Rawlsian equity and generalised utilitarianism with an infinite population

Summary This note provides axiomatizations of the discounting utilitarian rule and the Rawlsian infimum rule for an infinite setting. Contrary to the finite case, an objective function cannot at the same time satisfy anonymity and the strong Pareto principle. The Rawlsian rule is based on anonymity, and the discounting rule on the strong Pareto axiom. Furthermore, the infimum rule is compatible with an infinite version of ordinal level comparability, and the discounting rule with cardinal unit comparability.This text presents research results of the Belgian Programme of Interuniversity Poles of attraction initiated under the Science Policy Programmes of the Prime Minister's Office, Belgium. The scientific responsibility is assumed by its authors.I would like to thank two referees for their most helpful comments. I am grateful to Bart Capéau, Erik Schokkaert, John Sweeney, Dirk Van de gaer, and Luc Van Liedekerke for their remarks.     

Utility in Case-Based Decision Theory

This paper provides two axiomatic derivations of a case-based decision rule. Each axiomatization shows that, if preference orders over available acts in various contexts satisfy certain consistency requirements, then these orders can be numerically represented by maximization of a similarity-weighted utility function. In each axiomatization, both the similarity function and the utility function are simultaneously derived from preferences, and the axiomatic derivation also suggests a way to elicit these theoretical concepts from in-principle observable preferences. The two axiomatizations differ in the type of decisions that they assume as data. Journal of Economic Literature Classification Number: D80.     

Reallocation of an infinitely divisible good

Summary. We consider the problem of reallocating the total initial endowments of an infinitely divisible commodity among agents with single-peaked preferences. With the uniform reallocation rule we propose a solution which satisfies many appealing properties, describing the effect of population and endowment variations on the outcome. The central properties which are studied in this context are population monotonicity, bilateral consistency, (endowment) monotonicity and (endowment) strategy-proofness. Furthermore, the uniform reallocation rule is Pareto optimal and satisfies several equity conditions, e.g., equal-treatment and envy-freeness. We study the trade-off between properties concerning variation and properties concerning equity. Furthermore, we provide several characterizations of the uniform reallocation rule based on these properties. Received: August 29, 1995; revised version June 26, 1996     

Generalized contest success functions

The key element of models of contest is the contest success function (CSF) which specifies the winning probabilities of agents. The existing axiomatizations of CSFs assume that contestants can make only one type of investment. This paper generalizes these axiomatizations to the case where each agent can have multiple types of investments. This allows us to provide a unified framework to extend and interpret the results of Skaperdas (Econ Theory 7:283–290, 1996) and Clark and Riis (Econ Theory 11:201–204, 1998), and rationalize some seemingly ad hoc CSFs used by applied researchers. This paper has benefitted from the comments of Aicke Hinrichs, Rene Levinsky, and two anonymous referees. We are especially thankful to one of the referees for detailed comments which have helped improve every aspect of this paper.     

Precaution under mixed uncertainty: Implications for environmental management

A model of pollution control subject to two types of uncertainty is presented. First, the natural decay of the pollution stock follows stochastic dynamics that drives a diffusion pollution process (“stochastic uncertainty”). Moreover, the damage coefficient which determines the amount of damage corresponding to each pollution stock can undergo an abrupt increase at some uncertain future time (“event uncertainty”). The model admits an explicit and simple dynamic characterization of the optimal emission rule and the ensuing pollution process. When only one type of uncertainty is permitted (by setting either the variance of the stochastic process or the hazard rate for the damage increase to zero) it acts to promote the intuitive response of precaution. However, allowing the two types to interact gives rise to a non-monotonic behavior, whereby increasing the stochastic variance first enhances, then diminishes the response to the hazard. The results confirm and expand recent findings based on discrete-time formulations.     

A note on linear utility

This note shows how a spectrum of specific axiomatizations for linear utility for decision-making under risk can be integrated under a generalized axiom scheme in which one of the two binary relations used in the axioms parameterizes specific applications.     

Exchange Rules and the Incentive Compatibility of Choice Experiments

The incentive properties of stated-preference surveys continue to be a central debate in the valuation of public goods. The majority of empirical studies have focused on incentive properties of contingent valuation questions in relation to situations where answers have monetary consequences. This research explores the incentive properties of repeated, attribute-based choice questions when subjects are provided with an explicit connection between choices and outcomes. Two market/provision-rules are investigated: a posted-price market and a plurality-rule vote. These two provision rules are contrasted to treatments in which no provision rule is discussed—subjects are simply asked to choose their preferred alternative. These three hypothetical choice treatments are compared with a binding choice treatment. While none of the public good treatments are theoretically incentive compatible, we include a comparison of hypothetical and binding choices for a private-good that is incentive compatible. The private good experiments indicate that marginal willingness to pay (WTP) estimates from the hypothetical treatment are larger, but not statistically different than corresponding estimates in the binding choice treatment. Results for the public good experiments indicate that marginal WTP estimates from the hypothetical treatments are much larger, and statistically different than corresponding estimates in the binding choice treatment. The bias is largest when no provision rule is discussed. The bias is reduced with the inclusion of a provision rule, but surprisingly, there was no difference across provision rule treatments. Overall, our results indicate that choice experiments involving a public good should include a provision rule to reduce bias, but the resulting marginal WTP estimates may still be more biased, on average, than those arising from contingent valuation survey formats.     

Range convexity and ambiguity averse preferences

We show that range convexity of beliefs, a `technical' condition that appears naturally in axiomatizations of preferences in a Savage-like framework, imposes some unexpected restrictions when modelling ambiguity averse preferences. That is, when it is added to a mild condition, range convexity makes the preferences collapse to subjective expected utility as soon as they satisfy structural conditions that are typically used to characterize ambiguity aversion. Received: February 25, 2000; revised version: April 17, 2000     

Does the monetary policy committee still care about inflation? Some evidence from a small macroeconomic model

This article derives an optimal Taylor rule for the UK economy using a simple estimated model based on data prior to the financial crisis of 2008. Optimal policy rules are calculated using simulation of the model over a long time period coupled with a search for optimal Taylor rule parameters using the Newton-Raphson loss minimization algorithm. The weights in the pre-crisis loss function are then inferred from the Taylor rule parameters estimated from the period corresponding to Bank of England independence, i.e. 1997–2008. These estimates are consistent with a low weight on inflation relative to output stabilization even before the crisis. The model is therefore consistent with the hypothesis that there has been no change in Bank of England preferences and that the Bank has responded to the crisis in a way which would have been predicted on the basis of its pre-crisis behaviour.     

Voluntary Participation and Provision of Public Goods in Large Finite Economies

We consider a public good provision game with voluntary participation. Agents participating in the game provide a public good and pay the fees according to a mechanism (allocation rule), while nonparticipants can free‐ride on the participants. We examine how the equilibrium public good provision level is affected by enlarging the population of an economy. We introduce a condition for an allocation rule, the asymptotic uniform continuity in replication (AUCR), which requires that small changes in the population must yield only small changes in the public good provision and which is satisfied by many mechanisms. We show that under AUCR, the equilibrium level of the public good converges to zero as the economy is replicated in the sense of Milleron.     

Spatial Heterogeneity and the Choice of Instruments to Control Nonpoint Pollution

Because it is difficult to monitor emissions and to implement differentialtaxes and standards, uniform taxes and standards on agricultural chemicaluse are often proposed and used to control nonpoint-source pollution. Thisarticle analyzes the relative efficiency of these uniform instruments in thepresence of spatial heterogeneity. We show that the relative slopes of themarginal pollution cost and marginal profit of chemical use are only one ofthe factors that affect the relative efficiency. Other factors includecorrelation between marginal pollution costs and marginal profits and theslope and variability of marginal profits. In addition, a uniform tax mayresult in some farmers not using the chemical, and a uniform standard mayhave no effect on low-input land. We show empirically that the presence ofcorner solutions can reverse a conventional finding of tax or standardsuperiority based on the relative-slop rule.     

Subjective expected utility theory with costly actions

This paper explores alternative axiomatizations of subjective expected utility theory for decision makers with direct preferences over actions; including a general subjective expected utility representation with action-dependent utility, and separately additive representations. In the context of the state-space formulation of agency theory, the results of this paper constitute axiomatic foundations of the agent's behavior.     

Sustainability and its relation to efficiency under uncertainty

Evaluating the long-run consequences of present actions, as in the context of sustainability, requires information about the actions’ outcomes and about future preferences that is often uncertain. We analyze a risk-based criterion of sustainability and a corresponding efficiency concept that cover these uncertainties. We derive several properties of these criteria and formally characterize the trade-off between sustainability and efficiency. Furthermore, we show that maximizing the probability of ex post efficiency under a sustainability constraint provides an interesting choice rule and that, for a special case, this rule is connected to portfolio theory. We are indebted to an anonymous referee whose comments helped substantially to improve the article.     

Maximal Domain of Preferences in the Division Problem

The division problem consists of allocating an amount of a perfectly divisible good among a group of n agents. Sprumont (1991) showed that if agents have single-peaked preferences over their shares, then the uniform allocation rule is the unique strategy-proof, efficient, and anonymous rule. We identify the maximal set of preferences, containing the set of single-peaked preferences, under which there exists at least one rule satisfying the properties of strategy-proofness, efficiency, and strong symmetry. In addition, we show that our characterization implies a slightly weaker version of Ching and Serizawa's (1998) result. Journal of Economic Literature Classification Numbers: D71, D78, D63.     

Efficiency and prices in economies of overlapping generations

In a general economy of overlapping generations, I introduce a notion of uniform inefficiency, corresponding to the occurrence of a Pareto improvement with a small uniform destruction of resources [G. Debreu, The coefficient of resource utilization, Econometrica 19 (1951) 273-292]. I provide a necessary and sufficient condition for uniform inefficiency in terms of prices at a competitive equilibrium: an allocation is uniformly inefficient if and only if the relative price of the aggregate endowment in a given period into the aggregate endowment up to that period does not vanish over periods of trade, a sort of Modified Cass Criterion [D. Cass, On capital overaccumulation in the aggregative neoclassical model of economic growth: a complete characterization, J. Econ. Theory 4 (1972) 200-223]. Minimal assumptions on fundamentals are needed for such a complete characterization. Furthermore, proofs reduce to simple and short direct arguments. Finally, I verify that uniform inefficiency is preserved under perturbations, a property that might fail for the canonical notion of inefficiency. Remarkably, an allocation is uniformly inefficient if and only if a non-vanishing redistribution, like a social security mechanism, is welfare improving.     

Uniqueness of stationary equilibria in bargaining one-dimensional policies under (super) majority rules

We consider negotiations selecting one-dimensional policies. Individuals have instantaneous preferences represented by continuous, concave and single-peaked utility functions, and they are impatient. Decisions arise from a bargaining game with random proposers and (super) majority approval, ranging from the simple majority up to unanimity. We provide sufficient conditions that guarantee the existence of a unique stationary subgame perfect equilibrium, and we provide its explicit characterization. The uniqueness of the equilibrium permits an analysis of the set of Pareto optimal voting rules. For symmetric distributions of peaks and uniform recognition probabilities unanimity is the unanimously preferred majority rule.     

N-PERSON NASH BNARGAINING WITH VARIABEL THREATS*

We consider two models of n-person bargaining problems with the endogenous determination of disagreement points. In the first model, which is a direct extension of Nash's variable threat bargaining model, the disagreement point is determined as an equilibrium threat point. In the second model, the disagreement point is given as a Nash equilibrium of the underlying noncooperative game. These models are formulated as extensive games, and axiomatizations of solutions are given for both models. It is argued that for games with more than two players, the first bargaining model does not preserve some important properties valid for two-person games, e.g., the uniqueness of equilibrium payoff vector. We also show that when the number of players is large, any equilibrium threat point becomes approximately a Nash equilibrium in the underlying noncooperative game, and vice versa. This result suggests that the difference between the two models becomes less significant when the number of players is large.     

Game theoretic analysis of a bankruptcy problem from the Talmud

For three different bankruptcy problems, the 2000-year old Babylonian Talmud prescribes solutions that equal precisely the nucleoli of the corresponding coalitional games. A rationale for these solutions that is independent of game theory is given in terms of the Talmudic principle of equal division of the contested amount; this rationale leads to a unique solution for all bankruptcy problems, which always coincides with the nucleolus. Two other rationales for the same rule are suggested, in terms of other Talmudic principles. (Needless to say, the rule in question is not proportional division).     

Symmetry and impartial lotteries

A prize is to be awarded, so each candidate designates one of his peers on a ballot. The ballots determine the lottery that selects the winner, and impartiality requires that no candidate's choice of designee impacts his own chance of winning, removing incentives for strategic ballot submission. The primary results are (1) a characterization of all impartial rules that treat agents symmetrically as voters, and (2) a characterization of all impartial rules that treat agents symmetrically as candidates. Each rule in either class may be represented as a randomization over a finite set of simple rules. These results have immediate interpretation in a second context: the division of surplus among team members. Corollaries include the constant rule impossibility of Holzman and Moulin (2013), a new dictatorship impossibility, and the first axiomatic characterization of uniform random dictatorship.     

Common knowledge and quantification

Summary. The paper consists of two parts. The first one is a concise introduction to epistemic (both propositional and predicate) logic with common knowledge operator. As the full predicate logics of common knowledge are not even recursively enumerable, in the second part we introduce and investigate the monodic fragment of these logics which allows applications of the epistemic operators to formulas with at most one free variable. We provide the monodic fragments of the most important common knowledge predicate logics with finite Hilbert-style axiomatizations, prove their completeness, and single out a number of decidable subfragments. On the other hand, we show that the addition of equality to the monodic fragment makes it not recursively enumerable. Received: March 7, 2001; revised version: April 4, 2001