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.     

Designer path independent choice functions

Summary. This paper provides an algorithm for the construction of all PICFs on a finite set of alternatives, V, designed by an a priori given set I of initial choices as well as the determination of whether the initial set I is consistent with path independence. The algorithm is based on a new characterization result for path independent choice functions (PICF) on finite domains and uses that characterization as the basis of the algorithm. The characterization result identifies two properties of a partition of the Boolean algebra as necessary and sufficient for a choice function C to be a PICF: (i): For every subset A of V the set is an interval in the Boolean algebra 2 ^V . (ii): If A/B is an interval in the Boolean algebra such that C(A) = C(B) and if M/N is an upper transpose of A/B then C(M) = C(N). The algorithm proceeds by expanding on the implications of these two properties.Received: 5 November 2003, Revised: 20 July 2004, JEL Classification Numbers: D00, D70.     

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.     

How to gamble against all odds

We compare the power of betting strategies (aka martingales) whose wagers take values in different sets of reals. A martingale whose wagers take values in a set A is called an A-martingale. A set of reals B anticipates a set A, if for every A-martingale there is a countable set of B-martingales, such that on every binary sequence on which the A-martingale gains an infinite amount at least one of the B-martingales gains an infinite amount, too.We show that for two important classes of pairs of sets A and B, B anticipates A if and only if the closure of B contains rA, for some positive r. One class is when A is bounded and B is bounded away from zero; the other class is when B is well ordered. Our results generalize several recent results in algorithmic randomness and answer a question posed by Chalcraft et al. (2012).     

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(·).     

Reciprocity and Giving in a Consecutive Three‐Person Dictator Game with Social Interaction

We study pure indirect reciprocity by setting up a modified dictator game with three players A, B, and C acting sequentially. Subject A takes a share of a pie and passes the rest to subject B, while B divides the rest between herself and C. We find that this consecutive three‐person dictator game increases generosity compared with the traditional two‐person dictator game. We analyze the influence of social interaction and uncertainty. In treatments with certainty we observe pure indirect reciprocity: B indirectly reciprocates for A's behavior in the decision on how generous to be to C.     

Lexicographic compositions of multiple criteria for decision making

This paper considers two distinct procedures to lexicographically compose multiple criteria for social or individual decision making. The first procedure composes M binary relations into one, and then selects its maximal elements. The second procedure first selects the set of maximal elements of the first binary relation, and then within that set, chooses the maximal elements of the second binary relation, and iterates the procedure until the Mth binary relation. We show several distinct sets of conditions for the choice functions representing these two procedures to satisfy non-emptiness and choice-consistency conditions such as contraction consistency and path independence. We also examine the relationships between the outcomes of the two procedures. Finally we investigate under what conditions the outcomes of each procedure are independent of the order of lexicographic application of the criteria.     

Technology adoption in markets with network effects: Theory and experimental evidence

We examine a technology-adoption game with network effects in which coordination on either technology A or technology B constitutes a Nash equilibrium. Coordination on technology B is assumed to be payoff dominant. We define a technology’s critical mass as the minimal share of users, which is necessary to make the choice of this technology the best response for any remaining user. We show that the technology with the lower critical mass implies risk dominance and selection by the maximin criterion. We present experimental evidence that both payoff dominance and risk dominance explain participants’ choices in the technology-adoption game. The relative riskiness of a technology can be proxied using either technologies’ critical masses or stand-alone values absent any network effects.     

Increases in risk aversion and the distribution of portfolio payoffs

Oliver Hart proved the impossibility of deriving general comparative static properties in portfolio weights. Instead, we derive new comparative statics for the distribution of payoffs: A is less risk averse than B iff A?s payoff is always distributed as B?s payoff plus a non-negative random variable plus conditional-mean-zero noise. If either agent has nonincreasing absolute risk aversion, the non-negative part can be chosen to be constant. The main result also holds in some incomplete markets with two assets or two-fund separation, and in multiple periods for a mixture of payoff distributions over time (but not at every point in time).     

When is multidimensional screening a convex program?

A principal wishes to transact business with a multidimensional distribution of agents whose preferences are known only in the aggregate. Assuming a twist (= generalized Spence-Mirrlees single-crossing) hypothesis, quasi-linear utilities, and that agents can choose only pure strategies, we identify a structural condition on the value b(x,y) of product type y to agent type x — and on the principal?s costs c(y) — which is necessary and sufficient for reducing the profit maximization problem faced by the principal to a convex program. This is a key step toward making the principal?s problem theoretically and computationally tractable; in particular, it allows us to derive uniqueness and stability of the principal?s optimal strategy — and similarly of the strategy maximizing the expected welfare of the agents when the principal?s profitability is constrained. We call this condition non-negative cross-curvature: it is also (i) necessary and sufficient to guarantee convexity of the set of b-convex functions, (ii) invariant under reparametrization of agent and/or product types by diffeomorphisms, and (iii) a strengthening of Ma, Trudinger and Wang?s necessary and sufficient condition (A3w) for continuity of the correspondence between an exogenously prescribed distribution of agents and of products. We derive the persistence of economic effects such as the desirability for a monopoly to establish prices so high they effectively exclude a positive fraction of its potential customers, in nearly the full range of non-negatively cross-curved models.     

Learnability and rationality of choice

In this paper we study the learnability of the class of rationalizable choice functions using the basic concept of PAC-learnability from statistical learning theory. We prove that the class of rationalizable choice functions on N alternatives is learnable from O(N) examples and is optimal in terms of PAC-learnability among classes which are invariant under permutations of the elements.     

A characterization of quasi-concave functions

This paper provides a simple and short proof that the following two properties are equivalent under ▽f(x) ≠ 0: (A) ▽f(x) h = 0 implies h^T▽²f(x) h ? 0 for any x; (B) f(x¹) ? f(x⁰) implies f(tx¹ + (1?t)x⁰) ?f(x⁰) for 0? t ? 1.     

Characterization of optimal plans for stochastic dynamic programs

This paper provides general techniques for the characterization of optimal plans resulting from stochastic dynamic programming. We show that under standard assumptions the optimal plans in both finite and infinite horizon problems can be obtained by an application of the Implicit Function Theorem to first order conditions. Further, we show that under certain checkable conditions, optimal plans and value functions are p-times differentiable for any integer p ? 0. Finally, we apply our technique to obtain a C^p plan and value function in a one sector infinite horizon growth problem under uncertainty.     

Set-rationalizable choice and self-stability

Rationalizability and similar notions of consistency have proved to be highly problematic in the context of social choice, as witnessed by a range of impossibility results, among which Arrow?s is the most prominent. We propose to rationalize choice functions by preference relations over sets of alternatives (set-rationalizability) and introduce two consistency conditions, and , which are defined in analogy to Sen?s α and γ. We find that a choice function satisfies if and only if it is set-rationalizable and that it satisfies and if and only if it is self-stable, a new concept based on earlier work by Dutta. The class of self-stable social choice functions contains a number of appealing Condorcet extensions.     

Minimal conditions for upper-hemicontinuity of the choice correspondence

We denote by ?(M, r) the set of all maximal elements in the constraint set M with respect to a (partial) order r and find the coarsest topologies on the domain of ? such that ? is u.h.c.     

Numerical representation of intransitive preferences on a countable set

Let > be a preference relation on a countable set X. We prove that if > is acyclic (that is, has irreflexive transitive closure), then there exists a mapping u of X into $\text{R}$ such that x > y entails u(x)>u(y). We also give a simple proof of a representation theorem of Fishburn when > is an interval order.     

Efficiency First or Equity First? Two Principles and Rationality of Social Choice

This paper formalizes two different principles to socially rank allocations under efficiency-equity trade off. The efficiency-first social preference relation ranks an allocation x higher than an allocation y if and only if (i) x is Pareto-superior to y or (ii) x and y are Pareto-noncomparable and x is equity-superior to y. The equity-first social preference relation reverses the order of application of the two criteria. We show that the efficiency-first relation may have a cycle, whereas the equity-first relation is transitive. We also characterize and compare the sets of maximal elements for these social preference relations in the Edgeworth box and examine the nonemptiness of the sets. Journal of Economic Literature Classification Numbers: D71, D63, D61.     

Majority committees

Suppose an m-member committee is to be selected by a set of voters from a set X of M > m nominees. If A is an m-member committee, and if as many voters prefer A to B as prefer B to A for every other committee B of m nominees, then A is a majority committee of size m. Although the existence of majority nominees (m = 1) has been extensively analyzed, little attention has been given to conditions that imply the existence of majority committees of size m ? 2. Known restrictions on voters' preferences on X that guarntee the existence of a majority nominee could be applied directly to voters' preferences on m-member committees. However, this definitional exercise lacks intuitive appeal, and a different approach is taken in this paper. The paper presumes that profiles of voters' preferences on X are either dichotomous or single peaked. Both cases ensure the existence of a majority nominee. Independence-monotonicity assumptions are then used to connect voters' preferences on X to their preferences on committees of equal size. Although these assumptions guarantee the existence of majority committees when m = 1 and when m = M ? 1, they do not generally do so when 1 < m < M ? 1. The latter observation motivates additional restrictions on profiles. In the dichotomous case, we consider profiles in which all voters have the same number k of nominees in their preferred subsets, and show that this restriction guarantees the existence of a majority committee of size m for 1 < m < M ? 1 only when k = 1 or k = M ? 1. In the single-peaked case, we consider profiles in which all voters have the same most-preferred (peak) nominee, and prove that this guarantees the existence of a majority committee of size m for every m between 1 and M ? 1.     

The network of weighted majority rules and its geometric realizations

In a previous work the first author considered the network G(n,p) of weighted majority rules (WMR) for n decision makers whose competencies are given by their probabilities p=(p₁,…,p_n) of making a correct decision. On this paper we consider chains of decision profiles, which must occur in G(n,p) in a fixed order, and show that they can be mapped onto straight lines in a low-dimensional geometric realization. The minimal number of directions which must used to separate all edges is given as the chromatic number of a certain incidence graph. We also define degenerate networks in which several nodes coalesce.     

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.