Ratul Lahkar

Abstract: The projection dynamic is an evolutionary dynamic for population games. It is derived from a model of individual choice in which agents abandon their current strategies at rates inversely proportional to the strategies' current levels of use. The dynamic admits a simple geometric definition, its rest points coincide with the Nash equilibria of the underlying game, and it converges globally to Nash equilibrium in potential games and in stable games.

Abstract: We investigate a variety of connections between the projection dynamic and the replicator dynamic. At interior population states, the standard microfoundations for the replicator dynamic can be converted into foundations for the projection dynamic by replacing imitation of opponents with “revision driven by insecurity” and direct choice of alternative strategies. Both dynamics satisfy a condition called inflow–outflow symmetry, which causes them to select against strictly dominated strategies at interior states; still, because it is discontinuous at the boundary of the state space, the projection dynamic allows strictly dominated strategies to survive in perpetuity. The two dynamics exhibit qualitatively similar behavior in strictly stable and null stable games. Finally, the projection and replicator dynamics both can be viewed as gradient systems in potential games, the latter after an appropriate transformation of the state space.

Abstract: We adopt an evolutionary framework to explain price dispersion as a time varying phenomenon. By developing a finite strategy analogue of the Burdett and Judd (1983) price dispersion model, we show that all dispersed price equilibria are unstable under the class of perturbed best response dynamics. Instead, numerical simulations using the logit dynamic show that price dispersion manifests itself as a limit cycle. We verify that limit cycles persist even when the finite strategy model approaches the original continuous strategy model. For a particularly simple case of the model, we prove the existence of a limit cycle.

Abstract: We define the logit dynamic for games with continuous strategy sets and establish its fundamental properties, namely, the existence of a logit equilibrium, its convergence to a Nash equilibrium as the perturbation factor becomes small, and existence, uniqueness and continuity of solution trajectories. We apply the dynamic to the analysis of potential games and negative semidefinite games. We show that in a restricted state space of probability measures with bounded density functions, solution trajectories of the logit dynamic converge to logit equilibria in these two classes of games.

Abstract: The expansion of microfinance has triggered concerns of rising indebtedness, and higher default and interest rates. Using a screening model, we show that even if interest and default rates increase due to expansion, borrower welfare may improve. This is because: (i) all borrowers previously denied credit can obtain loans, and (ii) screening costs for pre-existing borrowers go down. Hence, policies that seek to regulate interest and screening levels can be counterproductive.

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Abstract: In the Hamadryas baboon, males are substantially larger than females. A troop of baboons is subdivided into a number of ‘one-male groups’, consisting of one adult male and one or more females with their young. The male prevents any of ‘his’ females from moving too far from him. Kummer (1971) performed the following experiment. Two males, A and B, previously unknown to each other, were placed in a large enclosure. Male A was free to move about the enclosure, but male B was shut in a small cage, from which he could observe A but not interfere. A female, unknown to both males, was then placed in the enclosure. Within 20 minutes male A had persuaded the female to accept his ownership. Male B was then released into the open enclosure. Instead of challenging male A , B avoided any contact, accepting A’s ownership.

Abstract: This Review presents basic facts regarding the long-run evolution of income and wealth inequality in Europe and the United States. Income and wealth inequality was very high a century ago, particularly in Europe, but dropped dramatically in the first half of the 20th century. Income inequality has surged back in the United States since the 1970s so that the United States is much more unequal than Europe today. We discuss possible interpretations and lessons for the future.

Abstract: We study a class of population games called stable games . These games are characterized by self-defeating externalities : when agents revise their strategies, the improvements in the payoffs of strategies to which revising agents are switching are always exceeded by the improvements in the payoffs of strategies which revising agents are abandoning. We prove that the set of Nash equilibria of a stable game is globally asymptotically stable under a wide range of evolutionary dynamics. Convergence results for stable games are not as general as those for potential games: in addition to monotonicity of the dynamics, integrability of the agents' revision protocols plays a key role.