Ratul Lahkar

The dynamic instability of dispersed price equilibria

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.

The logit dynamic for games with continuous strategy sets

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.

Reinforcement learning in population games

We study reinforcement learning in a population game. Agents in a population game revise mixed strategies using the Cross rule of reinforcement learning. The population state—the probability distribution over the set of mixed strategies—evolves according to the replicator continuity equation which, in its simplest form, is a partial differential equation. The replicator dynamic is a special case in which the initial population state is homogeneous, i.e. when all agents use the same mixed strategy. We apply the continuity dynamic to various classes of symmetric games. Using 3×3 coordination games, we show that equilibrium selection depends on the variance of the initial strategy distribution, or initial population heterogeneity. We give an example of a 2×2 game in which heterogeneity persists even as the mean population state converges to a mixed equilibrium. Finally, we apply the dynamic to negative definite and doubly symmetric games.

The Continuous Logit Dynamic and Price Dispersion

We define the logit dynamic for games with continuous strategy spaces and establish its fundamental properties , i.e. the existence, uniqueness and continuity of solutions. We apply the dynamic to the analysis of the Burdett and Judd (1983) model of price dispersion. Our objective is to assess the stability of the logit equilibrium corresponding to the unique Nash equilibrium of this model. Although a direct analysis of local stability is difficult due to technical difficulties, an appeal to finite approximation techniques suggest that the logit equilibrium is unstable. Price dispersion, instead of being an equilibrium phenomenon, is a cyclical phenomenon. We also establish a result on the Lyapunov stability of logit equilibria in negative definite games.

Large Population Aggregative Potential Games

We consider population games in which payoff depends upon the aggregate strategy level and which admit a potential function. Examples of such aggregative potential games include the tragedy of the commons and the Cournot competition model. These games are technically simple as they can be analyzed using a one-dimensional variant of the potential function. We use such games to model the presence of externalities, both positive and negative. We characterize Nash equilibria in such games as socially inefficient. Evolutionary dynamics in such games converge to socially inefficient Nash equilibria.

The dynamics of generalized reinforcement learning

We consider reinforcement learning in games with both positive and negative payoffs. The Cross rule is the prototypical reinforcement learning rule in games that have only positive payoffs. We extend this rule to incorporate negative payoffs to obtain the generalized reinforcement learning rule. Applying this rule to a population game, we obtain the generalized reinforcement dynamic which describes the evolution of mixed strategies in the population. We apply the dynamic to the class of Rock–Scissor–Paper (RSP) games to establish local convergence to the interior rest point in all such games, including the bad RSP game.

Risk diversification through multiple group membership in microfinance

We consider group formation in the joint liability setting in microfinance. Joint liability imposes additional liability of having to repay for group partners should they fail to repay. Multiple group membership allows diversification of that risk, and therefore, is welfare enhancing for risk averse agents. Welfare enhancement occurs even when the total loan of an agent is unchanged. Therefore, multiple borrowing is not synonymous with over-borrowing.

Does multiple borrowing necessarily mean over-borrowing?

Using a survey dataset collected from Andhra Pradesh, India, we test if multiple borrowing is equivalent to over-borrowing. Results suggest that over-borrowing and multiple loans are not necessarily synonymous. As the number of credit agencies in a village increases, the average loan burden of villagers does not increase. We also find evidence of substitution of formal sources of credit for informal ones with increased presence of formal credit institutions. Such substitution is greater with addition of microcredit institutions than with other formal lending agencies. Our results indicate that joint liability setup seems to ensure that individuals at a greater risk of non-repayment are discouraged from obtaining microcredit.

Evolution in Bayesian Supermodular Population Games

We study evolution in Bayesian supermodular population games. We define such games in the context of large populations of agents and establish the existence of a maximum and a minimum pure strategy Nash equilibria which are monotone in types. To study evolution in such games, we introduce Bayesian perturbed best response dynamics and the corresponding aggregate perturbed best response dynamic. Using the theory of cooperative dynamical systems, we show that solution trajectories of the aggregate perturbed best response dynamic converge to the set of perturbed equilibrium distributions. Using results from Ely and Sandholm (2005), we then conclude that the L1 solution trajectories of the Bayesian perturbed best response dynamic also converge to the set of Bayesian perturbed equilibria.

Evolutionary game theory: an exposition

Purpose - This article aims to provide an exposition of evolutionary game theory which can be used for pedagogical purposes. Design/methodology/approach - The exposition is presented as a mathematical model in order to cover the formal underpinnings of evolutionary game theory. The paper aims to illustrate the theory using some simple examples. Findings - The paper discusses population games and describes the notion of revision protocols that agents use to change strategies. As an example of an evolutionary dynamic, the paper discusses the replicator dynamic in detail. It shows convergence of this dynamic to Nash equilibrium in simple 2 strategy games. The paper then applies this dynamic to a particular class of 3 strategy games to establish the possibility on cyclical behavior around a Nash equilibrium. Originality/value - The paper can serve as an educational briefing for students and researchers who are new to the field of evolutionary game theory.