL STUDY

We study first price asymmetric private value auctions with resale opportunities presented in seller's and buyer's markets. We offer experimental evidence on bidding behavior, prices, and resource allocation. Building upon the Hafalir and Krishna (2008) model, we find that bidders will bid higher in an auction if the resale market is a seller's market than a buyer's market. There is a price/revenue‐efficiency trade‐off established theoretically between these two resale regimes. In equilibrium, however, final efficiency is high irrespective of the resale market structure. Evidence of bid symmetrization and higher final efficiency is found in the buyer‐advantaged resale case. (JEL D44, C92)     

Bounded rationality and consumer choice: an evaluation of consumer choice of mobile bundles

Although numerous prior studies have examined consumer valuations in bundling transactions from a theoretical perspective, relatively few have focused on large datasets. In addition, little prior research has targeted developing telecommunication markets. To address these gaps in the literature, we examine the rationality of consumers' bundling choices in the mobile telecommunication industry, using operational data from China Telecom. We compare the current prices paid by consumers to the optimal prices of alternative bundling plans, based on the consumers' actual usage. We determine whether bundling choice is affected by risk aversion, the complexity of the bundle, and learning effects. The findings suggested that many users chose sub-optimal bundles that were more expensive than others were. The results showed that consumers were risk averse, and they were more likely to make non-optimal bundle selections with more complex usage patterns. No significant differences were found according to gender, age, or usage level.     

Analysing the effects of tax-benefit reforms on income distribution: a decomposition approach

To assess the impact of tax-benefit policy changes on income distribution over time, we suggest a methodology based on counterfactual tax-benefit simulations. Changes in inequality/poverty indices are decomposed into three contributions: changes in the tax-benefit structure, changes in nominal levels of market incomes and tax-benefit money parameters, and all other changes, including shifts in market income inequality and demographic composition. The policy effect can be evaluated conditionally on base-period data or end-period data; it is also possible to average the two measures, which corresponds to an application of the Shapley value method as reinterpreted by Shorrocks (Decomposition Procedures for Distributional Analysis: A Unified Framework Based on the Shapley Value, University of Essex and Institute for Fiscal Studies, Wivenhoe Park, 1999). The decomposition is used to quantify the relative role of policy changes on inequality/poverty trends in France and Ireland in the 1990s. When end-period data are not available, e.g., for forward looking analysis of possible reforms, the base weighted decomposition helps to extract an absolute measure of the impact of tax-benefit changes on income distribution as evaluated against a distributionally neutral benchmark; in our application, it is not significantly different from the policy effect stemming from the Shorrocks-Shapley decomposition. Estimates of this type are derived to assess recent policy changes in twelve European countries.     

Lorenz undominated allocations for TU-games: The weighted Coalitional Lorenz Solutions

In this paper we introduce and study the w-Coalitional Lorenz Solutions to identify the similarities and differences between the prenucleolus and the Shapley value. The similarity is that they both use egalitarian criteria over coalitions. The two main differences are: the prenucleolus and the Shapley value use different egalitarian criteria, and they weight the coalitions differently when applying the criteria. Received: 27 October 2000/Accepted: 2 October 2001     

DO HMO PLANS REDUCE HEALTH CARE EXPENDITURE IN THE PRIVATE SECTOR?

The cost containment performance of health maintenance organization (HMO) plans relative to non‐HMOs is examined using data from the 2000 Medical Expenditure Panel Survey. When various compounding factors are controlled for, among the privately insured, nonelderly population, HMO enrollment is found to contain neither total health care spending nor total insurance payment, though it reduces total out‐of‐pocket expenditure. We further find that this result is not attributed to selectivity in health plan choice due to health risk. The favorable cost sharing for enrollees and the distinct reimbursement schemes in HMO plans seem to account for no significant overall cost saving. (JEL I11, C25)     

FIXED REVENUE AUCTIONS: THEORY AND BEHAVIOR

In this paper, we study auctions in which the revenue is fixed but the quantity is determined by the auction mechanism. Specifically, we investigate the theory and behavior of English quantity clock, Dutch quantity clock, last‐quantity sealed bid, and penultimate‐quantity sealed bid auctions. For theoretically equivalent fixed quantity and fixed revenue auctions, we find that fixed revenue auctions are robust to all the previously observed empirical regularities in fixed quantity auctions. (JEL C9, D4, L2)     

ENTRY COORDINATION AND AUCTION DESIGN WITH PRIVATE COSTS OF INFORMATION ACQUISITION

Auction design with endogenous entry is complicated by entry coordination among bidders due to multiple entry equilibria issue. This article studies auction design when information acquisition costs are private information of bidders. We show that this problem can be resolved by sufficient dispersion in these costs. First, we find that a simple second‐price auction with no entry fee and a reserve price equal to the seller's valuation is ex ante efficient, while a revenue‐maximizing auction involves personalized entry fees, which are determined by the hazard rates of their information acquisition cost distribution. Second, we show that sufficient dispersion in the information acquisition costs (more dispersion than a particular uniform distribution by the Bickel‐Lehman dispersive order) can coordinate bidders and implement uniquely the desirable entry. The dispersion in information acquisition costs is also necessary for this “unique implementation” result. (JEL D44, D82)     

DETERMINANTS OF REVENUE IN SPORTS LEAGUES: AN EMPIRICAL ASSESSMENT

This study investigates determinants of revenue in North America's four major professional sports leagues. Revenue is positively associated with on‐field success in baseball (MLB), basketball (NBA), and hockey (NHL), but not in football (NFL). The returns to success are not diminishing as commonly assumed, which casts doubt on the uncertainty of outcome hypothesis, and differences across leagues are consistent with revenue sharing arrangements. Estimates indicate a strong negative but diminishing relationship between stadium age and revenue. Teams in larger markets generate more revenue than smaller markets, but the returns to success do not differ according to market size. (JEL Z21)     

NON‐MONETARY INCENTIVES AND OPPORTUNISTIC BEHAVIOR: EVIDENCE FROM A LABORATORY PUBLIC GOOD GAME

This study reports data from a laboratory experiment that investigates the incentive effect of three distinct social communication schemes on free‐riding behavior. We use performance‐based approval and disapproval ratings and a linear public good game to address the above issues. The treatments vary in terms of subjects' opportunities to anonymously assign (1) only the approval ratings to other group members, (2) only the disapproval ratings to other group members, and (3) either the approval or the disapproval ratings to other group members (but not both to the same group member), after they play a standard linear public good game. Despite the Nash prediction of zero individual contribution in all three treatments, the data show that the disapproval points generate significantly higher contribution than the approval points. The treatment in which subjects could communicate either the approval or the disapproval points produces the highest level of contribution. We discuss the implications that these findings may have for efficient design of organizations. (JEL D03, H41, C72, C92)     

The apex power measure for directed networks

In this paper we provide an axiomatization of the Shapley value restricted to the class of apex games using an equal loss property which states that the payoff of an apex player and a non-apex player decrease by the same amount if we make this particular non-apex player a null player. We also generalize this axiomatization to the class of games that can be obtained as sums of apex games. After discussing these axiomatizations we apply apex games and their Shapley values in measuring relational power in directed networks. We conclude by mentioning how these results can be adapted to give axiomatizations of the Banzhaf value. Received: 17 February 1999/Accepted: 2 October 2002     

DEBT BAILOUTS AND CONSTITUTIONS

A demand‐based theory of subnational debt bailouts is presented. It is shown that demand for bailouts among politicians with regional constituencies is affected by revenue sharing arrangements as a bailout implies a shift of taxation toward the federal tier. Automatic revenue sharing may lead to the formation of pro‐bailout coalitions formed by indebted states and states that are net recipients of the revenue sharing arrangement. The model shows that the state debt bailouts approved by the Brazilian Senate prior to the enactment of the Fiscal Responsibility Act were consistent with politicians who maximize the proceeds accruing to their constituencies. (JEL H70, D70)