Implementation in generalized matching problems

We search for (Nash) implementable solutions on a class of one-to-one matching problems which includes both the housing market (Shapley and Scarf, Journal of Mathematical Economics, 1974, 1, 23–28) and marriage problems (Gale and Shapley, American Mathematical Monthly, 1962, 69, 9–15). We show that the core correspondence is implementable. We show, furthermore, that any solution that is Pareto efficient, individually rational, and implementable is a supersolution of the core correspondence. That is, the core correspondence is the minimal solution that is Pareto efficient, individually rational, and implementable. A corollary of independent interest in the context of the housing market is that the core correspondence is the only single-valued solution that is Pareto efficient, individually rational, and implementable.     

Limited arbitrage is necessary and sufficient for the existence of an equilibrium

In Chichilnisky (Working Paper No. 586, 1991), Chichilnisky (Working Paper No. 650, 1992) and Chichilnisky (Economic Theory, 1995, 5, 79–108), I introduced the concept of a global cone and used it to define a condition on endowments and preferences, ‘limited arbitrage’, which I showed to be necessary and sufficient for the existence of a competitive equilibrium. In response to a comment (Monteiro et al., Journal of Mathematical Economics, 1997, 26, 000-000), I show here that the authors misunderstood my results by focussing on brief announcements which cover other areas, social choice (Chichilnisky, American Economic Review, 1994, 427–434 and algebraic topology (Chichilnisky, Bulletin of the American Mathematical Society, 1993, 29, 189–207), rather than on the publication which contains may proofs on equilibrium. The comment's example is irrelevant to my results in Chichilnisky (Economic Theory, 1995, 5, 79–108) because it starts from different conditions. Limited arbitrae is always necessary and sufficient for the existence of a competitive equilibrium, with or without short sales, with the global cones as I defined them, and exactly as proved in Chichilnisky (Economic Theory, 1995, 5, 79–108).     

Strategy-proofness in many-to-one matching problems

We search forstrategy-proof solutions in the context of (many-to-one) matching problems (Gale and Shapley, 1962). In this model, whenever the firms can hire as many workers as they want (the capacities are unlimited) the stable set is a singleton. There exists aPareto efficient, individually rational, andstrategy-proof matching rule if and only if the capacities are unlimited. Furthermore, whenever the capacities unlimited, the matching rule which selects the unique stable matching is the only matching rule that isPareto efficient, individually rational, andstrategy-proof.     

The coordinate-wise core for multiple-type housing markets is second-best incentive compatible

We consider the generalization of Shapley and Scarf’s (1974) [Shapley, L., Scarf’s, H., 1974. On cores and indivisibility. Journal of Mathematical Economics 1, 23–37.] model of trading indivisible objects (houses) to so-called multiple-type housing markets. We show that the prominent solution for these markets, the coordinate-wise core rule, is second-best incentive compatible.     

A note on the multiple partners assignment game

In the assignment game of Shapley and Shubik [Shapley, L.S., Shubik, M., 1972. The assignment game. I. The core, International Journal of Game Theory 1, 11–130] agents are allowed to form one partnership at most. That paper proves that, in the context of firms and workers, given two stable payoffs for the firms there is a stable payoff which gives each firm the larger of the two amounts and also one which gives each of them the smaller amount. Analogous result applies to the workers. Sotomayor [Sotomayor, M., 1992. The multiple partners game. In: Majumdar, M. (Ed.), Dynamics and Equilibrium: Essays in Honor to D. Gale. Mcmillian, pp. 322–336] extends this analysis to the case where both types of agents may form more than one partnership and an agent's payoff is multi-dimensional. Instead, this note concentrates in the total payoff of the agents. It is then proved the rather unexpected result that again the maximum of any pair of stable payoffs for the firms is stable but the minimum need not be, even if we restrict the multiplicity of partnerships to one of the sides.     

Arbitrage, equilibrium, and gains from trade: A counterexample

We present a counterexample to a theorem due to Chichilnisky (Bulletin of the American Mathematical Society, 1993, 29, 189–207; American Economic Review, 1994, 84, 427–434). Chichilnisky's theorem states that her condition of limited arbitrage is necessary and sufficient for the existence of an equilibrium in an economy with unbounded short sales. Our counterexample shows that the condition defined by Chichilnisky is not sufficient for existence of equilibrium. We also discuss difficulties in Chichilnisky (Economic Theory, 1995, 5, 79–107).     

Myopic utility functions on sequential economies

In Brown and Lewis (1981) continuity in the Mackey topology of (l^∞, l¹) is related to myopic (or impatient) economic behavior. They also show that finer (locally convex) topologies admit continuous non-myopic utility functions. In that work the space of bounded sequences, l^∞, is interpreted as all time sequences of bounded consumption plans. In Brown (1981) the analysis is extended to study the theory of interest on related sequence spaces.This note applies our simple technique for ‘computing’ Mackey continuity of real-valued functions defined on l^∞. Our first result is motivated by Bewley's (1972, app. II) theorem, but extends it in several important ways (on sequential economies). First, Beweley's examples (specialized to the sequential setting) are all ‘temporally separable’,that is, consumption in one time period does not affect indifference sets in another. We give new explicit examples of Cobb-Douglass-like utility functions and show that the ‘obvious’ infinite-dimensional Cobb-Douglass functions are non-myopic. Known equilibrium theory [from Bewley (1972), but pre-dating him in the sequential case] applies to these new examples. Second, we remove the assumptions of concavity and monotony from the proof of continuity.Our second result shows that some of the ‘stationary’ utility functions studied by Koopmans, Diamond and Williamson (1964) are also myopic in the sense of Brown and Lewis. In general their work is based on the finer uniform topology.Finally, we show how to transform our technique so that it applies to Brown's more general sequential economies. A change of variables transfers our examples to these spaces.     

A topological invariant for competitive markets

This paper summarizes the concepts of global cones and limited arbitrage introduced in Chichilnisky (Economic Theory, 1995, 5, 79–108), and the corresponding results establishing that limited arbitrage is necessary and sufficient for the existence of a competitive equilibrium and for the compactness of Pareto frontier (announced in Chichilnisky (American Economic Review, 1992, 84, 427–434, and Chichilnisky (Bulletin of the American Mathematical Society, 1993, 29, 189–207). Using the same global cones I extend my earlier results to encompass ‘mixed economies’ based on Chichilnisky (CORE Discussion Paper No. 9527, 1995). I introduce a topological invariant for competitive markets which deepens the concept of limited arbitrage. This invariant encodes exact information on the equilibria and on the social diversity of the economy and all its subeconomies, and predicts a failure of effective demand.     

Evolving Game Theory

Jörgen W. Weibull (1995) Evolutionary Game Theory. MIT Press. £18.50 $30 xv+265 pp. Ken Binmore, Alan Kirman and Piero Tani (eds) (1993) Frontiers of Game Theory. MIT Press. £40.50 $47.50 350 pp.     

Linear-quadratic approximation, external habit and targeting rules

We examine the linear-quadratic approximation of nonlinear dynamic stochastic optimization problems. A discrete-time version of Magill [1977a. A local analysis of N-sector capital accumulation under uncertainty. Journal of Economic Theory 15(2), 211–219] is generalized to models with forward-looking variables paying special attention to second-order conditions. This is the ‘large distortions’ case in the literature. We apply the approach to monetary policy in a DSGE model with external habit in consumption. We then develop a condition for ‘target-implementability’, a concept related to ‘targeting rules’. Finally, we extend the approach to a comparison between cooperative and non-cooperative equilibria in a two-country model and show that the ‘small distortions’ approximation is inappropriate for this exercise.     

Benchmark priors for Bayesian model averaging

In contrast to a posterior analysis given a particular sampling model, posterior model probabilities in the context of model uncertainty are typically rather sensitive to the specification of the prior. In particular, ‘diffuse’ priors on model-specific parameters can lead to quite unexpected consequences. Here we focus on the practically relevant situation where we need to entertain a (large) number of sampling models and we have (or wish to use) little or no subjective prior information. We aim at providing an ‘automatic’ or ‘benchmark’ prior structure that can be used in such cases. We focus on the normal linear regression model with uncertainty in the choice of regressors. We propose a partly non-informative prior structure related to a natural conjugate g-prior specification, where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter g_0j. The consequences of different choices for g_0j are examined. We investigate theoretical properties, such as consistency of the implied Bayesian procedure. Links with classical information criteria are provided. More importantly, we examine the finite sample implications of several choices of g_0j in a simulation study. The use of the MC³ algorithm of Madigan and York (Int. Stat. Rev. 63 (1995) 215), combined with efficient coding in Fortran, makes it feasible to conduct large simulations. In addition to posterior criteria, we shall also compare the predictive performance of different priors. A classic example concerning the economics of crime will also be provided and contrasted with results in the literature. The main findings of the paper will lead us to propose a ‘benchmark’ prior specification in a linear regression context with model uncertainty.     

Generic properties of the core and equilibria of pure exchange economies

We show that for an open dense set of markets with a continuum of traders the number of equilibrium allocations [which by the celebrated theorem of ^{Aumann (Econometrica. 1964, 32. 39–50)} coincide with the core allocations for such markets are finite. This presents a limiting case result that complements similar asymptotic theorems for cores of large economies proved by ^{Bewley (Econometrica 1973, 41, 425–454)}, and ^{Dierker (Journal of Mathematical Economics 1975, 2. 155–169)}. If we require that the measure on the space of agents be one with a finite number of atoms of equal weight, our reasoning recovers the classical results obtained by ^{Debreu (Econometrica. 1970. 38, 387–392)} for economies with a finite number of agents.     

Locally lipschitz cooperative games

A locally Lipschitz cooperative generalized game is described by its coalition worth function v defined on the set [0, 1]ⁿ of generalized (or fuzzy) coalitions of n players. We assume that v is positively homogeneous and locally Lipschitz. We propose the Clarke's generalized gradient ∂v(c^N) of v at the coalition c^N=(1,…,1) of all players as a set of solutions, and we study its property. We point out that it coincides with the core when v is super–additive and to the Shapley value when v is smooth.     

Structural instability of the core

Let σ be a q-rule, where any coalition of size q, from the society of size n, is decisive. Let w(n,q)= 2q-n+1 and let W be a smooth ‘policy space’ of dimension w. Let U(W)^N be the space of all smooth profiles on W, endowed with the Whitney topology. It is shown that there exists an ‘instability dimension’ w^*(σ) with 2w^*(σ)w(n,q) such that:

1. (i) if ww^*(σ), and W has no boundary, then the core of σ is empty for a dense set of profiles in U(W)^N (i.e., almost always),

2. (ii) if ww^*(σ)+1, and W has a boundary, then the core of σ is empty, almost always,

3. (iii) if ww^*(σ)+1 then the cycle set is dense in W, almost always,

4. (iv) if ww^*(σ)+2 then the cycle set is also path connected, almost always.

The method of proof is first of all to show that if a point belongs to the core, then certain generalized symmetry conditions in terms of ‘pivotal’ coalitions of size 2q−n must be satisfied. Secondly, it is shown that these symmetry conditions can almost never be satisfied when either W has empty boundary and is of dimension w(n,q) or when W has non-empty boundary and is of dimension w(n,q)+1.     

Theoretical tests of the rational expectations hypothesis in economic dynamical models

This paper examines different theoretical stability tests of infinite-horizon rational expectations equilibria. These ‘tests’ have different status: two of them express that the considered equilibrium is ‘isolated’ [neither (non-sunspot) equilibria (test 1) nor (well-behaved) sunspot equilibria exist in a neighbourhood (test 2)] and two of them are learning criteria [either standard ‘evolutive learning’ (test 3) or game-theoretical ‘eductive’ learning (test 4)]. Surprisingly, these four tests select the same steady state equilibria in the class of one-dimensional one-step-forward looking economic models. The extension of this equivalence theorem to n-dimensional and then more complex systems is discussed.     

Stochastic processes of temporary equilibria

It is the aim of the paper to study within the framework of an ‘overlapping generation model’ the evolution of temporary equilibria. At date t, there are ‘newborn’ agents and ‘old’ agents who were born in previous periods; the old agents hold cash balances (fiat money) that they carried over from the previous period. At the beginning of period t, all agents receive a random endowment of consumption goods. Then the agents exchange these endowments and money on spot markets at date t (trading in future markets is not considered). Once a temporary equilibrium is reached, the economy move to the next date. Agents who were born at date t then become old and meet agents born at period t+1.It is shown that the evolution of temporary equilibria in this model leads to analyse the ergodic properties of a certain class of Markov processes with stationary transition probability.     

Innovativeness and involvement as determinants of website loyalty: II. Determinants of consumer loyalty in B2C e-commerce

This paper is the second of the series of studies entitled “Innovativeness and Involvement as Determinants of Website Loyalty”, which was designed to test Foxall's [(1995). Cognitive styles of consumer initiators. Technovation 15(5), 269–288] style/involvement model in the context of Internet buyer behaviours. In this paper, a consumer Website loyalty model was proposed to describe how consumer transfer their existing brand loyalty in the traditional retail market to the same brand's Website in the B2C e-commerce market and how their perceived risk at the brand's Website mediates this loyalty transformation. Data were collected via an email invitation and Web-based questionnaire. One thousand and fourty four Taiwan Internet buyers randomly selected from the database of a well-known brand's Website have completed the survey. The multiple regression technique indicated the robustness of this loyalty transformation model (adjusted R²=0.50). Findings further indicated the impact that consumers’ cognitive style/involvement have on their loyalty transformation model. After segmenting consumers via their DSI and PDI scores, the distinct loyalty transformation models are revealed: the adjusted R² of more-involved innovators’ loyalty transformation model was the highest (0.60), followed by more involved adaptors’ model (0.45), less involved innovators’ model (0.45), and finally, less involved adaptors’ model (0.42). Discussions of how consumers’ cognitive style and involvement level interact with each other and impact on the predictors of the Website loyalty are discussed. Proposals are made of how Website managers can use this knowledge to build marketing strategies.     

The Delphi technique as a forecasting tool: issues and analysis

This paper systematically reviews empirical studies looking at the effectiveness of the Delphi technique, and provides a critique of this research. Findings suggest that Delphi groups outperform statistical groups (by 12 studies to two with two ‘ties’) and standard interacting groups (by five studies to one with two ‘ties’), although there is no consistent evidence that the technique outperforms other structured group procedures. However, important differences exist between the typical laboratory version of the technique and the original concept of Delphi, which make generalisations about ‘Delphi’ per se difficult. These differences derive from a lack of control of important group, task, and technique characteristics (such as the relative level of panellist expertise and the nature of feedback used). Indeed, there are theoretical and empirical reasons to believe that a Delphi conducted according to ‘ideal’ specifications might perform better than the standard laboratory interpretations. It is concluded that a different focus of research is required to answer questions on Delphi effectiveness, focusing on an analysis of the process of judgment change within nominal groups.     

Representing preferences with nontransitive indifference by a single real-valued function

Let be an interval order on a topological space (X, τ), and let x _˜^* y if and only if [y z x z], and x _˜^** y if and only if [z x z y]. Then _˜^* and _˜^** are complete preorders. In the particular case when is a semiorder, let x _˜⁰ y if and only if x _˜^* y and x _˜^** y. Then _˜⁰ is a complete preorder, too. We present sufficient conditions for the existence of continuous utility functions representing _˜^*, _˜^** and _˜⁰, by using the notion of strong separability of a preference relation, which was introduced by Chateauneuf (Journal of Mathematical Economics, 1987, 16, 139–146). Finally, we discuss the existence of a pair of continuous functions u, υ representing a strongly separable interval order on a measurable topological space (X, τ, μ, ).     

Consistency of assessments in infinite signaling games

In this paper we investigate possible ways to define consistency of assessments in infinite signaling games, i.e. signaling games in which the sets of types, messages and answers are complete, separable metric spaces. Roughly speaking, a consistency concept is called appropriate if it implies Bayesian consistency and copies the original idea of consistency in finite extensive form games as introduced by ^{Kreps and Wilson (Econometrica 1982, 50, 863–894)}. We present a particular appropriate consistency concept, which we call strong consistency and give a characterization of strongly consistent assessments. It turns out that all appropriate consistency concepts are refinements of strong consistency. Finally, we define and characterize structurally consistent assessments in infinite signaling games.