Showing papers in "Theoretical Economics in 2010"

Strategic Experimentation with Poisson Bandits

Abstract: We study a game of strategic experimentation with two-armed bandits where the risky arm distributes lump-sum payoffs according to a Poisson process. Its intensity is either high or low, and unknown to the players. We consider Markov perfect equilibria with beliefs as the state variable and show that all such equilibria exhibit an “encouragement effect” relative to the single-agent optimum. There is no equilibrium in which all players use cutoff strategies. Owing to the encouragement effect, asymmetric equilibria in which players take turns playing the risky arm before all experimentation stops Pareto dominate the unique symmetric equilibrium. Rewarding the last experimenter with a higher continuation value increases the range of beliefs where players experiment, but may reduce the intensity of experimentation at more optimistic beliefs. This suggests that there is no equilibrium that uniformly maximizes the players’ average payoff.

A theory of regular Markov perfect equilibria in dynamic stochastic games: Genericity, stability, and purification

Abstract: This paper studies generic properties of Markov perfect equilibria in dynamic stochastic games. We show that almost all dynamic stochastic games have a finite number of locally isolated Markov perfect equilibria. These equilibria are essential and strongly stable. Moreover, they all admit purification. To establish these results, we introduce a notion of regularity for dynamic stochastic games and exploit a simple connection between normal form and dynamic stochastic games.

Efficiency and marginal cost pricing in dynamic competitive markets with friction

Abstract: This paper examines a dynamic general equilibrium model with supply friction. With or without friction, the competitive equilibrium is efficient. Without friction, the market price is completely determined by the marginal production cost. If friction is present, no matter how small, then the market price fluctuates between zero and the "choke-up" price, without any tendency to converge to the marginal production cost, exhibiting considerable volatility. The distribution of the gains from trading in an efficient allocation may be skewed in favor of the supplier, although every player in the market is a price taker.

Local stability under evolutionary game dynamics

Abstract: We prove that any regular evolutionarily stable strategy (ESS) is asymptotically stable under any impartial pairwise comparison dynamic, including the Smith dynamic; under any separable excess payoff dynamic, including the BNN dynamic; and under the best response dynamic. Combined with existing results for imitative dynamics, our analysis validates the use of regular ESS as a blanket sufficient condition for local stability under evolutionary game dynamics.

Uphill self-control

Abstract: This paper extends the theory of temptation and self-control introduced by Gul and Pesendorfer (2001) to allow for increasing marginal costs of resisting temptation, that is, convex self-control costs. It also proves a representation theorem that admits a general class of self-control cost functions. Both models maintain the Order, Continuity, and Set Betweenness axioms but violate Independence.

Competing auctions: Finite markets and convergence

Abstract: The literature on competing auctions offers a model where sellers compete for buyers by setting reserve prices. An outstanding conjecture (e.g., Peters and Severinov 1997) is that the sellers post prices close to their marginal costs when the market becomes large. This conjecture is confirmed in this paper: we show that if all sellers have zero costs, then the equilibrium reserve price converges to 0 in distribution. Under further conditions there is a symmetric pure strategy equilibrium. In this equilibrium, if the ratio of buyers to sellers increases, then the equilibrium reserve price increases, and the reserve price is decreasing in the size of the market. Convergence of reserve prices occurs at the fast rate of 1/n if the ratio of buyers to sellers is held constant.

Orders of limits for stationary distributions, stochastic dominance, and stochastic stability

Abstract: A population of agents recurrently plays a two-strategy population game. When an agent receives a revision opportunity, he chooses a new strategy using a noisy best response rule that satisfies mild regularity conditions; best response with mutations, logit choice, and probit choice are all permitted. We study the long run behavior of the resulting Markov process when the noise level η is small and the population size N is large. We obtain a precise characterization of the asymptotics of the stationary distributions μ N�η as η approaches zero and N approaches infinity, and we establish that these asymptotics are the same for either order of limits and for all simultaneous limits. In general, different noisy best response rules can generate different stochastically stable states. To obtain a robust selection result, we introduce a refinement of risk dominance called stochastic dominance, and we prove that coordination on a given strategy is stochastically stable under every noisy best response rule if and only if that strategy is stochastically dominant.

Symmetry of evidence without evidence of symmetry

Abstract: The de Finetti Theorem is a cornerstone of the Bayesian approach. Bernardo (1996, p. 5) writes that its “message is very clear: if a sequence of observations is judged to be exchangeable, then any subset of them must be regarded as a random sample from some model, and there exists a prior distribution on the parameter of such model, hence requiring a Bayesian approach.” We argue that although exchangeability, interpreted as symmetry of evidence, is a weak assumption, when combined with subjective expected utility theory, it also implies complete confidence that experiments are identical. When evidence is sparse and there is little evidence of symmetry, this implication of de Finetti’s hypotheses is not intuitive. This motivates our adoption of multiple-priors utility as the benchmark model of preference. We provide two alternative generalizations of the de Finetti Theorem for this framework. A model of updating is also provided.

Uniform Topologies on Types

Abstract: We study the robustness of interim correlated rationalizability to perturbations of higher-order beliefs. We introduce a new metric topology on the universal type space, called uniform-weak topology, under which two types are close if they have similar first-order beliefs, attach similar probabilities to other players having similar first-order beliefs, and so on, where the degree of similarity is uniform over the levels of the belief hierarchy. This topology generalizes the now classic notion of proximity to common knowledge based on common p-beliefs (Monderer and Samet 1989). We show that convergence in the uniform-weak topology implies convergence in the uniform-strategic topology (Dekel et al. 2006). Moreover, when the limit is a finite type, uniform-weak convergence is also a necessary condition for convergence in the strategic topology. Finally, we show that the set of finite types is nowhere dense under the uniform strategic topology. Thus, our results shed light on the connection between similarity of beliefs and similarity of behaviors in games.

Supermodular mechanism design

Abstract: This paper introduces a mechanism design approach that allows dealing with the multiple equilibrium problem, using mechanisms that are robust to bounded rationality. This approach is a tool for constructing supermodular mechanisms, i.e., mechanisms that induce games with strategic complementarities. In quasilinear environments, I prove that if a social choice function can be implemented by a mechanism that generates bounded strategic substitutes—as opposed to strategic complementarities—then this mechanism can be converted into a supermodular mechanism that implements the social choice function. If the social choice function also satisfies some efficiency criterion, then it admits a supermodular mechanism that balances the budget. Building on these results, I address the multiple equilibrium problem. I provide sufficient conditions for a social choice function to be implementable with a supermodular mechanism whose equilibria are contained in the smallest interval among all supermodular mechanisms. This is followed by conditions for supermodular implementability in unique equilibrium. Finally, I provide a revelation principle for supermodular implementation in environments with general preferences.

Nash implementation with little communication

Abstract: The paper considers the communication complexity (measured in bits or real numbers) of Nash implementation of social choice rules A key distinction is whether we restrict to the traditional one-stage mechanisms or allow multistage mechanisms For one-stage mechanisms, the paper shows that for a large and important subclass of monotonic choice rules—called intersection monotonic— the total message space size needed for one-stage Nash implementation is essentially the same as that needed for “verification” (with honest agents who are privately informed about their preferences) According to Segal (2007), the latter is the size of the space of minimally informative budget equilibria verifying the choice rule However, multistage mechanisms allow a drastic reduction in communication complexity Namely, for an important subclass of intersectionmonotonic choice rules (which includes rules based on coalitional blocking such as exact or approximate Pareto efficiency, stability, and envy-free allocations), we propose a two-stage Nash implementation mechanism in which at most 5 alternatives plus 4N log2 N bits are announced in any play Such two-stage mechanisms bring about an exponential reduction in the communication complexity of Nash implementation for discrete communication measured in bits or a reduction from

Capitalist investment and political liberalization

Abstract: We consider a simple political-economic model where capitalist investment is constrained by the government’s temptation to expropriate. Political liberalization can relax this constraint, increasing the government’s revenue, but also increasing the ruler’s political risks. We analyze the ruler’s optimal liberalization, where our measure of political liberalization is the probability of the ruler being replaced if he tried to expropriate private investments. Poorer endowments can support reputational equilibria with more investment, even without liberalization, so we find a resources curse, where larger resource endowments can decrease investment and reduce the ruler’s revenue. The ruler’s incentive to liberalize can be greatest with intermediate resource endowments. Strong liberalization becomes optimal in cases where capital investment yields approximately constant returns to scale. Adding independent revenue decreases optimal liberalization and investment. Mobility of productive factors that complement capital can increase incentives to liberalize, but equilibrium prices may adjust so that liberal and authoritarian regimes coexist.

Dynamic monopoly with relational incentives

Abstract: This paper studies the price-setting problem of a monopoly that in each time period has the option of failing to deliver its good after receiving payment. The monopoly may be induced to deliver the good if consumers expect that the monopoly will not deliver in the future if it does not deliver today. If the good is nondurable and consumers are anonymous, the monopoly's optimal strategy is to set a price equal to the static monopoly price each period if the discount factor is high enough, and otherwise to set the lowest price at which it can credibly promise to deliver the good. If the good is durable, we derive an intuitive lower bound on the monopoly's optimal profit for any discount factor and show that it converges to the optimal static monopoly profit as the discount factor converges to 1, in contrast to the Coase conjecture. We also show that rationing the good is never optimal for the monopoly if there is an efficient resale market and that the best equilibrium in which the monopoly always delivers involves a strictly decreasing price path that asymptotes to a level strictly above the ratio of the monopoly's marginal cost to the discount factor.