Class Of Equilibria Research Articles

Evolutionary Dynamics and Phi-Regret Minimization in Games

Regret has been established as a foundational concept in online learning, and likewise has important applications in the analysis of learning dynamics in games. Regret quantifies the difference between a learner’s performance against a baseline in hindsight. It is well known that regret-minimizing algorithms converge to certain classes of equilibria in games; however, traditional forms of regret used in game theory predominantly consider baselines that permit deviations to deterministic actions or strategies. In this paper, we revisit our understanding of regret from the perspective of deviations over partitions of the full mixed strategy space (i.e., probability distributions over pure strategies), under the lens of the previously-established Φ-regret framework, which provides a continuum of stronger regret measures. Importantly, Φ-regret enables learning agents to consider deviations from and to mixed strategies, generalizing several existing notions of regret such as external, internal, and swap regret, and thus broadening the insights gained from regret-based analysis of learning algorithms. We prove here that the well-studied evolutionary learning algorithm of replicator dynamics (RD) seamlessly minimizes the strongest possible form of Φ-regret in generic 2 × 2 games, without any modification of the underlying algorithm itself. We subsequently conduct experiments validating our theoretical results in a suite of 144 2 × 2 games wherein RD exhibits a diverse set of behaviors. We conclude by providing empirical evidence of Φ-regret minimization by RD in some larger games, hinting at further opportunity for Φ-regret based study of such algorithms from both a theoretical and empirical perspective.

Belief-invariant and quantum equilibria in games of incomplete information

Drawing on ideas from game theory and quantum physics, we investigate nonlocal correlations from the point of view of equilibria in games of incomplete information. These equilibria can be classified in decreasing power as general communication equilibria, belief-invariant equilibria and correlated equilibria, all of which contain the familiar Nash equilibria.The notion of belief-invariant equilibrium appeared in game theory in the 90s. However, the class of non-signalling correlations associated to belief-invariance arose naturally already in the 80s in the foundations of quantum mechanics.In the present work, we explain and unify these two origins of the idea and study the above classes of equilibria, together with quantum correlated equilibria, using tools from quantum information but the language of (algorithmic) game theory. We present a general framework of belief-invariant communication equilibria, which contains correlated equilibria and quantum correlated equilibria as special cases. Our framework also contains the theory of Bell inequalities and their violations due to non-locality, which is a question of intense interest in the foundations of quantum mechanics, and it was indeed the original motivation for the aforementioned studies. Moreover, in our framework we can also model quantum games where players have conflicting interests, a recent developing topic in physics.We then use our framework to show new results related to the social welfare of equilibria. Namely, we exhibit a game where belief-invariance is socially better than any correlated equilibrium, and a game where all non-belief-invariant communication equilibria have a suboptimal social welfare. We also show that optimal social welfare can in certain cases be achieved by quantum mechanical correlations, which do not need an informed mediator to be implemented, and go beyond the classical “sunspot” or shared randomness approach.

Normal forms for the Laplace resonance

We describe a comprehensive model for systems locked in the Laplace resonance. The framework is based on the simplest possible dynamical structure provided by the Keplerian problem perturbed by the resonant coupling truncated at second order in the eccentricities. The reduced Hamiltonian, constructed by a transformation to resonant coordinates, is then submitted to a suitable ordering of the terms and to the study of its equilibria. Henceforth, resonant normal forms are computed. The main result is the identification of two different classes of equilibria. In the first class, only one kind of stable equilibrium is present: the paradigmatic case is that of the Galilean system. In the second class, three kinds of stable equilibria are possible and at least one of them is characterised by a high value of the forced eccentricity for the ‘first planet’: here, the paradigmatic case is the exo-planetary system GJ-876, in which the combination of libration centres admits triple conjunctions otherwise not possible in the Galilean case. The normal form obtained by averaging with respect to the free eccentricity oscillations describes the libration of the Laplace argument for arbitrary amplitudes and allows us to determine the libration width of the resonance. The agreement of the analytic predictions with the numerical integration of the toy models is very good.

Many-player games of optimal consumption and investment under relative performance criteria

We study a portfolio optimization problem for competitive agents with CRRA utilities and a common finite time horizon. The utility of an agent depends not only on her absolute wealth and consumption but also on her relative wealth and consumption when compared to the averages among the other agents. We derive a closed form solution for the n-player game and the corresponding mean field game. This solution is unique in the class of equilibria with constant investment and continuous time-dependent consumption, both independent of the wealth of the agent. Compared to the classical Merton problem with one agent, the competitive model exhibits a wide range of highly nonlinear and non-monotone dependence on the agents’ risk tolerance and competitiveness parameters. Counter-intuitively, competitive agents with high risk tolerance may behave like non-competitive agents with low risk tolerance.

Linear Stability Analysis of the Relativistic Vlasov--Maxwell System in an Axisymmetric Domain

We consider the plasma confined in a general axisymmetric spatial domain with perfect conducting boundary which reflects particles specularly, and look at a certain class of equilibria, assuming axisymmetry in the problem. We prove a sharp criterion of spectral stability under these settings. Moreover, we provide several explicit families of stable/unstable equilibria using this criterion.

New inertial algorithm for a class of equilibrium problems

The article introduces a new algorithm for solving a class ofequilibrium problems involving strongly pseudomonotone bifunctions with Lipschitz-type condition. We describe how to incorporate the proximal-like regularized technique with inertial effects. The main novelty of the algorithm is that it can be done without previously knowing the information on the strongly pseudomonotone and Lipschitz-type constants of cost bifunction. A reasonable explain for this is that the algorithm uses a sequence of stepsizes which is diminishing and non-summable. Theorem of strong convergence is proved. In the case, when the information on the modulus of strong pseudomonotonicity and Lispchitz-type constant is known,the rate of linear convergence of the algorithm has been established. Several of numerical experiments are performed to illustrate the convergence of the algorithm and also compare it with other algorithms.

Modelling heterogeneity and an open-mindedness social norm in opinion dynamics

We study heterogeneous interactions in a time-continuous bounded confidence model for opinion formation. The key new modelling aspects are to distinguish between open-minded and closed-minded behaviour and to include an open-mindedness social norm. The investigations focus on the equilibria supported by the proposed new model; particular attention is given to a novel class of equilibria consisting of multiple connected opinion clusters, which does not occur in the absence of heterogeneity. Various rigorous stability results concerning these equilibria are established. We also incorporate the effect of media in the model and study its implications for opinion formation.

Information Disclosure in Dynamic Lemon Market

This paper studies the incentive of a long run seller to disclose past offers when trading with a sequence of short-run buyers. Compared with the models of mandatory disclosure or mandatory non-disclosure, there is a new set of equilibria generated by allowing flexibility in the disclosure option. In this new class of equilibria, the seller adopts a threshold rule in disclosure and only discloses rejected price offer above the threshold to the future buyers. In a two-period case I show that the welfare-maximizing equilibrium in the optional disclosure model could generate a strictly higher social surplus than any equilibria in models with mandatory restrictions on the disclosure policy. Mandatory disclosure or concealment of past offers may not be necessary for market information structure. Moreover, a policy maker can adopt non-disclosure policy for lower prices and voluntary (or mandatory) disclosure policy for higher prices to enhance trading efficiency.

Magnetohydrostatic modelling of stellar coronae

We introduce to the stellar physics community a method of modelling stellar coronae that can be considered to be an extension of the potential field. In this approach, the magnetic field is coupled to the background atmosphere. The model is magnetohydrostatic (MHS) and is a balance between the Lorentz force, the pressure gradient and gravity. Analytical solutions are possible and we consider a particular class of equilibria in this paper. The model contains two free parameters and the effects of these on both the geometry and topology of the coronal magnetic field are investigated. A demonstration of the approach is given using a magnetogram derived from Zeeman-Doppler imaging of the 0.75 M$_{\odot}$ M-dwarf star GJ 182.

Speculative runs on interest rate pegs

We analyze a new class of equilibria that emerges when a central bank conducts monetary policy by setting an interest rate (as an arbitrary function of its available information) and letting the private sector set the quantity traded. These equilibria involve a run on the central bank׳s interest target, whereby money grows fast, private agents borrow as much as possible against the central bank, and the shadow interest rate is different from the policy target. We argue that these equilibria represent a particular danger when banks hold large excess reserves, such as is the case following periods of quantitative easing. Our analysis suggests that successfully managing the exit strategy requires additional tools beyond setting interest-rate targets and paying interest on reserves; in particular, freezing excess reserves or fiscal-policy intervention may be needed to fend off adverse expectations.

On Existence and Uniqueness of Equilibrium in a Class of Noisy Rational Expectations Models: Figure 1

I study a general class of noisy rational expectations models that nests the standard Grossman and Stiglitz (1980) and Hellwig (1980) models, but relaxes the usual assumption of joint normality of asset pay-offs and supply, and allows for more general signal structures. I provide a constructive proof of existence of equilibrium, characterize the price function, and provide sufficient conditions for uniqueness within the class of equilibria with continuous price functions, which are met by both the Grossman and Stiglitz (1980) models and the Hellwig (1980) models with a continuum of investors. My solution approach does not rely on the typical “conjecture and verify” method, and I exhibit a number of non-normal examples in which asset prices can be characterized explicitly and in a closed form. The results presented here open up a broad class of models for applied work. To illustrate the usefulness of generalizing the standard model, I show that in settings with non-normally distributed pay-offs, shocks to fundamentals may be amplified purely due to learning effects, price drifts can arise naturally, and the disagreement–return relation depends in a novel way on return skewness.

Inducing Leaders to Take Risky Decisions: Dismissal, Tenure, and Term Limits

How can a principal (employer or voter) induce an agent (worker or politician) to choose the 'right' actions if risky actions reveal the agent's decision-making competence and only dismissal can be used as an incentive instrument? The principal wishes to replace incompetent agents, but dismissal-fear can cause the agent to choose excessively safe actions that are uninformative. We first show that if the principal can commit to a replacement strategy in advance, then optimal mechanisms involve either (i) a probationary period and then indefinite tenure, or (ii) a dismissal strategy that replaces incompetent agents but also randomly replaces agents who take safe actions even if they are thought to be competent and may be taking an appropriate action. We characterize when each of these mechanisms is optimal. If the principal (e.g., a set of voters) regularly chooses whether to replace a leader, then whether or not efficiency is possible depends on the class of equilibria. Commitment-like behavior and approximate efficiency can be enforced in equilibria that use grim trigger-like strategies off-path, but efficiency is not possible in equilibria in which a new leader always provides a positive expected value, regardless of how competent a current leader is thought to be, and incentives can be improved by imposing term limits on leaders.

Inefficient sales delays by a durable‐good monopoly facing a finite number of buyers

AbstractThis article offers a new explanation for unscheduled price cuts and slow adoption of durable goods. We study a standard durable‐good monopoly model with a finite number of buyers and show that this game can have multiple subgame perfect equilibria in addition to the Pacman outcome—including the Coase conjecture. Of particular interest is a class of equilibria where the seller first charges a high price and only lowers that price once some—but not all—high‐valuation buyers purchase. This price structure creates a war of attrition between those buyers, which delays market clearing and rationalizes unscheduled purchase and price cut dates.

Give peace a chance: The effect of ownership and asymmetric information on peace

We study the possibility of peace when two countries fight a war over the ownership of a resource. War is always the outcome of the game played by rational countries – under complete or asymmetric information – when there is no pre-established distribution of the resource among countries. When there is such a distribution of the resource, under complete information peace is feasible for some initial distributions of the resource, whereas under asymmetric information there are two classes of equilibria: Peaceful Equilibria, in which peace has a positive probability, and Aggressive Equilibria, which assign probability one to war. Surprisingly, a little asymmetric information may yield war.

Open and Closed Loop Nash Equilibria in Games with a Continuum of Players

In this paper, the problem of relations between closed loop and open loop Nash equilibria is examined in the environment of discrete time dynamic games with a continuum of players and a compound structure encompassing both private and global state variables. An equivalence theorem between these classes of equilibria is proven, important implications for the calculation of these equilibria are derived and the results are presented on models of a common ecosystem exploited by a continuum of players. An example of an analogous game with finitely many players is also presented for comparison.

On Existence and Uniqueness of Equilibrium in Noisy Rational Expectations Economies

I study a general class of noisy rational expectations models that nests the standard Grossman and Stiglitz (1980) and Hellwig (1980) models, but relaxes the usual assumption of joint normality of asset payoffs and supply, and allows for more general signal structures. I provide a constructive proof of existence of equilibrium, characterize the price function, and provide sufficient conditions for uniqueness within the class of equilibria with continuous price functions, which are met by both the Grossman and Stiglitz (1980) model and the Hellwig (1980) model with a continuum of investors. My solution approach does not rely on the typical conjecture and verify method, and I exhibit a number of non-normal examples in which asset prices can be characterized explicitly and in closed form. The results presented here open up a broad class of models for applied work. To illustrate the usefulness of generalizing the standard model, I show that in settings with non-normally distributed payoffs, shocks to fundamentals may be amplified purely due to learning effects, price drifts can arise naturally, and the disagreement-return relation depends in a novel way on return skewness.

Give Peace a Chance: The Effect of Asymmetric Information on Peace

We study the possibility of peace when two countries fight a war over the ownership of a resource. War is always the outcome of the game played by rational countries - under complete or asymmetric information - when there is no pre-established distribution of the resource among countries. When there is such a distribution of the resource, under complete information peace is feasible for some initial distributions of the resource, whereas under asymmetric information there are two classes of equilibria: Peaceful Equilibria, in which peace has a positive probability, and Aggressive Equilibria, which assign probability one to war. Surprisingly, a little asymmetric information may yield war.

Long time evolution of atherosclerotic plaques

The evolution of atherosclerosis in general, and the influence of wall shear stress on the growth of atherosclerotic plaques in particular, is an intricate phenomenon which is still only partly understood. We therefore propose a qualitative mathematical model which consists of a number of ordinary differential equations for the concentrations of the most relevant constituents of the atherosclerotic plaque. These equations were studied both for the case that the wall shear stress is a parameter (model A), and for the case in which the plaque evolution is coupled to the blood flow (model B) which results in a time dependent wall shear stress. We find that both models exhibit a class of marginally stable equilibria, all reflecting states in which the plaque only grows for a short period of time after a perturbation. The uncoupled model A, however, shows bi-stability between this class of equilibria and another equilibrium state in which the plaque experiences unlimited growth in time, if the LDL cholesterol intake exceeds a threshold value. In model B the bi-stability vanishes, but we find that there is still a critical value of the LDL cholesterol intake beyond which the lumen radius drastically decreases. We show that this decrease is quite sensitive to the value of the wall shear stress.

Conditionally Stationary Equilibria in Discounted Dynamic Games

This paper introduces simple conditionally stationary strategies and related stationary penal codes for dynamic games with perfect monitoring. The class of equilibria that these strategies support is defined—they are called conditionally stationary equilibria. The main results are necessary and sufficient conditions for extremal strategies, or corresponding penal codes, which give the minimal payoffs to players among conditionally stationary equilibria. Results are applied to resource extraction games.

Optimal response functions in global pollution problems can be upward-sloping: accounting for adaptation

The standard model of global pollution predicts that countries’ optimal emission response functions are downward-sloping: if one country or group of countries raises (reduces) its emissions, the optimal response for other countries is to reduce (raise) their emissions. We add to the standard model the possibility of taking adaptation measures and find that the optimal emission response function can be upward-sloping when adaptation capacities are sufficiently high. We investigate the existence and stability of a certain class of equilibria under alternative configurations of response functions.