Continuity and Equilibrium Stability

This paper discusses the problem of stability of equilibrium points in normal form games in the tremling-hand framework. An equilibrium point is called perffect if it is stable against at least one seqence of trembles approaching zero. A strictly perfect equilibrium point is stable against every such sequence. We give a sufficient condition for a Nash equilibrium point to be strictly perfect in terms of the primitive characteristics of the game (payoffs and strategies), which is new and not known in the literature. In particular, we show that continuity of the best response correspondence (which can be stated in terms of the primitives of the game) implies strict perfectness; we prove a number of other useful theorems regarding the structure of best responce correspondence in normal form games.Strictly perfect equilibrium, best responce correspondence, unit simplex, face of a unit simplex

A Refinement of Perfect Equilibria Based On Substitute Sequences

We propose an equilibrium refinement of strict perfect equilibrium for the finite normal form games, which is not known in the literature. Okada came up with the idea of strict perfect equilibrium by strengthening the main definition of a perfect equilibrium, due to Selten [14]. We consider the alternative (and equivalent) definition of perfect equilibrium, based on the substitute sequences, as appeared in Selten [14]. We show that by strengthening and modifiyng this definition slightly, one can obtain a refinement stronger than strict perfectness. We call the new refinement strict substitute perfect equilibrium. The main advantage of this solution concept is that it reflects the local dominance of an equilibrium point. An example is provided to show that a strict perfect equilibrium may fail to be strict substitute perfect.Perfect equilibrium, strictly perfect equilibrium, substitute sequence, substitute perfect equilibrium, unit simplex

THE CHEAPEST HEDGE:A PORTFOLIO DOMINANCE APPROACH

Investors often wish to insure themselves against the payoff of their portfolios falling below a certain value. One way of doing this is by purchasing an appropriate collection of traded securities. However, when the derivatives market is not complete, an investor who seeks portfolio insurance will also be interested in the cheapest hedge that is marketed. Such insurance will not exactly replicate the desired insured-payoff, but it is the cheapest that can be achieved using the market. Analytically, the problem of finding a cheapest insuring portfolio is a linear programming problem. The present paper provides an alternative portfolio dominance approach to solving the minimum-premium insurance portfolio problem. This affords remarkably rich and intuitive insights to determining and describing the minimum-premium insurance portfolios.

An Overlapping Generations Model Core Equivalence Theorem

The classical Debreu-Scarf core equivalence theorem asserts that in an exchange economy with a finite number of agents art allocation (under certain conditions) is a Walrasian equilibrium if and only if it belongs to the core of every replica of the exchange economy. The pioneering work of P. Samuelson has shown that such a result fails to be true in exchange economies with a countable number of agents. This paper presents a Debreu-Scarf type core equivalence theorem for the overlapping generations (OLG) model. Specifically, the notion of a short-term core allocation for the overlapping generations model is introduced and it is shown that (under some appropriate conditions) an OLG model allocation is a Walrasian equilibrium if and only if it belongs to the short-term core of every replica of the OLG economy

When is the Core Equivalence Theorem Valid?

In 1983 L. E. Jones exhibited a surprising example of a weakly Pareto optimal allocation in a two consumer pure exchange economy that failed to be supported by prices. In this example the price space is not a vector lattice (Riesz space). Inspired by Jones' example, A. Mas-Colell and S. F. Richard proved that this pathological phenomenon cannot happen when the price space is a vector lattice. In particular, they established that (under certain conditions) in a pure exchange economy the lattice structure of the price space is sufficient to guarantee the supportability of weakly Pareto optimal allocations by prices-i.e., they showed that the second welfare theorem holds true in an exchange economy whose price space is a vector lattice. In addition, C. D. Aliprantis, D. J. Brown and O. Burkinshaw have shown that when the price space of an exchange economy is a certain vector lattice, the Debreu-Scarf core equivalence theorem holds true, i.e., the sets of Walrasian equilibria and Edgeworth equilibria coincide. (An Edgeworth equilibrium is an allocation that belongs to the core of every replica economy of the original economy.) In other words, the lattice structure of the price space is a sufficient condition for avoiding the pathological situation occurring in Jones' example. This work shows that the lattice structure of the price space is also a necessary condition. That is, "optimum" allocations in an exchange economy are supported by prices (if and) only if the price space is a vector lattice. Specifically, the following converse-type result of the Debreu-Scarf core equivalence theorem is established: If in a pure exchange economy every Edgeworth equilibrium is supported by prices, then the price space is necessarily a vector lattice

Economies With Many Commodities

We discuss the two fundamental theorems of welfare economics in the context of the Arrow-Debreu-McKenzie model with an infinite dimensional commodity space. As an application, we prove the existence of competitive equilibrium in the standard single agent growth model

Equilibria in Markets with a Reisz Space of Commodities

Using the theory of Riesz spaces, we present a new proof of the existence of competitive equilibria for an economy having a Riesz space of commodities

On dominant contractions and a generalization of the zero-two law

Zaharopol proved the following result: let T,S:L^1(X,{\cf},\m)\to L^1(X,{\cf},\m) be two positive contractions such that $T\leq S$. If $\|S-T\|<1$ then $\|S^n-T^n\|<1$ for all n\in\bn. In the present paper we generalize this result to multi-parameter contractions acting on $L^1$. As an application of that result we prove a generalization of the "zero-two" law.Comment: 10 page

The Non-Archimedean Theory of Discrete Systems

In the paper, we study behavior of discrete dynamical systems (automata) w.r.t. transitivity; that is, speaking loosely, we consider how diverse may be behavior of the system w.r.t. variety of word transformations performed by the system: We call a system completely transitive if, given arbitrary pair $a,b$ of finite words that have equal lengths, the system $\mathfrak A$, while evolution during (discrete) time, at a certain moment transforms $a$ into $b$. To every system $\mathfrak A$, we put into a correspondence a family $\mathcal F_{\mathfrak A}$ of continuous maps of a suitable non-Archimedean metric space and show that the system is completely transitive if and only if the family $\mathcal F_{\mathfrak A}$ is ergodic w.r.t. the Haar measure; then we find easy-to-verify conditions the system must satisfy to be completely transitive. The theory can be applied to analyze behavior of straight-line computer programs (in particular, pseudo-random number generators that are used in cryptography and simulations) since basic CPU instructions (both numerical and logical) can be considered as continuous maps of a (non-Archimedean) metric space $\mathbb Z_2$ of 2-adic integers.Comment: The extended version of the talk given at MACIS-201

Bilateral Matching with Latin Squares

We develop a general procedure to construct pairwise meeting processes characterized by two features. First, in each period the process maximizes the number of matches in the population. Second, over time agents meet everybody else exactly once. We call this type of meetings absolute strangers. Our methodological contribution to economics is to offer a simple procedure to construct a type of decentralized trading environments usually employed in both theoretical and experimental economics. In particular, we demonstrate how to make use of the mathematics of Latin squares to enrich the modeling of matching economies