The self-dual serial cost-sharing rule

In this study, we present a cost-sharing rule for cost-sharing problems. This rule prescribes the same allocations in a problem and in its dual one. Moreover, in some specific problems it gives the same allocations as the serial cost-sharing rule (Moulin and Shenker, Econometrica, 60, 1009–1037, 1992) does in a related problem. That is why we call it as the self-dual serial cost-sharing rule. We give two axiomatizations of this new rule and another one for the serial cost-sharing rule.     

A Note on Implementation of Bargaining Solutions

Miyagawa (Games and Economics Behavior 41(2), 292–308 [2002]) provides a simple extensive game form that implements a large class of two-agent bargaining solutions in subgame-perfect equilibrium. This class includes all of the Nash, Kalai–Smorodinsky, and relative utilitarian solutions. This note extends Miyagawa’s result to multi-agent bargaining problems.     

Envy Freeness in Experimental Fair Division Problems

Envy is sometimes suggested as an underlying motive in the assessment of different economic allocations. In the theoretical literature on fair division, following Foley [Foley, D. (1967), Yale Economic Essays, 7, 45–98], the term “envy” refers to an intrapersonal comparison of different consumption bundles. By contrast, in its everyday use “envy” involves interpersonal comparisons of well-being. We present, discuss results from free-form bargaining experiments on fair division problems in which inter-and intrapersonal criteria can be distinguished. We find that interpersonal comparisons play the dominant role. The effect of the intrapersonal criterion of envy freeness is limited to situations in which other fairness criteria are not applicable.      

The Value of a Probability Forecast from Portfolio Theory

A probability forecast scored ex post using a probability scoring rule (e.g. Brier) is analogous to a risky financial security. With only superficial adaptation, the same economic logic by which securities are valued ex ante – in particular, portfolio theory and the capital asset pricing model (CAPM) – applies to the valuation of probability forecasts. Each available forecast of a given event is valued relative to each other and to the “market” (all available forecasts). A forecast is seen to be more valuable the higher its expected score and the lower the covariance of its score with the market aggregate score. Forecasts that score highly in trials when others do poorly are appreciated more than those with equal success in “easy” trials where most forecasts score well. The CAPM defines economically rational (equilibrium) forecast prices at which forecasters can trade shares in each other’s ex post score – or associated monetary payoff – thereby balancing forecast risk against return and ultimately forming optimally hedged portfolios. Hedging this way offers risk averse forecasters an “honest” alternative to the ruse of reporting conservative probability assessments.     

Strategic games with security and potential level players

This paper examines the existence of strategic solutions to finite normal form games under the assumption that strategy choices can be described as choices among lotteries where players have security- and potential level preferences over lotteries (e.g., Cohen, Theory and Decision, 33, 101–104, 1992, Gilboa, Journal of Mathematical Psychology, 32, 405–420, 1988, Jaffray, Theory and Decision, 24, 169–200, 1988). Since security- and potential level preferences require discontinuous utility representations, standard existence results for Nash equilibria in mixed strategies (Nash, Proceedings of the National Academy of Sciences, 36, 48–49, 1950a, Non-Cooperative Games, Ph.D. Dissertation, Princeton University Press, 1950b) or for equilibria in beliefs (Crawford, Journal of Economic Theory, 50, 127–154, 1990) do not apply. As a key insight this paper proves that non-existence of equilibria in beliefs, and therefore non-existence of Nash equilibria in mixed strategies, is possible in finite games with security- and potential level players. But, as this paper also shows, rationalizable strategies (Bernheim, Econometrica, 52, 1007–1028, 1984, Moulin, Mathematical Social Sciences, 7, 83–102, 1984, Pearce, Econometrica, 52, 1029–1050, 1984) exist for such games. Rationalizability rather than equilibrium in beliefs therefore appears to be a more favorable solution concept for games with security- and potential level players.      

Differential marginality,van den Brink fairness,and the Shapley value

We revisit the characterization of the Shapley value by van den Brink (Int J Game Theory, 2001, 30:309–319) via efficiency, the Null player axiom, and some fairness axiom. In particular, we show that this characterization also works within certain classes of TU games, including the classes of superadditive and of convex games. Further, we advocate some differential version of the marginality axiom (Young, Int J Game Theory, 1985, 14: 65–72), which turns out to be equivalent to the van den Brink fairness axiom on large classes of games.     

On Decomposing Net Final Values: Eva,Sva and Shadow Project

A decomposition model of Net Final Values (NFV), named Systemic Value Added (SVA), is proposed for decision-making purposes, based on a systemic approach introduced in Magni [Magni, C. A. (2003), Bulletin of Economic Research 55(2), 149–176; Magni, C. A. (2004) Economic Modelling 21, 595–617]. The model translates the notion of excess profit giving formal expression to a counterfactual alternative available to the decision maker. Relations with other decomposition models are studied, among which Stewart’s [Stewart, G.B. (1991), The Quest for Value: The EVA™ Management Guide, Harper Collins, Publishers Inc]. The index here introduced differs from Stewart’s Economic Value Added (EVA) in that it rests on a different interpretation of the notion of excess profit and is formally connected with the EVA model by means of a shadow project. The SVA is formally and conceptually threefold, in that it is economic, financial, accounting-flavoured. Some results are offered, providing sufficient and necessary conditions for decomposing NFV. Relations between a project’s SVA and its shadow project’s EVA are shown, all results of Pressacco and Stucchi [Pressacco, F. and Stucchi, P. (1997), Rivista di Matematica per le Scienze Economiche e Sociali 20, 165–185] are proved by making use of the systemic approach and the shadow counterparts of those results are also shown.     

The <Emphasis Type="Italic">γ</Emphasis>-core in Cournot oligopoly TU-games with capacity constraints

In cooperative Cournot oligopoly games, it is known that the β-core is equal to the α-core, and both are non-empty if every individual profit function is continuous and concave (Zhao, Games Econ Behav 27:153–168, 1999b). Following Chander and Tulkens (Int J Game Theory 26:379–401, 1997), we assume that firms react to a deviating coalition by choosing individual best reply strategies. We deal with the problem of the non-emptiness of the induced core, the γ-core, by two different approaches. The first establishes that the associated Cournot oligopoly Transferable Utility (TU)-games are balanced if the inverse demand function is differentiable and every individual profit function is continuous and concave on the set of strategy profiles, which is a step forward beyond Zhao’s core existence result for this class of games. The second approach, restricted to the class of Cournot oligopoly TU-games with linear cost functions, provides a single-valued allocation rule in the γ-core called Nash Pro rata (NP)-value. This result generalizes Funaki and Yamato’s (Int J Game Theory 28:157–171, 1999) core existence result from no capacity constraint to asymmetric capacity constraints. Moreover, we provide an axiomatic characterization of this solution by means of four properties: efficiency, null firm, monotonicity, and non-cooperative fairness.     

Achievable Hierarchies In Voting Games

Previous work by Diffo Lambo and Moulen [Theory and Decision 53, 313–325 (2002)] and Felsenthal and Machover [The Measurement of Voting Power, Edward Elgar Publishing Limited (1998)], shows that all swap preserving measures of voting power are ordinally equivalent on any swap robust simple voting game. Swap preserving measures include the Banzhaf, the Shapley–Shubik and other commonly used measures of a priori voting power. In this paper, we completely characterize the achievable hierarchies for any such measure on a swap robust simple voting game. Each possible hierarchy can be induced by a weighted voting game and we provide a constructive proof of this result. In particular, the strict hierarchy is always achievable as long as there are at least five players.     

Political influence in multi-choice institutions: cyclicity,anonymity, and transitivity

We study political influence in institutions where each member chooses a level of support for a collective goal. These individual choices determine the degree to which the goal is reached. Influence is assessed by newly defined binary relations, each of which ranks members on the basis of their relative performance at a corresponding level of participation. For institutions with three options (e.g., voting games in which each voter may vote “yes”, “abstain”, or vote “no”), we obtain three influence relations, and show that their strict components may be cyclic. This latter property describes a “paradox of power” which contrasts with the transitivity of the unique influence relation of binary voting games. Weak conditions of anonymity suffice for each of these relations to be transitive. We also obtain a necessary and sufficient condition for each of these relations to be complete. Further, we characterize institutions in which the rankings induced by these relations, and the Banzhaf–Coleman and Shapley–Shubik power indices coincide. We argue that extending the influence relations to firms would be useful in efficiently assigning workers to different units of production. Finally, we provide applications to various forms of political and economic organizations.     

Equivalent comparisons of information channels

This article constructs a static model of information acquisition when the agent does not know exactly what pieces of information he is missing. A representation of preferences over information channels and menus of lotteries is shown by adapting the model of unforeseen contingencies by Dekel et al. (Econometrica 69:891–934, 2001; Econometrica 75:591–600, 2007), which is an extension of Kreps (Econometrica 47:565–576, 1979; Economic analysis of markets and games: essays in honor of Frank Hahn, 1992). Also, characterisation of informativeness of an information channel analogous to the one by Blackwell (Ann Math Stud 24:265–272, 1953) is examined in conjunction with the preference for flexibility by applying the structure of Shapley value.     

Multidimensional bargains and the desirability of ex post inefficiency

In multidimensional bargaining situations where individuals possess relevant private information, say about preferences, allocational efficiency is a central concern. Even if there is no squabbling over distribution—for example, if contingent commitments on allocations can be made before private information is secured—honest revelation comes only by sacrificing efficiency. Indeed, the incentive-compatible, second-best outcomes generally require that some allocations be off the contract curve (ex post inefficient). The potential for recontracting, by ruling out such inefficient allocations and the second-best equilibria they support, would hurt matters further.Support was received from the Decision, Risk and Management Science Program of the National Science Foundation. Rachel Algaze Croson provided able assistance; referees provided helpful comments.     

Coalitional stability and efficiency of partitions in matching problems

?zkal-Sanver (Theory Decis 59:193–205, 2005) studies stability and efficiency of partitions of agents in two-sided matching markets in which agents can form partitions by individual moves only, and a matching rule determines the matching in each coalition in a partition. In this study, we present the relationship between stability and efficiency of partitions that is analyzed for several matching rules and under various membership property rights codes, now allowing coalitional moves.     

Sleeping Beauty and the absent-minded driver

The Sleeping Beauty problem is presented in a formalized framework which summarizes the underlying probability structure. The two rival solutions proposed by Elga (Analysis 60:143–147, 2000) and Lewis (Analysis 61:171–176, 2001) differ by a single parameter concerning her prior probability. They can be supported by considering, respectively, that Sleeping Beauty is “fuzzy-minded” and “blank-minded”, the first interpretation being more natural than the second. The traditional absent-minded driver problem is reinterpreted in this framework and sustains Elga’s solution.     

Games of Competition in a Stochastic Environment

The paper presents a set of games of competition between two or three players in which reward is jointly determined by a stochastic biased mechanism and players’ choices. More specifically, a resource can be found with unequal probabilities in one of two locations. The first agent is rewarded only if it finds the resource and avoids being found by the next agent in line; the latter is rewarded only if it finds the former. Five benchmarks, based on different psychological and game-theoretic assumptions are derived and their predictions compared to actual behavior of 120, 40, and 48 participants playing repeatedly. Of the five benchmarks—the unique (Nash) equilibrium, reinforcement learning, trust-based efficiency, maximum unpredictability, and regret-based (Impulse Balance) equilibrium—regret for missed opportunities best accounts for the qualitative aspect of participants’ behavior and regret attenuated by randomization best accounts for the quantitative aspect of behavior.     

Valuing fatal risks to children and adults: Effects of disease,latency, and risk aversion

We examine heterogeneity of willingness to pay (WTP) to reduce risks of fatal disease and trauma to adults and children. Using a stated-preference survey fielded to a large, nationally representative internet panel, we find that WTP to reduce fatal-disease risks (caused by consuming pesticide residues on foods) are similar for several types of cancer and non-cancer diseases and similar to WTP to reduce motor-vehicle crashes. WTP to reduce risk to one’s child is uniformly larger than to reduce risk to another adult or to oneself. Estimated values per statistical life are $6–10 million for adults and $6–10 million for adults and 12–15 million for children.     

Economic Darwinism

We define an evolutionary process of “economic Darwinism” for playing the field, symmetric games. The process captures two forces. One is “economic selection”: if current behavior leads to payoff differences, behavior yielding lowest payoff has strictly positive probability of being replaced by an arbitrary behavior. The other is “mutation”: any behavior has at any point in time a strictly positive, very small probability of shifting to an arbitrary behavior. We show that behavior observed frequently is in accordance with “evolutionary equilibrium”, a static equilibrium concept suggested in the literature. Using this result, we demonstrate that generally under positive (negative) externalities, economic Darwinism implies even more under- (over-)activity than does Nash equilibrium.     

Aspects concerning entropy and utility

Expected utility maximization problem is one of the most useful tools in mathematical finance, decision analysis and economics. Motivated by statistical model selection, via the principle of expected utility maximization, Friedman and Sandow (J Mach Learn Res 4:257–291, 2003a) considered the model performance question from the point of view of an investor who evaluates models based on the performance of the optimal strategies that the models suggest. They interpreted their performance measures in information theoretic terms and provided new generalizations of Shannon entropy and Kullback–Leibler relative entropy and called them U-entropy and U-relative entropy. In this article, a utility-based criterion for independence of two random variables is defined. Then, Markov’s inequality for probabilities is extended from the U-entropy viewpoint. Moreover, a lower bound for the U-relative entropy is obtained. Finally, a link between conditional U-entropy and conditional Renyi entropy is derived.     

Modus Ponens and Modus Tollens for Conditional Probabilities,and Updating on Uncertain Evidence

There are narrowest bounds for P(h) when P(e)  =  y and P(h/e)  =  x, which bounds collapse to x as y goes to 1. A theorem for these bounds – Bounds for Probable Modus Ponens – entails a principle for updating on possibly uncertain evidence subject to these bounds that is a generalization of the principle for updating by conditioning on certain evidence. This way of updating on possibly uncertain evidence is appropriate when updating by ‘probability kinematics’ or ‘Jeffrey-conditioning’ is, and apparently in countless other cases as well. A more complicated theorem due to Karl Wagner – Bounds for Probable Modus Tollens – registers narrowest bounds for P(∼h) when P(∼e) =  y and P(e/h)  =  x. This theorem serves another principle for updating on possibly uncertain evidence that might be termed ‘contraditioning’, though it is for a way of updating that seems in practice to be frequently not appropriate. It is definitely not a way of putting down a theory – for example, a random-chance theory of the apparent fine-tuning for life of the parameters of standard physics – merely on the ground that the theory made extremely unlikely conditions of which we are now nearly certain. These theorems for bounds and updating are addressed to standard conditional probabilities defined as ratios of probabilities. Adaptations for Hosiasson-Lindenbaum ‘free-standing’ conditional probabilities are provided. The extended on-line version of this article (URL: ) includes appendices and expansions of several notes. Appendix A contains demonstrations and confirmations of elements of those adaptations. Appendix B discusses and elaborates analogues of modus ponens and modus tollens for probabilities and conditional probabilities found in Elliott Sober’s “Intelligent Design and Probability Reasoning.” Appendix C adds to observations made below regarding relations of Probability Kinematics and updating subject to Bounds for Probable Modus Ponens.      

Hierarchies achievable in simple games

A previous work by Friedman et al. (Theory and Decision, 61:305–318, 2006) introduces the concept of a hierarchy of a simple voting game and characterizes which hierarchies, induced by the desirability relation, are achievable in linear games. In this paper, we consider the problem of determining all hierarchies, conserving the ordinal equivalence between the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices, achievable in simple games. It is proved that only four hierarchies are non-achievable in simple games. Moreover, it is also proved that all achievable hierarchies are already obtainable in the class of weakly linear games. Our results prove that given an arbitrary complete pre-ordering defined on a finite set with more than five elements, it is possible to construct a simple game such that the pre-ordering induced by the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices coincides with the given pre-ordering.