Kam-Chau Wong

Concave utility on finite sets

Abstract When does a preference relation on a finite set have a concave or a strictly concave utility function? We provide a complete answer. Our proof is an application of the Theorem of the Alternative, and constructs a concave utility if one exists.

Network formation and stable equilibrium

Abstract We study the non-cooperative formation of a (directed) network that builds upon one-way immediate links. Our novel feature is that we require circular paths of links for productive activities. This captures more clearly the higher level of coordination that is often required for network formations than models such as those of Jackson and Wolinsky [A strategic model of social and economic networks, J. Econ. Theory 71 (1996) 44–74] and Bala and Goyal [A non-cooperative model of network formation, Econometrica 68 (2000) 1181–1229], in that in our model a single agents unilateral action is limited in affecting the network structure. We completely characterize the class of Nash equilibria by showing that the minimality of a graph is both necessary and sufficient (Theorem 1). We also propose the concept of a stable network that is similar to the pairwise stability of Jackson and Wolinsky (1996). We show that stable equilibria are a wheel, a trivial network, or a sub-wheel partitioned network (Theorem 2). We modify the Bala and Goyal (2000) dynamic process with inertia, and obtain the convergence to stable equilibria regardless of the initial state (Theorem 3).

Revealed homothetic preference and technology

Abstract Using the revealed preference approach (Samuelson, P.A. (1938): A Note on the pure theory of consumers behavior, Economica, 5, 61–71; Houthakker, H. (1950): Revealed preference and the utility function, Economica, 17, 159–174), we obtain a new axiom that completely characterizes the consumer demand behavior generated by homothetic preferences. This axiom provides a unified basis for sharpening previous theorems on homothetic rationalizations: it relaxes Chipmans (Chipman, J.S. (1974): Homothetic preferences and aggregation, Journal of Economic Theory, 8, 26–38) differentiability assumption on demand functions, and strengthens Varians (Varian, H.R. (1983): Non-parametric tests of consumer behavior, Review of Economic Studies, 50: 99–110) rationalization notion into a stronger one (cf. Richter, M.K. (1966): Revealed preference theory, Econometrica, 34, 635–645). Our approach is also useful for homothetic production analysis; for example, it allows us to sharpen Varians (Varian, H.R. (1984): The non-parametric approach to production analysis, Econometrica, 52, 579–597) (non-parametric) theorem in several directions (stronger rationalization notion, fewer attributes of observations).

Computable preference and utility

Abstract We introduce computability definitions for both preference orderings and utility functions, on both Euclidean spaces and computable real spaces. We prove that computable preferences have computable utility functions. Computable preferences are interesting for three reasons: (i) they model one aspect of bounded rationality; (ii) they tell us when computational economics is practical; and (iii) they provide a framework for developing complexity classifications of economic theories. Our proof uses a new utility-construction algorithm, extending the methods of Debreu and Bridges in several directions.

Long-run equilibria with dominated strategies

We show that the predictions of long-run behavior under the evolutionary process are highly sensitive to the addition and elimination of strictly dominated strategies. In particular, for the best response dynamics [cf. Kandori, M., Rob, R., 1995. Evolution of equilibria in the long run: A general theory and applications. J. Econ. Theory 65 (2), 383-414] we prove that for any symmetric normal form game, any strict Nash equilibrium can be selected as the unique long-run equilibrium by appropriately adding only one single strategy which is strictly dominated by all original strategies. Moreover, if we further assume instantaneous adjustment, then any convex combination of strict Nash equilibria with rational number weights can be realized as the long-run distribution by appropriately adding strictly dominated strategies.

A fixed point theorem for o-minimal structures

We prove a definable analogue to Brouwers Fixed Point Theorem for o-minimal structures of real closed field expansions: A continuous definable function mapping from the unit simplex into itself admits a fixed point, even though the underlying space is not necessarily topologically complete. Our proof is direct and elementary; it uses a triangulation technique for o-minimal functions, with an application of Sperners Lemma. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)