Continuum Of Agents Research Articles

Equilibria in abstract economies with a continuum of agents with discontinuous and non-ordered preferences

This paper focus on the problem of the existence of an equilibrium in abstract economies and exchange economies. Spanning over the literature we have managed to extend and generalize some previous results. In particular, we generalize the main theorem of Yannelis (1987) on the existence of an equilibrium in an abstract economy with a continuum of agents, by allowing for discontinuous preferences. As a corollary of this result, we extend the finite agent Cournot–Nash equilibrium existence theorems with discontinuous preferences (e.g., Reny, 1999; Bareli and Meneghel, 2013; He and Yannelis, 2016; among others), to a continuum of agents. We also obtain an existence theorem for an abstract economy which allows for a convexifying effect on aggregation and nonconvex strategy and constraint sets. Furthermore, our new main theorem is used to prove the existence of a Walrasian equilibrium with a continuum of agents with discontinuous, non-ordered, interdependent and price-dependent preferences and thus extending the results of Aumman (1966) and Schmeidler (1969).

Stability in Large Markets

Abstract In matching models, pairwise-stable outcomes do not generally exist without substantial restrictions on both preferences and the topology of the network of contracts. We address the foundations of matching markets by developing a matching model with a continuum of agents that allows for arbitrary preferences and network structures. We show that pairwise-stable outcomes are guaranteed to exist. When agents can interact with multiple other counterparties, pairwise stability is too weak of a solution concept, and we argue that a refinement of it called tree stability is the most appropriate solution concept in this setting. Our main results show that tree-stable outcomes exist for arbitrary preferences and network topologies.

When (not) to publicize inspection results

We consider a dynamic inspection problem between a principal and several agents. The principal observes the full inspection history, whereas each agent only observes inspections imposed on himself. When inspection resources are limited, the inspection intensities for agents are negatively correlated, and hence each agent cares not only about his own inspection history, but also about the inspection histories of the other agents. In such cases, should the principal publicly reveal past inspection history, or should she conceal this information? We show that the principal benefits from concealing inspection history. Nevertheless, this benefit becomes less significant as the number of agents increases, and disappears in the limit case with a continuum of agents.

The Weak Hybrid Equilibria of an Exchange Economy with a Continuum of Agents and Externalities

Abstract Inspired by Zhao, J. 1996. “The Hybrid Equilibria and Core Selection in Exchange Economies with Externalities.” Journal of Mathematical Economics 26 (4): 387–407, Askoura, Y. 2011. “The Weak-Core of a Game in Normal Form with a Continuum of Players.” Journal of Mathematical Economics 47: 43–7, Askoura, Y. 2017. “On the Core of Normal Form Games with a Continuum of Players.” Mathematical Social Sciences 89: 32–42, Yang, Z. 2020. “The Weak α-core of Exchange Economies with a Continuum of Players and Pseudo-utilities.” Journal of Mathematical Economies 91: 43–50 and Yang, Z., and X. Zhang. 2021. “A Weak α-core Existence Theorem of Games with Nonordered Preferences and a Continuum of Agents.” Journal of Mathematical Economics 94: 102464, we establish an exchange economy with externalities and a continuum of agents. We define the weak hybrid equilibrium in this model and prove the existence theorem under the regular conditions. Furthermore, we analyze the relation between the set of hybrid equilibria and the set of competitive equilibria in an exchange economy without externalities and with a continuum of agents.

On rank dominance of tie‐breaking rules

Lotteries are a common way to resolve ties in assignment mechanisms that ration resources. We consider a model with a continuum of agents and a finite set of resources with heterogeneous qualities, where the agents' preferences are generated from a multinomial‐logit (MNL) model based on the resource qualities. We show that all agents prefer a common lottery to independent lotteries at each resource if every resource is popular, meaning that the mass of agents ranking that resource as their first choice exceeds its capacity. We then prove a stronger result where the assumption that every resource is popular is not required and agents' preferences are drawn from a (more general) nested MNL model. By appropriately adapting the notion of popularity to resource types, we show that a hybrid tie‐breaking rule in which the objects in each popular type share a common lottery dominates independent lotteries at each resource.

A social coalitional weak equilibrium existence theorem with a continuum of agents and applications

In this paper, we establish a social coalitional weak equilibrium existence theorem with a continuum of agents, which is an infinite-player extension of the fully cooperative version of Ichiishi (1981). As a unified approach, we obtain the weak α-core existence theorems of normal-form games and exchange economies with a continuum of agents by the social coalitional weak equilibrium existence theorem.

Incomplete‐information games in large populations with anonymity

The paper provides theoretical foundations for models of strategic interdependence under uncertainty that have a continuum of agents and a decomposition of uncertainty into a macro component and an agent‐specific micro component, with a law of large numbers for the latter. This macro–micro decomposition of uncertainty is implied by a condition of exchangeability of agents' types, which holds at the level of the prior if and only if it also holds at the level of agents' beliefs, i.e., posteriors. Under an additional condition of anonymity in payoffs, agents' behaviors are fully determined by their beliefs about the cross‐section distribution of types and other macro variables, and by their beliefs about the cross‐section distribution of other agents' strategies. Any probability distribution over cross‐section distributions of types and other macro variables is compatible with a fully specified belief system, but not every function from types to such probability distributions is compatible with a common prior. The paper gives necessary and sufficient conditions for compatibility of such a function with a common prior.

Public-Good Provision with Macro Uncertainty about Preferences: Efficiency, Budget Balance, and Robustness

The paper studies efficient public-good provision in a model with private values whose distribution depends on a macro shock; conditionally on this shock, values are independent and identically distributed. A generalization of the Bayesian mechanism of d'Aspremont and Gerard-Varet is shown to implement an efficient provision rule with budget balance. However, first-best implementation and budget balance are incompatible with a reqruirement of weak robustness whereby incentive compatibility of the mechanism is independent of the stochastic specification within the class of specifications defined by the structure of the model. Budget imbalances with robust implementation are small if there are many participants, as surplus from the Clarke-Groves mechanism converges to zero in probability when the number of participants becomes large. In the limit, with a continuum of agents, a first-best provision rule with equal cost sharing is robustly incentive-compatible. In this limit, information about the macro shock, which is the only thing that matters for public-good provision, can be elicited without any efficiency loss.

Dynamic assignment without money: Optimality of spot mechanisms

We study a large market model of dynamic matching with no monetary transfers and a continuum of agents. Time is discrete and horizon finite. Agents are in the market from the first date and, at each date, have to be assigned items (or bundles of items). When the social planner can only elicit ordinal preferences of agents over the sequences of items, we prove that, under a mild regularity assumption, incentive compatible and ordinally efficient allocation rules coincide with spot mechanisms. A spot mechanism specifies “virtual prices” for items at each date and, at the beginning of time, for each agent, randomly selects a budget of virtual money according to a (potentially non-uniform) distribution over [0,1]. Then, at each date, the agent is allocated the item of his choice among the affordable ones. Spot mechanisms impose a linear structure on prices and, perhaps surprisingly, our result shows that this linear structure is what is needed when one requires incentive compatibility and ordinal efficiency. When the social planner can elicit cardinal preferences, we prove that, under a similar regularity assumption, incentive compatible and Pareto efficient mechanisms coincide with a class of mechanisms we call Spot Menu of Random Budgets mechanisms. These mechanisms are similar to spot mechanisms except that, at the beginning of the time, each agent must pick a distribution in a menu. This distribution is used to initially draw the agent's budget of virtual money.

A weak [formula omitted]-core existence theorem of games with nonordered preferences and a continuum of agents

Inspired by Kajii (1992) and Askoura (2011, 2017), we introduce the notion of the weak α-core for games with nonordered preferences and a continuum of agents. First, we extend the work of Kajii (1992) to games with spaces of strategies defined on Hausdorff topological vector spaces. Furthermore, we prove the nonemptiness of the weak α-core. Finally, we establish the relations between normal-form games, games with nonordered preferences and games with pseudo-utilities.

On the Core, Bargaining Set and the Set of Competitive Allocations of Fuzzy Exchange Economy with a Continuum of Agents

In this paper, we present a new model closer to the real-life — called the fuzzy exchange economy with a continuum of agents (FXE-CA) — that combines fuzzy consumption and fuzzy initial endowment with the agent’s fuzzy preference in the fuzzy consumption set. To characterize the fuzzy competitive allocations of the FXE-CA, we define the indifference fuzzy core of a FXE-CA as the set of all fuzzy allocations that cannot be dominated by any coalition of agents. We also propose the Mas-Colell indifference fuzzy bargaining set, in which no coalition has a justified objection at a fuzzy allocation against any other coalition. Finally, we verify that the indifference fuzzy core and the indifference fuzzy bargaining set of a FXE-CA coincide with the set of all fuzzy competitive allocations under some conditions, respectively. This indicates that the agents unanimously distribute the fuzzy competitive allocations of a FXE-CA.

Efficiency in trading markets with multi-dimensional signals

There is a continuum of agents, each of whom trades a divisible asset via demand function competition. Individual valuations are determined by payoff shocks that are correlated across agents. Agents observe multi-dimensional signals about the payoff shocks; it is only assumed that the signals are normally and symmetrically distributed. We give three results about this economy. First, an equilibrium exists. Second, the equilibrium is constrained inefficient; a higher total surplus could be attained if agents submitted different demands. Third, a constrained-efficient outcome can be implemented by setting an appropriate capital-gains tax. The second result identifies a new type of inefficiency that only arises when agents observe multi-dimensional signals; the third result identifies the taxation policy that allows correcting this inefficiency.

Incomplete-Information Games in Large Populations with Anonymity

The paper provides theoretical foundations for models of strategic interdependence under uncertainty that have a continuum of agents and a decomposition of uncertainty into a macro component and an agent-specific micro component, with a law of large numbers for the latter. This macro-micro decomposition of uncertainty is implied by a condition of exchangeability of agents' types, which holds at the level of the prior if and only if it also holds at the level of agents' beliefs, i.e., posteriors. Under an additional condition of anonymity in payoffs, agents' behaviours are fully determined by their macro beliefs about the cross-section distribution of types and other macro variables and about the cross-section distribution of other agents' strategies. Any probability distribution over cross-section distributions of types and other macro variables is compatible with a fully specified belief system, but not every macro belief function is compatible with a common prior. The paper gives necessary and sufficient conditions for compatibility of a macro belief function with a common prior.

Recursive equilibrium in Krusell and Smith (1998)

This paper uses the tools developed in the literature on dynamically incomplete markets with finite agents to study the large economy with a continuum of agents and both aggregate and idiosyncratic shocks in Krusell and Smith (1998). It establishes the existence of sequential competitive equilibrium, generalized recursive equilibrium, recursive equilibrium with an extended-state space, and characterizes several important properties of the equilibrium variables. The equilibrium process admits an ergodic measure, which enables the application of the ergodic theorem for simulating and calibrating the model. Without aggregate shocks, the existence and some characterization results carry over to economies with only idiosyncratic shocks such as Huggett (1997)'s economy. However, the existence of recursive equilibrium with the natural minimal state space in Krusell and Smith's economy remains elusive, as in finite-agent incomplete markets economies.

Deep Equilibrium Nets

We introduce deep equilibrium nets---a deep learning-based method to compute approximate functional rational expectations equilibria of economic models featuring a substantial amount of heterogeneity, significant uncertainty, and occasionally binding constraints. Deep equilibrium nets are neural networks that directly approximate all equilibrium functions and that are trained in an unsupervised fashion to satisfy all equilibrium conditions along simulated paths of the economy. Since the neural network approximates the equilibrium functions directly, simulating the economy is computationally cheap, and training data can be generated at virtually zero cost. We demonstrate that deep equilibrium nets can solve rich and economically relevant models accurately by applying them to solve three different models, all featuring a very high-dimensional state space. Specifically, we solve two overlapping generations models with aggregate and idiosyncratic uncertainty, illiquid capital, a one-period bond, and occasionally binding constraints. Additionally, we solve a Bewley-style model with a continuum of agents, aggregate and idiosyncratic risk, borrowing constraints, and recursive preferences.

Pricing of Platforms in Two-Sided Markets with Heterogeneous Agents and Limited Market Size

We study a two-sided market represented by network platforms and heterogeneous agents. Our setup departs from Armstrong (2006)’s monopoly model by assuming both (1) a continuum of agents of limited size on each side of the market and (2) heterogeneous utility of agents with Hotelling specification. We show that the monopoly’s optimal pricing strategy always results in a corner solution in terms of the equilibrium market share. We also solve for the social planner’s optimization problem and obtain a similar corner solution result. In addition, the exact values for the equilibrium in the case of duopoly for a two-sided market on two platforms are obtained.

Incomplete-Information Games in Large Populations with Anonymity

The paper provides theoretical foundations for models of strategic interdependence under uncertainty that have a continuum of agents and a decomposition of uncertainty into a macro component and an agent-specific micro component, with a law of large numbers for the latter. This macro-micro decomposition of uncertainty is implied by a condition of exchangeability of agents' types, which holds at the level of the prior if and only if it also holds at the level of agents' beliefs, i.e., posteriors. Under an additional condition of anonymity in payoffs, agents' behaviours are fully determined by their macro beliefs about the cross-section distribution of types and other macro variables and about the cross-section distribution of other agents' strategies. Any probability distribution over cross-section distributions of types and other macro variables is compatible with a fully specified belief system, but not every macro belief function is compatible with a common prior. The paper gives necessary and sufficient conditions for compatibility of a macro belief function with a common prior.

Disagreement, Information Quality and Asset Prices

We solve analytically a pure exchange general equilibrium model with a continuum of agents and two key elements: Reference-dependent preferences (in the form of habit formation) and stochastic disagreement (from fluctuating information quality). The model explains the equity premium, the stock price volatility, and the empirical relation between forecast dispersion and different properties of asset prices. We evaluate its quantitative implications and show that the usual asset pricing channels of disagreement in the literature are not quantitatively important, while information quality is as important as reference-dependent preferences.

Prices and Allocations in a Trading Market with Multi-Dimensional Signals

There is a continuum of agents, each of whom trades a divisible asset via demand function competition. Individual valuations are determined by multiple idiosyncratic and common shocks, regarding which the agent observes multi-dimensional signals. The only assumptions are that the signals are normally and symmetrically distributed. We introduce two measures to quantify the distance between an outcome of the demand function competition game and the first-best outcome: the allocative distortion and the price distortion. We say that an equilibrium is “price-quantity efficient” if no outcome of the demand function competition game generates lower allocative and price distortions. We give three results about this economy. First, an equilibrium exists. Second, we give “lax” sufficient conditions (that is, an agent learns about different payoff shocks from different signals) so that every Nash equilibrium is price-quantity efficient. Third, every price-quantity efficient outcome can be implemented by setting an appropriate capital gains tax; because this tax reduces agents’ incentives to speculate on the asset’s price, it decreases the allocative distortion but increases the price distortion.

Informationally Robust Comparative Statics in Incomplete Information Games

A continuum of agents play an incomplete information game with continuous action. An agent’s best-response function is linear on his expectation of a common payoff shock and the average action taken by the other agents. The interaction parameter is the weight that an agent places on the actions of the other agents. We give informationally robust predictions of the impact that a marginal change in the interaction parameter has on the agents’ equilibrium expected utility for any given equilibrium outcome. Our predictions are informationally robust in the sense that they hold across all information structures that are consistent with the observed equilibrium outcome. We express our informationally robust predictions solely in terms of the equilibrium outcome; this gives a “Bayes-correlated-equilibrium approach” to comparative statics (cf. Bergemann and Morris (2013); Bergemann and Morris (2016)). The results are applied to give robust predictions on the impact that a marginal change in the interaction parameter has on any other statistics of the equilibrium outcome (e.g., other measures of total welfare or the actions’ volatility). We use our results to study a competitive market with dispersed information (cf. Guesnerie (1992); Vives (1993)); we give robust predictions of the impact that a marginal change in the elasticity of the demand has on the firms’ profits. We also use our results to study economies that are inefficient when agents have dispersed information (cf. Morris and Shin (2002); Angeletos and Pavan (2007)); we give robust predictions of the impact that a marginal change in a tax rate has on the total welfare.