The expected probability of Condorcet's paradox

Let P(n,p) be the probability that there is a Condorcet winner on three alternatives for n (odd) voters. The vector p defines the probability that a randomly selected voter will have any of the six linear rankings on three alternatives as his or her preference ranking. It is shown that if all p vectors are equally likely, the expected value of P(n,p) is given by 15(n+3)²/[16(n+2)(n+4)].     

A note on the fundamental theorem of exact aggregation

If individual i's demands for a commodity are a function of prices, p, income M_i and a vector of attributes A_i, then aggregate demand is

This paper derives the necessary and sufficient conditions of f_i, F and a system of functions g_k(M₁,…,M_N,A₁,…,A_N) symmetric in the M's and A's such that F can be written in the form F(p, g, (M₁,…,M_n,A₁,…,A_N),…, g_n(M₁,…,M_n,A₁,… ,A_n)) for all values of its arguments.     

Optimal control problems from second-order difference equations

We examine solutions x?l_∞(Rⁿ) of the equations $\text{F}{}_{\mathrm{t}}\text{(}\text{x}{}_{\mathrm{t?1}}\text{,}\text{x}{}_{\mathrm{t}}\text{,}\text{x}{}_{\mathrm{t+1}}\text{) = 0}$ and derive conditions for the existence of objective functions G_t so that $\text{x}$ solves max_x?l∞ΣG_t(x_t, x_t+1). We specialize the conditions to time autonomous equations and apply them to a competitive industry in temporary equilibrium. The objective function in the example is of the cost-benefit form: consumer surplus minus opportunity cost of labor minus industry-wide cost of investment.     

Zero-sum condition: A necessary and sufficient condition for a transitive voting system

A simple and quick way to ascertain whether or not any given majority voting system can always produce a transitive social preference orderings without imposing any restriction on the distribution of diverse individual preference orderings is to examine whether all individual voting (preference) vectors satisfy the Addition Rule or not. This conclusion was obtained by first reformulating the voting mechanism into that of a linear mapping from T^m defined by q = Σp_i. It was found that the subset P of T that can accommodate all possible individual preference ordering profiles and such that every sum vector q = Σ p_i of its member vectors p_i is contained in T can be expressed as P = {p: p∈ T, s(p) = 0}. It was also pointed out that this is equivalent to the requirement that all individual preference (voting) functions must satisfy the Addition Rule. Finally, Borda's Rule and Saposnik's Contributive Rule were shown to be examples of transitive voting rules which satisfy these necessary and sufficient conditions.     

A progress report on Kelly's majority conjectures

LetC(m, n) be the proportion of n-voter profiles on m alternatives that have a majority winner. Jerry Kelly conjectured that C(m, n) > C(m + 1, n) for m ? 3 and n = 3 and n ? 5, and C(m, n) > C(m, n + 2) for m ? 3 and n ? 3. We prove these for special cases.     

Unmediated communication in games with complete and incomplete information

We study the effects of adding unmediated communication to static, finite games of complete and incomplete information. We characterize S^U(G), the set of outcomes of a game G, that are induced by sequential equilibria of cheap talk extensions. A cheap talk extension of G is an extensive-form game in which players communicate before playing G. A reliable mediator is not available and players exchange private or public messages that do not affect directly their payoffs. We first show that if G is a game of complete information with five or more players and rational parameters, then S^U(G) coincides with the set of correlated equilibria of G. Next, we demonstrate that if G is a game of incomplete information with at least five players, rational parameters and full support (i.e., all profiles of types have positive probability), then S^U(G) is equal to the set of communication equilibria of G.     

Saddle points of Hamiltonian systems in convex Lagrange problems having a nonzero discount rate

Problems are studied in which an integral of the form ∫₀^+∞L(k(t),k(t))e^?ptdt is minimized over a class of arcs k: [0, +∞) → Rⁿ. It is assumed that L is a convex function on Rⁿ × Rⁿ and that the discount rate ? is positive. Optimality conditions are expressed in terms of a perturbed Hamiltonian differential system involving a Hamiltonian function H(k, q) which is concave in k and convex in q, but not necessarily differentiable. Conditions are given ensuring that, for ? sufficiently small, the system has a stationary point, in a neighborhood of which one has classical “saddle point” behavior. The optimal arcs of interest then correspond to the solutions of the system which tend to the stationary point as t → +∞. These results are motivated by questions in theoretical economics and extend previous work of the author for the case ? = 0. The case ? < 0 is also covered in part.     

Topological upper limits of mixed Nash equilibria

Our purpose in this article is to prove that given any integer n ≥ 2 and any non-empty compact Polish spaces S ₁, ..., S _n , if for any u ∈ C( S ₁ × ... × S _n , R) ⁿ , we denote by MNE(u) the set of mixed Nash equilibria of (S ₁, ..., S _n , u), then MNE(u) is a non-empty compact subset of P(S ₁) × ... × P(S _n ) and if u ^k → u in C(S ₁ × ... × S _n , R) ⁿ as k → ∞, then lim sup _{k → ∞} MNE (u ^k ) MNE(u). The author would like to thank the referee for offering critical comments on this paper.     

Transitive measurable utility

This paper continues a study of theories of preferences under risk that do not use the independence axiom of the von Neumann-Morgenstern theory. Unlike its predecessor, it assumes that preferences are transitive. The effects of transitivity are noted in two representations of preferences. The first, which also uses continuity and dominance axioms, involves a function u on a set P of probability measures for which u(p) > u(q) if and only if p is preferred to q. Although u might be nonlinear, it has other features of a von Neumann-Morgenstern linear utility function. The second representation has linear functions u and w on P, with w strictly positive except perhaps at preference-extreme measures—where it might vanish, such that u(p) w(q) > u(q) w(p) if and only if p is preferred to q. A symmetry axiom along with the axioms for the first representation are necessary and sufficient for the second representation.     

Sufficient conditions in free final time optimal control problems. A comment

By an application of sufficient conditions, assume that an optimal pair (χ_τ(·), u_τ(·)) and an adjoint function p_τ(·) were found in the control problem in question with the final time τ fixed but arbitrary. Then a sufficient condition for one of these pairs, say , to be optimal in the corresponding free final time problem is that the Hamiltonian, with (χ_τ(t), u_τ(τ?), p_τ(τ), τ) inserted, is nonnegative (nonpositive) to the left (right) of $\text{τ}{}^1$.     

Min, Max, and Sum

This paper provides characterization theorems for preferences that can be represented by U(x₁, …, x_n)=min{x_k}, U(x₁, …, x_n)=max{x_k}, U(x₁, …, x_n)=∑ u(x_k), or combinations of these functionals. The main assumption is partial separability, where changing a common component of two vectors does not reverse strict preferences, but may turn strict preferences into indifference. We discuss applications of our results to social choice. Journal of Economic Literature Classification Numbers: C0, D1, D6.     

A proposed convention for measuring the state of the art of products or processes

During 1980, The Futures Group performed a study under contract to the National Science Foundation to devise a convention for describing—in quantitative terms—technological state of the art of essential any technology. In designing this convention we hope that different analysts, working independently, would be able to arrive at a similar conclusion about the state of the art of a technology under study. With such a measure at hand, it would be possible, for example, to evaluate the effectiveness of investment in R&D by evaluating the improvement in the technological state of the art of a given technology, per unit investment.The form of the equation chosen for the state-of-the-art convention was SOA = K₁(P₁/P'₁ + K₂(P₂/P'₂) ? K_n(P_n/P'_n) where SOA = state of the art, K_n = the relative weight associated with each parameter describing the technology, P_n = the value of the particular parameter useful in describing the state of the art, and P'_n = a reference value of the parameter. Various approaches to the selection of the parameters and their weights were described. Of particular interest is a statistical approach which assumes that the state of the art, over time, is an S-shaped curve. In this instance it is possible to compute the value of the weights through an iterative numerical technique.The convention was applied to two technologies: computers and antibiotics, and state-of-the-art measures were developed for each. In the case of antibiotics, selected organism/antibiotic pairs were analyzed to identify the state of the art of a particular antibiotic as applied to the control of a particular microorganism, over time.     

Rising marginal cost and concavity of the production function

Suppose a production function, f, is continuous, quasi-concave and weakly monotone on the non-negative orthant of Euclidean n-space. Let c(·, ·) be the associated cost function. Then it is shown that f is concave if and only if for each w, c(w, ·) is convex.     

A global non-linear extension of the Le Chatelier-Samuelson Principle for linear Leontief models

The extension is concerned with non-linear mappings A(·) which map the space of n-vectors of total gross outputs into the space of n-vectors of final demands. Roughly speaking, it is shown that, if throughout the non-negative orthant the Jacobian matrix of A(·) is of the Leontief type and satisfies the Hawkins-Simon condition, and if A(·) satisfies a certain other condition, then propositions of a global nature entirely analogous to the Le Chatelier-Samuelson Principle in both the weak and strong forms for linear Leontief systems are valid for A(·).     

Environmental protection mechanisms and technological dynamics

The paper proposes a new financial mechanism that could be implemented to protect the environment of a tourist region. For this purpose, we investigate the potential consequences of two financial activities, issued by the local government (G) of a region R, which work like contracts between G and, respectively, visitors of R and firms operating in R. According to these contracts, agents who decide to visit R (firms that decide to adopt an environmental friendly technology) have to buy an option that entitle them to get a partial or total reimbursement if environmental quality in R turns out to be sufficiently low (high), namely, below (above) a given predetermined threshold level.     

Bayes prediction density and regression estimation — A semiparametric approach

This paper is concerned with the Bayes estimation of an arbitrary multivariate density,f(x), x ? R ^k. Such anf(x) may be represented as a mixture of a given parametric family of densities {h (x¦θ)} with support inR ^k, whereθ (inR ^d) is chosen according to a mixing distributionG. We consider the semiparametric Bayes approach in whichG, in turn, is chosen according to a Dirichlet process prior with given parameterα. We then specialize these results whenf is expressed as a mixture of multivariate normal densitiesΦ (x¦Μ, λ) whereΜ is the mean vector and λ is the precision matrix. The results are finally applied to estimating a regression parameter.     

Optimal scrutiny in multi-period promotion tournaments

Consider a principal who hires heterogeneous agents to work for him over T periods, without prior knowledge of their skills, and intends to promote one of them at the end. In each period the agents choose effort levels that influence their outputs, and are fully informed of the past history of outputs. The principal's major objective is to maximize the expected total output, but he may also put some weight on detecting the higher-skilled agent for promotion. To this end, he randomly samples n out of the T periods and promotes the agent who produces more on the sample. This determines an extensive form game Γ(T,n), which we analyze for its subgame perfect equilibria in behavioral strategies. We show that the principal will do best to always choose a small sample size n. More precisely, if η(T) is the maximal optimal sample size, then η(T)/T→0 as T→∞.     

Remarks on the application of statistical games to technological development forecasting

This work presents the probability of determining a quantitative forecast of technological development S(t) defined by a set m of parameters S⁽¹⁾(t),S⁽²⁾(t),…,S^(m)(t), based on statistical game theory. Assuming that the coordinates S⁽ⁱ⁾(t) (i = 1, 2,…,m) of a forecasted vector S(t) are stochastic processes with given probabilistic characteristics, a formula of a function forecasting the value of a coordinate S⁽ⁱ⁾(t) of this vector can be obtained. This formula permits to determine a vector of forecasts _τT(x) of technological development S(t) at a given moment t = τ+T.     

Budget preferences and direct utility

This paper extends a result of Sakai, who presented conditions for indirect preferences from which a utility function can be deduced if demand is single valued. It will be shown that—adding a “partial Lipschitz condition”—Sakai's theorem can be extended to multi-valued demand. Our result follows from the extension of a theorem of Hurwicz and Richter, who have shown that, under certain hypotheses on demand correspondences, an upper semicontinuous utility function f exists, so that the set of utility maximal elements μ_f(B) is contained in the choice set h(B) for every budget B. By our partial Lipschitz condition h(B) ?- μ_f(B) also follows.     

Lexicographic composition of simple games

Certain voting bodies can be modeled as a simple game where a coalition's winning depends on whether it wins, blocks or loses in two smaller simple games. There are essentially five such ways to combine two proper games into a proper game. The most decisive is the lexicographic rule, where a coalition must either win in G₁, or block in G₁ and win in G₂. When two isomorphic games are combined lexicographically, a given role for a player confers equal or more power when held in the first game than the second, if power is assessed by any semi-value. A game is lexicographically separable when the players of the two components partition the whole set. Games with veto players are not separable, and games of two or more players with identical roles are separable only if decisive. Some separable games are egalitarian in that they give players identical roles.