Monotone Equilibrium Research Articles

About the Subgradient Method for Equilibrium Problems

Convergence results of the subgradient algorithm for equilibrium problems were mainly obtained using a Lipschitz continuity assumption on the given bifunctions. In this paper, we first provide a complexity result for monotone equilibrium problems without assuming Lipschitz continuity. Moreover, we give a convergence result of the value of the averaged sequence of iterates beyond Lipschitz continuity. Next, we derive a rate convergence in terms of the distance to the solution set relying on a growth condition. Applications to convex minimization and min–max problems are also stated. These ideas and results deserve to be developed and further refined.

Convergence of inertial prox-penalization and inertial forward–backward algorithms for solving monotone bilevel equilibrium problems

The main focus of this paper is on bilevel optimization on Hilbert spaces involving two monotone equilibrium bifunctions. We present a new achievement consisting on the introduction of inertial methods for solving these types of problems. Indeed, two several inertial type methods are suggested: a proximal algorithm and a forward–backward one. Under suitable conditions and without any restrictive assumption on the trajectories, the weak and strong convergence of the sequence generated by the both iterative methods are established. Two particular cases illustrating the proposed methods are thereafter discussed with respect to hierarchical minimization problems and equilibrium problems under a saddle point constraint. Furthermore, numerical examples are given to demonstrate the implementability of our algorithms. The algorithms and their convergence results improve and develop previous results in the field.

Monotone equilibrium in matching markets with signaling

We introduce a notion of competitive signaling equilibrium (CSE) in one-to-one matching markets with a continuum of heterogeneous senders and receivers. We then study monotone CSE where equilibrium outcomes - sender actions, receiver reactions, beliefs, and matching - are all monotone in the stronger set order. We show that if the sender utility is monotone-supermodular and the receiver's utility is weakly monotone-supermodular, a CSE is stronger monotone if and only if it passes Criterion D1 (Cho and Kreps (1987), Banks and Sobel (1987)). Given any interval of feasible reactions that receivers can take, we fully characterize a unique stronger monotone CSE and establishes its existence with quasilinear utility functions.

Strategic disclosure with reputational concerns

I study a strategic disclosure model wherein an uninformed decision-maker (DM) consults an expert of uncertain types regarding the state before acting. The expert may be an honest type, who is committed to reporting the truth; or a strategic type, whose payoff increases in the DM’s action independent of the state and, thus, strategically discloses information to facilitate his agenda while also valuing a reputation for honesty. We find that if the expert fails to obtain information with positive probability, a monotone equilibrium exists that involves an interval wherein the strategic expert adopts a mixed strategy for disclosure, in contrast to a simple cutoff rule that cannot be sustained in equilibrium. The value that the strategic expert attaches to reputation serves as a commitment device to promote disclosure, as does the higher probability that the state is observed, whereas an honest expert’s greater presence may harm the strategic expert’s disclosure incentive.

Regularization iterative method of bilevel form for equilibrium problems in Hilbert spaces

We study the regularization method for a monotone equilibrium problem and propose a new iterative method for solving the problem with a Lipschitz‐type condition in a Hilbert space. The method is designed by the proximal mapping incorporated with regularization terms. The method uses variable stepsizes which are taken by simple rules without a linesearch procedure. The strong convergence of iterative sequences generated by the method is established under some suitable conditions imposed on control parameters. It turns out that the obtained asymptotic solution by the method is the solution of an equilibrium problem whose constraint is the solution set of the considered equilibrium problem. The computational effectiveness of the method is illustrated by several numerical examples.

A new algorithm for solving strongly monotone equilibrium and multivalued variational inequality problems

A new algorithm for solving strongly monotone equilibrium and multivalued variational inequality problems

Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule

In a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively.

Sequential auctions with ambiguity

This paper studies sequential auctions in the presence of ambiguity about the distribution of values. Using the multi-self approach to accommodate the possible time inconsistency that arises with dynamic bidding for maxmin bidders, we characterize the unique symmetric and monotone equilibrium in closed form. We show that the equilibrium prices are a supermartingale if the worst-case beliefs reverse hazard rate dominate the true distribution of values, providing an explanation for the well-documented puzzle “declining price anomaly” in sequential auctions. Importantly, the sufficient condition for declining prices allows for dynamic inconsistency. We also propose a non-parametric procedure to identify bidders' values and beliefs from auction data. The model delivers rich testable implications: on the practical side, declining prices imply that bidders' worst-case beliefs first-order stochastically dominate the true distribution of values; on the theoretical side, dynamic inconsistency generates history dependence in bidding strategies and possibly higher revenues for the seller.

Auxiliary Principle Technique for Hierarchical Equilibrium Problems

In this paper, building upon auxiliary principle technique and using proximal operator, we introduce a new explicit algorithm for solving monotone hierarchical equilibrium problems. The considered problem is a monotone equilibrium problem, where the constraint is the solution set of a set-valued variational inequality problem. The strong convergence of the proposed algorithm is studied under strongly monotone and Lipschitz-type assumptions of the bifunction. By combining with parallel techniques, the convergence result is also established for the equilibrium problem involving a finite system of demicontractive mappings. Several fundamental experiments are provided to illustrate the numerical behavior of the proposed algorithm and comparison with other known algorithms is studied.

All-pay auctions with private signals about opponents’ values

We study all-pay auctions where each player observes her private value as well as a noisy private signal about the opponent’s value, following Fang and Morris’s (J Econ Theory 126(1):1–30, 2006) analysis of winner-pay auctions with multidimensional private signals. A unique symmetric monotonic equilibrium exists if the signal is not informative enough. When the signal is sufficiently informative, there exists a symmetric non-monotonic equilibrium in which all types of players randomize in overlapping supports. The revenue is lower than that in the standard independent private value setting. Fixing the signal’s informativeness, the all-pay auction raises lower revenue than the second-price auction, whereas the revenue ranking between the all-pay and the first-price auction is ambiguous.

Delay and Learning in Coordination

This paper studies a coordination game with incomplete information and the option to delay. The delay option enables the agents to observe a binary signal depending on whether the early actions (e.g., investment) surpasses a threshold. The anticipation of information incentivizes agents to wait and free-ride on the information, whereas acting early can help to generate good news, and consequently induce more coordination. Relying on this trade-off, we find the unique monotone equilibrium, which features the inability of the delay option in facilitating coordination. If the waiting period is sufficiently short or the interest rate is sufficiently low, the delay option induces more inaction and makes coordination more difficult to achieve, as compared with the static case. We further discuss the theoretical implications for understanding the impact of monetary policy on economic recovery and how the accessibility of information affects financial stability.

Strong convergence theorem for fixed points of relatively nonexpansive multi-valued mappings and equilibrium problems in Banach spaces

In this paper, we study the problem of finding a common element of the solution set of monotone equilibrium problem and the fixed point set of relatively nonexpansive multi-valued mappings in uniformly convex and uniformly smooth Banach spaces. We introduce a Halpern-S-iteration for solving the problem and establish a strong convergence theorem. Some consequences and applications of our main results are discussed. Some numerical experiments are performed to illustrate the convergence and computational performance of our algorithm in comparison with others having similar features. The numerical results have confirmed that the proposed algorithm has a competitive advantage over the existing methods. Our results extend and generalize some results in the literature in this direction.

New projection methods for equilibrium problems over fixed point sets

In this paper, we introduce some new approximation projection algorithms for solving monotone equilibrium problems over the intersection of fixed point sets of demicontractive mappings. By combining subgradient projection methods and hybrid steepest descent methods, strong convergence of the algorithms to a solution is shown in a real Hilbert space. Some numerical illustrations and comparisons are also reported to show the efficiency and advantage of the proposed algorithms.

A New Linesearch Algorithm for Split Equilibrium Problems

In this paper, we propose a new algorithm for solving a split equilibrium problem involving nonmonotone and monotone equilibrium bifunctions in real Hilbert spaces by using a shrinking projection method with a general Armijo line search rule on the e-subdifferential. We obtain a strong convergence theorem for the new algorithm.

A viscosity-type proximal point algorithm for monotone equilibrium problem and fixed point problem in an Hadamard space

In this paper, we introduce a viscosity-type proximal point algorithm comprising of a finite composition of resolvents of monotone bifunctions and a generalized asymptotically nonspreading mapping recently introduced by Phuengrattana [Appl. Gen. Topol. 18 (2017) 117–129]. We establish a strong convergence result of the proposed algorithm to a common solution of a finite family of equilibrium problems and fixed point problem for a generalized asymptotically nonspreading and nonexpansive mappings, which is also a unique solution of some variational inequality problems in an Hadamard space. We apply our result to solve convex feasibility problem and to approximate a common solution of a finite family of minimization problems in an Hadamard space.

Monotone equilibria in signaling games

We examine the monotonicity of the sender’s equilibrium strategy with respect to her type in signaling games. We show that when the sender’s return from the receiver’s action depends on her type, the Spence–Mirrlees condition cannot rule out equilibria in which a higher-type sender chooses a strictly lower action than a lower-type. We provide sufficient conditions under which all equilibria are monotone, which require that the sender’s payoff is decreasing in her action, is increasing in the receiver’s action, and has strictly increasing differences between her type and the action profile. We apply our sufficient conditions to education signaling, advertising, and warranty provision.

Subgradient projection methods extended to monotone bilevel equilibrium problems in Hilbert spaces

In this paper, by basing on the inexact subgradient and projection methods presented by Santos et al. (Comput. Appl. Math. 30: 91–107, 2011), we develop subgradient projection methods for solving strongly monotone equilibrium problems with pseudomonotone equilibrium constraints. The problem usually is called monotone bilevel equilibrium problems. We show that this problem can be solved by a simple and explicit subgradient method. The strong convergence for the proposed algorithms to the solution is guaranteed under certain assumptions in a real Hilbert space. Numerical illustrations are given to demonstrate the performances of the algorithms.

On monotone strategy equilibria in simultaneous auctions for complementary goods

We explore existence and properties of equilibrium when N≥2 bidders compete for L≥2 objects via simultaneous but separate auctions. Bidders have private combinatorial valuations over all sets of objects they could win, and objects are complements in the sense that these valuations are supermodular in the set of objects won. We provide a novel partial order on types under which best replies are monotone, and demonstrate that Bayesian Nash equilibria which are monotone with respect to this partial order exist on any finite bid lattice. We apply this result to show existence of monotone Bayesian Nash equilibria in continuous bid spaces when a single global bidder competes for L objects against many local bidders who bid for single objects only. We then consider monotone equilibrium with endogenous tiebreaking building on Jackson, Simon, Swinkels and Zame (2002), and demonstrate that these exist in general. These existence results apply to many auction formats, including first-price, second-price, and all-pay.

New subgradient extragradient methods for solving monotone bilevel equilibrium problems

ABSTRACTIn this paper, we propose new subgradient extragradient methods for finding a solution of a strongly monotone equilibrium problem over the solution set of another monotone equilibrium problem which usually is called monotone bilevel equilibrium problem in Hilbert spaces. The first proposed algorithm is based on the subgradient extragradient method presented by Censor et al. [Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl. 2011;148:318–335]. The strong convergence of the algorithm is established under monotone assumptions of the cost bifunctions with Lipschitz-type continuous conditions recently presented by Mastroeni in the auxiliary problem principle. We also present a modification of the algorithm for solving an equilibrium problem, where the constraint domain is the common solution set of another equilibrium problem and a fixed point problem. Several fundamental experiments are provided to illustrate the numerical behaviour of the algorithms and to compare with others.

On a Bregman regularized proximal point method for solving equilibrium problems

In this paper, we propose a Bregman regularized proximal point method for solving monotone equilibrium problems. Existence and uniqueness results as well as convergence of the sequence to a solution of an equilibrium problem is analyzed. We assume a coercivity condition on the Bregman function weaker than the one considered in the literature on equilibrium problems with Bregman regularization. Numerical experiments illustrate the efficiency of the method.