Outer measure and utility

In most economics textbooks there is a gap between the non-existence of utility functions and the existence of continuous utility functions, although upper semi-continuity is sufficient for many purposes. Starting from a simple constructive approach for countable domains and combining this with basic measure theory, we obtain necessary and sufficient conditions for the existence of upper semi-continuous utility functions on a wide class of domains. Although links between utility theory and measure theory have been pointed out before, to the best of our knowledge this is the first time that the present route has been taken.preferences; utility theory; measure theory; outer measure

Language, meaning and games: a model of communication, coordination and evolution

Language is arguably a powerful coordination device in real-life interactions. We here develop a game-theoretic model of two-sided pre-play communication that generalizes the cheap-talk approach by way of introducing a meaning correspondence between messages and actions, and postulating two axioms met by natural languages. Deviations from this correspondence are called dishonest and players have a lexicographic preference for honesty, second to material payoffs. The model is first applied to finite and symmetric two-player games and we establish that, in generic and symmetric n x n -coordination games, a Nash equilibrium component in such a lexicographic communication game is evolutionarily stable if and only if it results in the unique Pareto efficient outcome of the underlying game. We discus Aumann’s (1990) example of a Pareto efficient equilibrium that is not self-enforcing. We also extend the approach to one-sided communication.Communication, coordination, language, honesty, evolutionary stability.

Uniqueness in Infinitely Repeated Decision Problems

Dynamic decision-making without commitment is usually modelled as a game between the current and future selves of the decision maker. It has been observed that if the time-horizon is infinite, then such games may have multiple subgame-perfect equilibrium solutions. We provide a sufficient condition for uniqueness in a class of such games, namely infinitely repeated decision problems with discounting. The condition is two-fold: the range of possible utility levels in the decision problem should be bounded from below, and the discount factor between successive periods should be non-decreasing over time, a condition met by exponential, quasi-exponential and hyperbolic discounting.  Game Theory; Time Preference; Hyperbolic Discounting; Repeated Decision Problems

Prices and quality signals

We consider a market-for-lemons model where the seller is a price setter, and, in addition to observing the price, the buyer receives a private noisy signal of the product's quality, such as when a prospective buyer looks at a car or house for sale, or when an employer interviews a job candidate. Sufficient conditions are given for the existence of perfect Bayesian equilibria, and we analyze equilibrium prices, trading probabilities and gains of trade. In particular, we identify separating equilibria with partial and full adverse selection as well as pooling equilibria. We also study the role of the buyer's signal precision, from being completely uninformative (as in standard adverse-selection models) to being completely informative (as under symmetric information). The robustness of results for these two boundary cases is analyzed, and comparisons are made with established models of monopoly and perfect competition.lemons; noisy quality signal; adverse selection

Punctuality - A Cultural Trait as Equilibrium

A people's culture, norms and habits are important determinants not just of the quality of social life but of economic progress and growth. In this paper we take the view that while the importance of culture is undeniable, the innateness of culture is not. We work here with a single example and demonstrate how a human trait which is widely believed to be cultural is at the same time a matter of choice. The example that we shall work with concerns punctuality. We show that punctuality may be simply an equilibrium response of individuals to what they expect others to do. The same society can get caught in a punctual equilibrium or a non-punctual equilibrium. Punctuality; Coordination Games

Deterministic Approximation of Stochastic Evolution in Games

This paper provides deterministic approximation results for stochastic processes that arise when finite populations recurrently play finite games. The deterministic approximation is defined in continuous time as a system of ordinary differential equations of the type studied in evolutionary game theory. We establish precise connections between the long-run behavior of the stochastic process, for large populations, and its deterministic approximation. In particular, we show that if the deterministic solution through the initial state of the stochastic process at some point in time enters a basin of attraction, then the stochastic process will enter any given neighborhood of that attractor within a finite and deterministic time with a probability that exponentially approaches one as the population size goes to infinity. The process will remain in this neighborhood for a random time that almost surely exceeds an exponential function of the population size. During this time interval, the process spends almost all time at a certain subset of the attractor, its so-called Birkhoff center. We sharpen this result in the special case of ergodic processes.  Game Theory; Evolution; Approximation

Discounting and Future Selves

Is discounting of future instantaneous utilities consistent with altruism towards future selves? More precisely, can temporal preferences, expressed as a sum of discounted instantaneous utilities, be derived from a representation in the form of a sum of discounted total utilities? We find that a representation in the quasi-exponential (ß,d)-form in Phelps and Pollak (1968) and Laibson (1997) correspond to quasi-exponential altruism towards one's future selves. For ß=1/2, these welfare weights are exponential, while for ß 1/2 in favor of one's future selves. More generally, we establish a functional equation which relates welfare weights to instantaneous-utility weights and apply this equation to a number of examples. We also postulate five desiderata for instantaneous-utility discounting. None of the usual discount functions satisfy all desiderata, but we propose a simple class of discount functions which does.Altruism; Discounting; Dynamic Inconsistency; Welfare 

Welfare Foundations of Discounting

We investigate whether temporal preferences expressed as a sum of discounted consumption utilities can be derived from a welfare representation in the form of a sum of discounted total utilities. We find that a consumption-based representation in the usual exponential form corresponds to one-period "altruism" towards one's future selves: the current self gives positive weight to her total utility in the next period, and weight zero to her total utility in all subsequent periods. We also find that a consumption-based representation in the quasi-exponential (ß,d)-form suggested by Phelps and Pollak (1968) and Laibson (1997) correspond to quasi-exponential altruism towards one's future selves. For ß=1/2, the welfare weights are exponential, while for ß 1/2 in favor of one's future selves. More generally, we establish a functional equation which relates welfare weights to consumption-utility weights. We also postulate five desiderata for consumption-utility weights. None of the usual formalizations satisfy all desiderata, but we propose a simple formalization which does.Altruism; Discounting; Dynamic Inconsistency; Time Inconsistency; Welfare