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We consider convergence to Walrasian equilibrium in a situation where firms know only market price and their own cost function. We term this a situation of minimal information. We model the problem as a large population game of Cournot competition. The Nash equilibrium of this model is identical to the Walrasian equilibrium. We apply the best response (BR) dynamic as our main evolutionary model. This dynamic can be applied under minimal information as firms need to know only the market price and the their own cost to compute payoffs. We show that the BR dynamic converges globally to Nash equilibrium in an aggregative game like the Cournot model. Hence, it converges globally to the Walrasian equilibrium under minimal information. We extend the result to some other evolutionary dynamics using the method of potential games.

We compare equal treatment and affirmative action policies in Tullock contests. Equal treatment means that agents who exert equal effort have an equal probability of success. In affirmative action, agents who incur an equal cost of effort have an equal probability of success. Finite player contests with non-linearities in impact and cost functions cannot be solved in closed form. Instead, we approximate them with large population contests with measure zero agents. Affirmative action reduces aggregate effort in such contests, which can be solved. However, it ensures equality without any significant loss of aggregate welfare. We verify these findings for finite player contests through numerical simulations. For a sufficiently large number of players, the numerical simulations support the results of the large population analysis.

Tullock contests model rent seeking behavior where agents exert unproductive effort to probabilistically win a fixed prize. Rent dissipation measures the social loss involved in effort exertion in such contests. Tullock contests are characterized by an impact function, which measures how effort impacts success, and a cost of effort function. If these functions are asymmetric and non-linear, then the contest cannot be solved in closed form. Hence, we approximate such contests with a large population contest, for which Nash equilibria and rent dissipation can be explicitly calculated. Rent dissipation is then the ratio of the effort elasticity of impact to the effort elasticity of cost. Greater elasticity of impact incentivizes more exertion of unproductive effort generating higher social loss.

We consider dominant strategy implementation in a large population aggregative game. The model has strategic complementarities which generates multiple Nash equilibria. Moreover, externalities are positive due to which, all equilibria are socially inefficient. The planner, therefore, constructs a direct mechanism and assigns efficient strategies and transfer levels to agents. Truthful revelation then becomes strictly dominant, which implements efficiency. In our new evolutionary approach to this mechanism, the reported type distribution evolves under dynamics satisfying monotone percentage growth. Such dynamics eliminate dominated strategies thereby ensuring convergence to truthful revelation by all agents. Dominant strategy implementation is, therefore, robust under such evolutionary dynamics. Our evolutionary approach differs from existing models of evolutionary implementation based on potential games. That approach may fail to implement efficiency under strategic complementarities as a Pareto inferior Nash equilibrium can remain asymptotically stable under evolutionary dynamics. Our evolutionary approach is effective even under such strategic complementarities.

Abstract We consider Tullock contests where contestants can be divided into a finite set of types according to their strategy cost function. Solving such contests is intractable if the number of players is finite but large and there are nonlinearities and asymmetries present. But by approximating the finite player contest with a large population model that can be solved in closed form, we can approximate equilibrium behavior in the finite player model. We then characterize the optimal bias parameters of the large population contest and interpret them as approximations of optimal bias parameters in finite player contests. We also identify conditions under which those parameters are increasing or decreasing according to the cost parameters. The parameters are biased in favor of high-cost agents if the cost functions are strictly convex and the likelihood of success is sufficiently responsive to strategy.

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.

We consider the implementation of efficiency with minimum inequality in a large population model of negative externalities. Formally, the model is one of tragedy of the commons with the aggregate strategy at the efficient state being lower than at the Nash equilibrium. A planner can restore efficiency by imposing an externality equivalent tax and then redistribute the tax revenue as transfers to lower inequality. We characterize the transfer vector that minimizes inequality at the efficient state subject to incentive compatibility and budget balance. We then construct a mechanism that implements efficiency with minimum inequality in dominant strategies. We also show that minimizing inequality at the efficient state maximizes the minimum payoff at efficiency. But it is not equivalent to implementing the Rawlsian social choice function.

Evolutionary implementation is a standard method of implementation in large population games. Such implementation may, however, be ineffective in certain situations. We consider one such situation where strategic complementarities generate multiple Nash equilibria. The planner constructs an externality adjusted game by adding the positive externalities in the game to the original payoffs. However, strategic complementarities render the Pareto inferior Nash equilibrium evolutionarily stable. The society, therefore, fails to converge to the efficient state of the model leading to the failure of evolutionary implementation. We provide a new solution to this problem of implementation in large population games with multiple equilibria using dominant strategy implementation. Our main result is that the efficient state can be implemented in strictly dominant strategy by applying Pigouvian pricing calculated on the basis of the distribution of reported types.

Due to externalities, the equilibrium behavior in aggregative games is not efficient in the sense of maximizing aggregate payoff. We characterize conditions such that efficiency can be globally implemented in such games under evolutionary dynamics. If payoffs satisfy certain important concavity conditions, then the aggregate payoff function of these games has a unique maximizer. Once the planner imposes a transfer equal to the externality generated by agents, we obtain a new externality adjusted game. This is a potential game with the aggregate payoff function of the original game being its potential function. Evolutionary dynamics converge globally to the maximizer of this potential function, thereby implementing efficiency in the original game. Our earlier paper on public goods (Lahkar and Mukherjee [16]) emerges as an example of the present general analysis. Two new applications are public bads and the tragedy of the commons.

We consider implementation of the efficient state in a large population public goods game. Agents are divided into a finite set of types. The planner asks agents to report types, which generates a reported type distribution. Based on reported types and distribution, the planner calculates the efficient strategy level and a Pigouvian transfer for each type of agent. We show that this direct mechanism satisfies incentive compatibility in strictly dominant strategies, strong budget balance and ex–post individual rationality.