May’s theorem in an infinite setting

We generalize May’s theorem to an infinite setting, preserving the elementary character of the original theorem. We define voting scenarios and generalized voting scenarios, and prove appropriate versions of May’s theorem. The case of generalized voting scenarios specialized to a countably infinite set of voters and the collections of all coalitions that have asymptotic density, shows that majority rule is the only aggregation rule that satisfies neutrality, irrelevance of null coalitions, anonymity, and positive responsiveness.     

A topological approach to Wilson’s impossibility theorem

We will present a topological approach to Wilson’s impossibility theorem [Wilson, R.B., 1972. Social choice theory without the Pareto principle. Journal of Economic Theory 5, 478–486] that there exists no non-null binary social choice rule which satisfies transitivity, independence of irrelevant alternatives, non-imposition and has no dictator nor inverse dictator. Our research is in line with the studies of topological approaches to discrete social choice problems initiated by [Baryshnikov, Y., 1993. Unifying impossibility theorems: a topological approach. Advances in Applied Mathematics 14, 404–415]. This paper extends the result about the Arrow impossibility theorem shown in [Tanaka, Y., 2006. A topological approach to the Arrow impossibility theorem when individual preferences are weak orders. Applied Mathematics and Computation 174, 961–981] to Wilson’s theorem.     

Equilibrium in abstract economies without ordered preferences and with a measure space of agents

We report a generalization of Aumann's (1966) existence theorem to economies without ordered preferences and with externalities in consumption. Our work can alternatively be viewed as a generalization of the Shafer–Sonnenschein (1975) theorem to economies with a continuum of agents.     

Discrete fixed point theorem reconsidered

The aim of this note is to indicate an example that demonstrates the incorrectness of Iimura’s discrete fixed point theorem [Iimura, T., 2003. A discrete fixed point theorem and its applications. Journal of Mathematical Economics 39, 725–742] and to present a corrected statement using the concept of integrally convex sets.     

Equilibrium in abstract economies without ordered preferences

We extend a theorem of G. Debreu on the existence of equilibrium in a generalized N-person game (‘A Social Equilibrium Existence Theorem’). Applying standard techniques the result can be used to prove the existence of equilibrium in economies with interdependent preferences, price-dependent preferences and preferences which may be both non-transitive and non-complete. This generalizes a recent theorem of A. Mas-Colell.     

An existence theorem of intertemporal recursive utility in the presence of Lévy jumps

This paper presents an existence theorem for a class of backward stochastic integral equations. The main contribution is a generalization of Duffie and Epstein's [Duffie, D., Epstein, L., 1992. Stochastic differential utility, (Appendix C with Skiadas C.), Econometrica 60, 353–394.] existence theorem of intertemporal recursive utility to allow the information structure to be driven by a Lévy jump process. The existence theorem applies also for a more general class of utility functions, such as recursive utility with habit-formation, and can be used to prove the existence of an equilibrium asset price process as a unique solution to the stochastic Euler equation derived by Ma [Ma, C., 1993b. Valuation of Derivative Securities with Mixed Poisson–Brownian Information and Recursive Utility, McGill University, mimeo.].     

On games with incomplete information and the Dvoretsky–Wald–Wolfowitz theorem with countable partitions

It has remained an open question as to whether the results of Milgrom–Weber [Milgrom, P.R., Weber, R.J., 1985. Distributional strategies for games with incomplete information. Mathematics of Operations Research 10, 619–632] are valid for action sets with a countably infinite number of elements without additional assumptions on the abstract measure space of information. In this paper, we give an affirmative answer to this question as a consequence of an extension of a theorem of Dvoretzky, Wald and Wolfowitz (henceforth DWW) due to Edwards [Edwards, D.A., 1987. On a theorem of Dvoretsky, Wald and Wolfowitz concerning Liapunov measures. Glasgow Mathematical Journal 29, 205–220]. We also present a direct elementary proof of the DWW theorem and its extension, one that may have an independent interest.     

Some results on general assumptions about the existence of economic equilibrium

In the present paper we maximally use the possibilities provided by the Nash approach and the Kakutani fixed point theorem for proving the existence of an economic equilibrium. We obtain a general existence theorem which does not require a special form for income distribution functions and producer's objectives, independence of consumers' tastes, ordered preference and zero degree homogeneous price dependence. The role of the non-satiation assumption becomes more clear.     

The joint continuity of the expected payoff functions

Glicksberg [Glicksberg, I.L., 1952. A further generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points. In: Proceedings of the American Mathematical Society 3, pp. 170–174] generalized the Kakutani fixed point theorem to the setting of locally convex spaces and used it to prove that every k-person strategic game with action sets convex compact subsets of locally convex spaces and continuous payoff functions has a Nash equilibrium. He subsequently used this result to establish the following fundamental theorem of game theory: Every k-person strategic game with action sets metrizable compact topological spaces and continuous payoff functions has a mixed strategies equilibrium. However, in his proof of the latter result, Glicksberg did not show that the expected payoff functions were jointly continuous, something that was required for the existence of a mixed strategies equilibrium.     

Duality and maximum principle in multi-period convex programming

Starting from works in Rockafellar (1967), a duality theorem, existence theorem, reciprocity principle and a max–min principle are obtained in a class of dynamic programming which has a possibility of wide economic applications. These theorems serve to summarize the theory of the efficient price vectors developed in activity analysis and to provide a linkage between that theory and the maximum principle in the optimal control theory.     

A representation of risk measures

We provide a representation theorem for risk measures satisfying (1) monotonicity, (2) positive homogeneity and (3) translation invariance. As a simple corollary to our theorem, we obtain the usual representation of coherent risk measures (i.e., risk measures that are, in addition, sub-additive; see Artzner et al. in Math Finance 9:203–228, 1999).     

PRACTITIONERS' CORNER:Triangular Representation and Error Correction Mechanism in Cointegrated Systems

ABSTRACT In this paper we provide a proof of the ‘Granger representation theorem’ starting from Phillips' triangular representation of a cointegrated system. Proofs of the theorem have already been provided but starting either from the MA representation or from the AR representation. Conversely, we derive and characterize analytically the MA and the AR representations in terms of the parameters of the triangular system.     

Asymptotic linearity of minimax estimators

The celebrated local asymptotic minimax (LAM) theorem due to HÁjek (1972) also includes the statement that a LAM estimator Is necessarily asymptotically linear. A similar result. is true for semi-parametric models, but Hájek's result doesn't apply to this case as the efficient influence function is often not contained in the (proper) tangent space. This note gives a simple, elementary proof of both the LAM theorem and the necessity of asymptotic linearity of a LAM estimator sequence.     

Perron's stability theorem for non-linear mappings

Perron's theorem on positive matrices including its stability statement is extended to non-linear mappings which need neither additive nor homogeneous nor primitive. This generalizes known results and yields in particular a concave version of Perron's theorem. The theorem may be applied to balanced growth in non-linear systems and also to obtain a dynamic non-substitution theorem for general cost functions.     

No arbitrage conditions and liquidity

We extend the fundamental theorem of asset pricing to the case of markets with liquidity risk. Our results generalize, when the probability space is finite, those obtained by Kabanov et al. [Kabanov, Y., Stricker, C., 2001. The Harrison-Pliska arbitrage pricing theorem under transaction costs. Journal of Mathematical Economics 35, 185–196; Kabanov, Y., Rásonyi, M., Stricker, C., 2002. No-arbitrage criteria for financial markets with efficient friction. Finance and Stochastics 6, 371–382; Kabanov, Y., Rásonyi, M., Stricker, C., 2003. On the closedness of sums of convex cones in L⁰$L^0$ and the robust no-arbitrage property. Finance and Stochastics] and by Schachermayer [Schachermayer, W., 2004. The fundamental theorem of asset pricing under poportional transaction costs in finite discrete time. Mathematical Finance 14 (1), 19–48] for markets with proportional transaction costs. More precisely, we restate the notions of consistent and strictly consistent price systems and prove their equivalence to corresponding no arbitrage conditions. We express these results in an analytical form in terms of the subdifferential of the so-called liquidation function. We conclude the paper with a hedging theorem.     

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).     

Informativeness of experiments for MEU—A recursive definition

The well-known Blackwell theorem states the equivalence of statistical informativeness and economic valuableness. Çelen (2012) generalizes this theorem, which is well-known for subjective expected utility (seu), to maxmin expected utility (meu) preferences. We demonstrate that the underlying definition of the value of information used in Çelen (2012) is in contradiction with the principle of recursively defined utility. As a consequence, Çelen’s framework features dynamic inconsistency. Our main contribution consists in the definition of a value of information for meupreferences that is compatible with recursive utility and thus respects dynamic consistency.     

Cardinal utility, utilitarianism, and a class of invariance axioms in welfare analysis

This paper examines some problems of utility measurement and interpersonal comparisons. The following theorem is first established for real-valued mappings on a set of real numbers, B: If the closure of B is an interval, then such a mapping is positive affine iff it is a first-difference preserving transformation. This theorem is used to axiomatize cardinal utility. Standard invariance axioms in social choice theory are interpreted in an alternative framework and this gives new results in axiomatizing utilitarianism. It is shown that the conflict between utilitarianism and Rawlsianism arises, in some sense, earlier than traditionally supposed.     

On the dimensionality of bounds generated by the Shapley–Folkman theorem

The Shapley–Folkman theorem places a scalar upper bound on the distance between a sum of non-convex sets and its convex hull. We observe that some information is lost when a vector is converted to a scalar to generate this bound and propose a simple normalization of the underlying space which mitigates this loss of information. As an example, we apply this result to the Anderson (1978) core convergence theorem, and demonstrate how our normalization leads to an intuitive, unitless upper bound on the discrepancy between an arbitrary core allocation and the corresponding competitive equilibrium allocation.     

A ‘funnel’ turnpike theorem for optimal growth problems with discounting

A turnpike theorem for the optimal control problem with discounting is given. The optimal trajectory is shown to lie in an exponentially bounded region of the optimal steady-state. This region, referred to as a funnel, is determined by the discount rate of the problem. The funnel theorem reduces to the classical turnpike theorem when the discount rate is zero.