Conjectural Equilibrium Research Articles

Bilevel Optimization and Conjectural Equilibrium: Theoretical Results and Numerical Algorithms (An Invited Tutorial Paper)

We study a mixed oligopoly model, including a nonprivate firm that maximizes the convex combination of domestic social surplus and net profit, under the concepts of consistent conjectural variations (CCVE), Cournot–Nash, and perfect competition equilibria and analyze the behavior of these equilibria. We compare the results to develop an optimality criterion for a parameter, decided by the nonprivate firm, known as socialization level. To work with the CCVE, we need to define the concept of consistency; hence, we also consider a classic oligopoly, which only includes private firms that maximize net profit, and reformulated it as a non-cooperative game, named meta-game, where the players (the private firms) select a conjecture as their strategies; then, we prove that the Nash equilibrium for the meta-game implies the CCVE. After that, due to its similarities with the meta-game, we study the Tolls Optimization Problem (TOP). Taking advantage of the sensitivity analysis for convex quadratic optimization and the filled function method; then, we present an efficient algorithm to solve the TOP. Finally, we apply these results to a financial model in which a group of sectors buy, sell, or exchange a set of instruments; each sector faces some uncertainty when selecting its assets and liabilities to optimize its portfolio structure. The problem is modeled using the concepts of conjectural variations and meta-game and we present results concerning the CCVE.

Profitable Double Marginalization

When successive monopolies transact through noncooperative linear pricing, the resulting double markup decreases their joint profits relative to vertical integration. However, if there are downstream rivals (which are not double marginalized), the same noncooperative interaction often inadvertently raises their joint profits. Profit effects depend on how the well-understood harm from misaligned interests compares to the value of the resulting strategic effect. When profitable, vertical noncooperation incidentally approximates strategic delegation a la Bonanno & Vickers (1988), but avoids its credibility problem, suggesting an inability to bargain may be indirectly beneficial. The conjectural consistency concept helps to explain the disparate profit effects, and to synthesize the literature on strategic delegation and vertical control. The optimal way to distort a downstream firm's behavior is always to make it behave as if it has a conjecture, no matter the distortion mechanism. If upstream competitors do this in parallel, they induce a consistent conjectures equilibrium (CCE) -- or else a close analogue -- evincing a strong link between ordinary Nash games and the CCE.

Correlated-belief equilibrium

We introduce a new solution concept, called correlated-belief equilibrium. The difference to Nash equilibrium is that, while each player has correct marginal conjectures about each opponent, it is not necessarily the case that these marginal conjectures are independent. Then, we provide an epistemic foundation and we relate correlated-belief equilibrium with standard solution concepts, such as rationalizability, correlated equilibrium and conjectural equilibrium.

Analysis of Consistent Equilibria in a Mixed Duopoly

This paper examines a model of a mixed duopoly with conjectural variations equilibrium (CVE), in which one of the agents maximizes a convex combination of his/her net profit and domestic social surplus. The agents’ conjectures concern the price variations, which depend on their production output variations. Based on the already established existence and uniqueness results for the CVE (called theexterior equilibrium) for any set of feasible conjectures, the notion of interior equilibrium is introduced by developing a consistency criterion for the conjectures (referred to as influence coefficients), and the existence theorem for the interior equilibrium (understood as a CVE state withconsistent conjectures) is proven. When the convex combination coefficient tends to 1, thus transforming the model into the mixed duopoly in its extreme form, two trends are apparent. First, for the private company, the equilibrium with consistent conjectures becomes more proficient than the Cournot-Nash equilibrium. Second, there exists a (unique) value of the combination coefficient such that the private agent’s profit is the same in both of the above-mentioned equilibria, which makes subsidies to the producer or to consumers unnecessary.

A note on robust Nash equilibria with uncertainties

In this short note, we investigate the framework where agents or players have some uncertainties upon their payoffs or losses, the behavior (or the type, number or any other characteristics) of other players. More specifically, we introduce an extension of the concept of Nash equilibria that generalize different solution concepts called by their authors, and depending on the context, either as robust, ambiguous, partially specified or with uncertainty aversion. We provide a simple necessary and sufficient condition that guarantees its existence and we show that it is actually a selection of conjectural (or self-confirming) equilibria. We finally conclude by how this concept can and should be defined in games with partial monitoring in order to preserve existence properties.

Reciprocally opportunistic spectrum access

AbstractThe problem of opportunistic spectrum access in cognitive radio networks (CRNs) is considered in this paper. Especially, we discuss designing distributed learning protocols for non‐cooperative secondary users (SUs) to efficiently utilise the spectrum opportunities. It is well known that the SUs’ selfish behaviors will result in a network collapse. Each SU is thus enabled to form conjectures over how the competing SUs would respond to its decision‐makings and believes that they alter future spectrum access strategies in proportion to its own current strategy variation. The conjectural belief adapts in accordance with local observations. The interaction among SUs is modelled as a stochastic learning procedure, in which each autonomous SU behaves as an intelligent agent. In this way, all SUs can independently ‘learn’ the behaviors of their competitors, optimise the opportunistic spectrum access strategies and finally achieve the goal of reciprocity in CRNs. Based on the belief model, two learning mechanisms, that is, the best response learning algorithm and the gradient ascent learning algorithm, are proposed to stabilise the opportunistic CRNs. We also theoretically prove that the SUs’ stochastic behaviors and beliefs converge to the conjectural equilibrium under some conditions rising from the learning procedure. Numerical results are provided to evaluate the performance of the proposed mechanisms, and show that the achieved system performance gain outperforms the multi‐agent Q‐learning scheme and the Nash equilibrium scheme when there are numerous SUs in the CRN. Copyright © 2014 John Wiley & Sons, Ltd.

Conjecture-Based Load Balancing for Delay-Sensitive Users Without Message Exchanges

In this paper, we study how multiple users can balance their traffic loads to share common resources in an efficient and distributed manner, without message exchanges. Specifically, we study a deployment scenario where users deploy delay-sensitive applications over a wireless multipath network and aim to minimize their own expected delays. Since the performance of a user's load balancing strategy depends on the strategies that are deployed by other users, it becomes important that a user considers the multiuser coupling when making its own load balancing decisions. We model this multiuser interaction as a load balancing game (LBG) and show that users can converge to a ε-consistent conjectural equilibrium by building near-accurate beliefs about the remaining capacities on each path. Based on these beliefs, users can make load balancing decisions without explicitly knowing the actions of the other users. In such a conjecture-based LBG, we analytically show that, if a leader is elected to build beliefs about how the users' aggregate transmission strategies affect the remaining resources, then this leader can use this knowledge to shape its traffic such that the multiuser interaction can achieve an efficient allocation across paths. Even if no leader is present in the game, as long as the users follow a set of prescribed rules for building beliefs, they can reach efficient outcomes in a distributed manner. Importantly, the proposed distributed load balancing solution can be also applied to other multiuser communication and networking problems where message exchanges are prohibited (or prohibitively expensive in terms of delay or bandwidth), ranging from multichannel selection in wireless networks to relay assignment in multivehicle networks.

Rationalizable conjectural equilibrium: A framework for robust predictions

I introduce a new framework to study environments with both structural and strategic uncertainty, different from Harsanyi’s ( 1967–1968) “Bayesian games”, that allows a researcher to test the robustness of Nash predictions while maintaining certain desirable restrictions on players’ beliefs. The solution concept applied to this environment is rationalizable conjectural equilibrium (RCE), which integrates both learning from feedback (in the spirit of self-confirming equilibrium) and from introspection (in the spirit of rationalizability). I provide an epistemic definition of RCE and obtain a characterization in terms of a procedure that generalizes iterated deletion of strategies that are not a best response.

Expectational bottlenecks and the emerging of new organizational forms

In this article we discuss the dynamics of organizational change when agents have heterogeneous initial conjectures and do learn. In this framework, conjectural equilibrium is defined as a steady state of the learning process, and all the adjustment occurs in disequilibrium. We discuss the properties of the system under different “rationality” assumptions, and using well-known learning algorithms. We prove analytically that multiplicity of equilibria, and failure of good organizational routines, cannot be ruled out: better, they are fairly probable. Stability is a crucial matter: it is shown to depend on initial conjectures. Finally, learning does not necessarily select the best.

Scale economies, consistent conjectures and teams

This paper models the behavior of team members in a consistent conjectures equilibrium. When subject to scale economies, team members produce more than Nash and when subject to scale diseconomies, they produce less than Nash. Moreover, even when effort levels of team members are perfect substitutes in production, they can be strategic complements in the face of scale economies. Finally, with sufficient scale economies, the complementarity eliminates free-riding and the team optimum is obtained.

THE ASSIGNMENT GAME WITH NEGATIVE EXTERNALITIES AND BOUNDED RATIONALITY

We introduce negative externalities in the form of ill will among the players of the classic two-sided assignment game of Shapley and Shubik, by letting each player's utility be negatively correlated with the payoff of all the players in his group. The new game is very complex, but under a certain assumption of bounded rationality we derive a straightforward notion of stable outcomes as certain conjectural equilibria. We prove that several well-known properties of the set of stable outcomes in the assignment game carry over to this new game.

Linearly Coupled Communication Games

This paper discusses a special type of multi-user communication scenario, in which users' utilities are linearly impacted by their competitors' actions. First, we explicitly characterize the Nash equilibrium and Pareto boundary of the achievable utility region. Second, the price of anarchy incurred by the non-collaborative Nash strategy is quantified. Third, to improve the performance in the non-cooperative scenarios, we investigate the properties of an alternative solution concept named conjectural equilibrium, in which individual users compensate for their lack of information by forming internal beliefs about their competitors. The global convergence of the best response and Jacobi update dynamics that achieve various conjectural equilibria is analyzed. It is shown that the Pareto boundaries of the investigated linearly coupled games can be sustained as stable conjectural equilibria if the belief functions are properly initialized. The investigated models apply to a variety of realistic applications encountered in the multiple access design, including wireless random access and flow control.

Consistent Conjectural Variations Equilibrium in a Mixed Duopoly

In this paper, we consider a model of mixed duopoly with Conjectured Variations Equilibrium (CVE). The agents’ conjectures concern the price variations depending on the increase or decrease in their production outputs. We establish existence and uniqueness results for the conjectured variations equilibrium (called an exterior equilibrium) for any set of feasible conjectures. To introduce the notion of an interior equilibrium, we develop a consistency criterion for the conjectures (referred to as influence coefficients) and prove the existence theorem for the interior equilibrium (understood as a CVE with consistent conjectures). To prepare the base for the extension of our results to the case of non-differentiable demand functions, we also investigate the behavior of the consistent conjectures in dependence upon a parameter representing the demand function derivative with respect to the market price.

FROM IMPERFECT TO PERFECT COMPETITION: A PARAMETRIC APPROACH THROUGH CONJECTURAL VARIATIONS

This paper features perfect competition under a parametric approach. We consider conjectural variations in a pure exchange economy where strategic interactions hold. We show, through an example, that the conjectural general equilibrium can coincide with the competitive equilibrium, independently of a replication procedure or an asymptotic identification. Additionally, the competitive equilibrium is a locally consistent conjectural general equilibrium. Perfect competition is defined at equilibrium.

Optimal privatization in a mixed duopoly with consistent conjectures

We show that partially privatizing a public firm alters underlying conjectures, in turn, changing the optimal degree of privatization. The consistent conjectures equilibrium (CCE) generates substantially greater optimal privatization than does any conjecture shared between the firms including the standard Cournot–Nash equilibrium (CNE). Yet, when the private rival is foreign, the CCE generates substantially less privatization than the CNE. The optimal extent of privatization with a domestic rival exceeds that with a foreign rival in the CCE as well as in the CNE.

Dynamic conjectures in random access networks using bio-inspired learning

Inspired by the biological entities' ability to achieve reciprocity in the course of evolution, this paper considers a conjecture-based distributed learning approach that enables autonomous nodes to independently optimize their transmission probabilities in random access networks. We model the interaction among multiple self-interested nodes as a game. It is well-known that the Nash equilibria in this game result in zero throughput for all the nodes if they take myopic best-response, thereby leading to a network collapse. This paper enables nodes to behave as intelligent entities which can proactively gather information, form internal conjectures on how their competitors would react to their actions, and update their beliefs according to their local observations. In this way, nodes are capable to autonomously "learn" the behavior of their competitors, optimize their own actions, and eventually cultivate reciprocity in the random access network. To characterize the steady-state outcome of this "evolution", the conjectural equilibrium is introduced. Inspired by the biological phenomena of "derivative action" and "gradient dynamics", two distributed conjecture-based action update mechanisms are proposed to stabilize the random access network. The sufficient conditions that guarantee the proposed conjecture-based learning algorithms to converge are derived. Moreover, it is analytically shown that all the achievable operating points in the throughput region are stable conjectural equilibria corresponding to different conjectures. We also investigate how the conjectural equilibrium can be selected in heterogeneous networks and how the proposed methods can be extended to ad-hoc networks. Numerical simulations verify that the system performance significantly outperforms existing protocols, such as IEEE 802.11 Distributed Coordination Function (DCF) protocol and priority-based fair medium access control (P-MAC) protocol, in terms of throughput, fairness, convergence, and stability.

Conjectural Equilibrium in Multiuser Power Control Games

This paper considers a noncooperative game in which competing users sharing a frequency-selective interference channel selfishly optimize their power allocation in order to improve their achievable rates. Previously, it was shown that a user having the knowledge of its opponents' channel state information can make foresighted decisions and substantially improve its performance compared with the case in which it deploys the conventional iterative water-filling algorithm, which does not exploit such knowledge. This paper discusses how a foresighted user can acquire this knowledge by modeling its experienced interference as a function of its own power allocation. To characterize the outcome of the multiuser interaction, the conjectural equilibrium is introduced, and the existence of this equilibrium for the investigated water-filling game is proven. Importantly, we show that both the Nash equilibrium and the Stackelberg equilibrium are special cases of the conjectural equilibrium. We also develop practical algorithms to form accurate beliefs and select desirable power allocation strategies. Numerical simulations indicate that a foresighted user without any a priori knowledge of its competitors' private information can effectively learn how the other users will respond to its actions, and induce the entire system to an operating point that improves both its own achievable rate as well as the rates of the other participants in the water-filling game.

Monopolistic competition and new products: a conjectural equilibrium approach

In this paper we generalize the heterogeneous risk adverse agents model of diffusion of new products in a multi-firm, heterogeneous and interacting agents environment. We use a model of choice under uncertainty based on Bayesian theory. We discuss the possibility of product failures, the set of equilibria, their stability and some welfare properties depending on the degree of diversification.

Monopolistic Competition and New Products: A Conjectural Equilibrium Approach

In this paper we generalize the heterogeneous risk adverse agents model of diffusion of new products in a multi-firm, heterogeneous and interacting agents environment. We use a model of choice under uncertainty based on Bayesian theory. We discuss the possibility of product failures, the set of equilibria, their stability and some welfare properties.

On pure conjectural equilibrium with non-manipulable information

An information structure in a non-cooperative game determines the signal that each player observes as a function of the strategy profile. Such information structure is called non-manipulable if no player can gain new information by changing his strategy. A Conjectural Equilibrium (CE) (Battigalli in Unpublished undergraduate dissertation, 1987; Battigalli and Guaitoli 1988; Decisions, games and markets, 1997) with respect to a given information structure is a strategy profile in which each player plays a best response to his conjecture about his opponents’ play and his conjecture is not contradicted by the signal he observes. We provide a sufficient condition for the existence of pure CE in games with a non-manipulable information structure. We then apply this condition to prove existence of pure CE in two classes of games when the information that players have is the distribution of strategies in the population.