Cost Monotonicity Research Articles

Is Shapley cost sharing optimal?

A general approach to the design of budget-balanced cost-sharing mechanisms is to use the Shapley value, applied to the given cost function, to define payments from the players to the mechanism. Is the corresponding Shapley value mechanism “optimal” in some sense? We consider the objective of minimizing worst-case inefficiency subject to a revenue constraint, and prove results in three different regimes. First, for the public excludable good problem, the Shapley value mechanism minimizes the worst-case efficiency loss over all truthful, deterministic, and budget-balanced mechanisms that satisfy equal treatment. Second, even with randomization and approximate budget-balance allowed and dropping equal treatment, the worst-case efficiency loss of the Shapley value mechanism is within a constant factor of the minimum possible. Third, for no-deficit mechanisms, we prove a general positive result: for every monotone cost function, a suitable blend of the VCG and Shapley value mechanisms is no-deficit and enjoys good approximate efficiency guarantees.

Contractual Design in Agency Problems with Non-Monotonic Cost and Correlated Information

We model an agency relationship in which the agent's cost is non-monotonic with respect to type and the type is correlated with a public ex-post signal. The principal can use lotteries to exploit the type-signal correlation within the limit of the agent's liability. We establish conditions for first-best implementation, highlighting two effects on contractual design. First, the structure of the optimal lottery varies across types and, for each type, it depends on whether the cost is U shaped or reverse U shaped with respect to type. Second, as compared to the case of monotonic cost, the design of incentive compatible lotteries is easier when the cost is U shaped, more difficult when the cost is reverse U shaped. The root of the second effect is that incentives are non-monotonic either below or above some interior types. The two effects involve that non-monotonicity is unfavorable to the principal when the cost is reverse U shaped. This conclusion is at odds with the wisdom, concerning settings without correlated information, that non-monotonicity, which triggers countervailing incentives, enhances contracting.

English

We investigate the cost allocation strategy associated with the problem of providing some network service from source to a number of users, via the Minimum Cost Steiner Tree Network that spans the source and all the receivers. cost of such a Steiner tree network, is distributed among its receivers. objective of this paper is to develop a reasonably fair and computationally efficient cost allocation rule associated with the above cost allocation problem. Since finding the optimal Steiner tree is an NP-hard problem, the input to our cost allocation problem is the best known solution obtained using some heuristic. In order to allocate the cost of this Steiner tree to the users (receiver nodes), we formulate the associated Steiner Tree Network (STN) game in characteristic function form. It is well known that the core of the general STN game might be empty. We propose a new cost allocation rule for the modified STN game which might be attractive to network users due to its monotonic properties, associated with network growth. ACM CCS (2012) Classification : Networks→Network Algorithms→Network Economics; Mathematics of computing→Discretemathematics→Combinatorics→Combinatorial optimization; Theory of computation → Theory and algorithms for application domains → Algorithmic game theory and mechanism design→Algorithmic game theory; Applied computing→Operations research→Decision analysis → Multi-criterion optimization and decisionmaking *To cite this article: D. Skorin-Kapov , The Monotonic Cost Allocation Rule in Steiner Tree Network Games, CIT. Journal of Computing and Information Technology , vol. 24, no. 1, pp. 17-29, 2016.

Mean field stochastic games: Monotone costs and threshold\\ policies

This paper considers mean field games in a multi-agent Markov decision process framework. Each player has a continuum state and binary action. By active control, a player can bring its state to a resetting point. All players are coupled through their cost functions. The structural property of the individual strategies is characterized in terms of threshold policies when the mean field game admits a solution. We further introduce a stationary equation system of the mean field game and analyze uniqueness of its solution under positive externalities.

Egalitarian Equivalent Capital Allocation

We apply Moulin’s notion of egalitarian equivalent cost sharing of a public good to the problem of insurance capitalization and capital allocation where the liability portfolio is fixed. We show that this approach yields overall capitalization and cost allocations that are Pareto efficient, individually rational, and, unlike other mechanisms, stable in the sense of adhering to cost monotonicity.

Risk-Sensitive Control of Pure Jump Process on Countable Space with Near Monotone Cost

In this article, we study risk-sensitive control problem with controlled continuous time pure jump process on a countable space as state dynamics. We prove multiplicative dynamic programming principle, elliptic and parabolic Harnack’s inequalities. Using the multiplicative dynamic programing principle and the Harnack’s inequalities, we prove the existence and a characterization of optimal risk-sensitive control under the near monotone condition.

Constrained predictive synchronization of discrete-time chaotic Lur’e systems with time-varying delayed feedback control

This paper presents a predictive synchronization method for discrete-time chaotic Lur’e systems with input constraints by using time-varying delayed feedback control. Based on the model predictive control scheme, a delay-dependent stabilization criterion is derived for the synchronization of chaotic systems that is represented by Lur’e systems with input constraints. By constructing a suitable Lyapunov–Krasovskii functional and combining with a reciprocally convex combination technique, a delay-dependent stabilization condition for synchronization is obtained via linear matrix inequality (LMI) formulation. The control inputs are obtained by solving a min-max problem subject to cost monotonicity, which is expressed in terms of LMIs. The effectiveness of the proposed method will be verified throughout a numerical example.

An axiomatic approach in minimum cost spanning tree problems with groups

We study minimum cost spanning tree problems with groups, where agents are located in different villages, cities, etc. The groups are formed by agents living in the same village. In Bergantinos and Gomez-Rua (Economic Theory 43:227–262, 2010) we define the rule F as the Owen value of the irreducible game with groups and we prove that F generalizes the folk rule of minimum cost spanning tree problems. Bergantinos and Vidal-Puga (Journal of Economic Theory 137:326–352, 2007a) give two characterizations of the folk rule. In the first one they characterize it as the unique rule satisfying cost monotonicity, population monotonicity and equal share of extra costs. In the second characterization of the folk rule they replace cost monotonicity by independence of irrelevant trees and population monotonicity by separability. In this paper we extend such characterizations to our setting. Some of the properties are the same (cost monotonicity and independence of irrelevant trees) and the other need to be adapted. In general, we do it by claiming the property twice: once among the groups and the other among the agents inside the same group.

Output Feedback Model Predictive Tracking Control Using a Slope Bounded Nonlinear Model

In this paper, an output feedback model predictive tracking control method is proposed for constrained nonlinear systems, which are described by a slope bounded model. In order to solve the problem, we consider the finite horizon cost function for an off-set free tracking control of the system. For reference tracking, the steady state is calculated by solving by quadratic programming and a nonlinear estimator is designed to predict the state from output measurements. The optimized control input sequences are obtained by minimizing the upper bound of the cost function with a terminal weighting matrix. The cost monotonicity guarantees that tracking and estimation errors go to zero. The proposed control law can easily be obtained by solving a convex optimization problem satisfying several linear matrix inequalities. In order to show the effectiveness of the proposed method, a novel slope bounded nonlinear model-based predictive control method is applied to the set-point tracking problem of solid oxide fuel cell systems. Simulations are also given to demonstrate the tracking performance of the proposed method.

시변 시간지연을 가지는 입력제한 시스템의 모델예측제어

This paper considers a model predictive control problem of discrete-time constrained systems with time-varying delay. For this problem, a delay dependent state feedback control approach is used to achieve asymptotic stabilization of systems with input constraints. Based on Lyapunov stability theory, a new stability condition is obtained via linear matrix inequality formulation to find cost monotonicity condition of the model predictive control algorithm which guarantee the closed loop stability. Finally, the proposed method is applied to a numerical example in order to show the effectiveness of our results.

Predictive control for sector bounded nonlinear model and its application to solid oxide fuel cell systems

In this paper, a nonlinear model predictive control method is presented for solid oxide fuel cell systems. For realistic modeling, a sector bounded nonlinear model with input constraint is considered. As a performance index, we consider one horizon cost function that is represented by the weighted sum of state, control input and nonlinear function. By minimizing the upper bound of the cost function, the optimized control input sequences are obtained. For cost monotonicity, a terminal inequality condition is derived in terms of a finite number of linear matrix inequalities (LMIs) by using augmented vector feedback law which consists of state and sector bounded nonlinear function. In order to show the effectiveness of the proposed method, the sector bounded nonlinear model is derived for the solid oxide fuel cell systems and the system performance is verified by applying the proposed MPC algorithm to the systems.

Receding horizon H<inf>∞</inf> control for discrete-time Markovian jump linear systems

Receding horizon H ∞ control scheme which can deal with both the H ∞ disturbance attenuation and mean square stability is proposed for a class of discrete-time Markovian jump linear systems when minimizing a given quadratic performance criteria. First, a control law is established for jump systems based on pontryagin's minimum principle and it can be constructed through numerical solution of iterative equations. The aim of this control strategy is to obtain an optimal control which can minimize the cost function under the worst disturbance at every sampling time. Due to the difficulty of the assurance of stability, then the above mentioned approach is improved by determining terminal weighting matrix which satisfies cost monotonicity condition. The control move which is calculated by using this type of terminal weighting matrix as boundary condition naturally guarantees the mean square stability of the closed-loop system. A sufficient condition for the existence of the terminal weighting matrix is presented in linear matrix inequality (LMI) form which can be solved efficiently by available software toolbox. Finally, a numerical example is given to illustrate the feasibility and effectiveness of the proposed method.

Connection situations under uncertainty and cost monotonic solutions

This paper deals with cost allocation problems arising from connection situations where edge costs are closed intervals of real numbers. To solve such problems, we extend to the interval uncertainty setting the obligation rules from the theory of minimum cost spanning tree problems, and study their cost monotonicity and stability properties. We also present an application to a simulated ad hoc wireless network using a software implementation of an appealing obligation rule, the P-value.

A generalization of obligation rules for minimum cost spanning tree problems

Tijs et al. [23] introduce the family of obligation rules for minimum cost spanning tree problems. We give a generalization of such family. We prove that our family coincides with the set of rules satisfying an additivity property and a cost monotonicity property. We also provide two new characterizations for the family of obligation rules using the previous properties. In the first one, we add a property of separability; and in the second one, we add core selection.

Erratum to: Risk-Sensitive Control with Near Monotone Cost

We have used Theorem 3 of [2] in the proof of Theorem 2.1 in order to claim ρ ≤ β , and also in a remark preceding Theorem 2.2. The proof thereof in [2], however, is flawed. Our own proof of the reverse inequality ρ ≥ β also has gaps. We give an alternative proof of ρ = β below that sidesteps both difficulties. Let χn denote a nonnegative smooth function such that χk ≡ 1 in Bk := {x : ‖x‖ ≤ k}, χk ≡ 0 in B k+1 and 0 ≤ χk ≤ 1. Let rk = rχk . Define for α > 0, uαk (θ, x) := inf v∈M1 Ex[e ∫ ∞ 0 e −αt rk(Xt ,vt )dt ].

Maximum gradient embeddings and monotone clustering

Let (X,d X ) be an n-point metric space. We show that there exists a distribution over non-contractive embeddings into trees f: X → T such that for every x ∈ X, where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.

Erratum to: Risk-Sensitive Control with Near Monotone Cost

Erratum to: Risk-Sensitive Control with Near Monotone Cost

Consistency and monotonicity in sequencing problems

We explore the implications of consistency and monotonicity in sequencing problems. We first identify all rules satisfying Pareto indifference, individual rationality from random arrival, and consistency. Next, we ask whether there is a rule satisfying the three axioms together with either one of two monotonicity requirements, time monotonicity and cost monotonicity. As it turns out, the minimal transfer rule is the only rule satisfying Pareto indifference, individual rationality from random arrival, consistency together with either time monotonicity or cost monotonicity. We also investigate how the maximal transfer rule responds to changes in the service time and waiting cost.

On obligation rules for minimum cost spanning tree problems

Tijs et al. (2006) introduce the family of obligation rules in minimum cost spanning tree problems. We prove that obligation rules are closely related with the marginalistic values of the irreducible game. We also provide axiomatic characterizations of obligation rules with two basic monotonicity properties, namely population monotonicity (if new agents join a “society”, no agent from the “initial society” can be worse off) and strong cost monotonicity (if a number of connection costs increase, no agent can be better off). In this class, the folk rule is the only allocation rule satisfying equal treatment of equals.

Selfish Traffic Allocation for Server Farms

We study the price of selfish routing in noncooperative networks like the Internet. In particular, we investigate the price of selfish routing using the price of anarchy (a.k.a. the coordination ratio) and other (e.g., bicriteria) measures in the recently introduced game theoretic parallel links network model of Koutsoupias and Papadimitriou. We generalize this model toward general, monotone families of cost functions and cost functions from queueing theory. A summary of our main results for general, monotone cost functions is as follows: 1. We give an exact characterization of all cost functions having a bounded/unbounded price of anarchy. For example, the price of anarchy for cost functions describing the expected delay in queueing systems is unbounded. 2. We show that an unbounded price of anarchy implies an extremely high performance degradation under bicriteria measures. In fact, the price of selfish routing can be as high as a bandwidth degradation by a factor that is linear in the network size. 3. We separate the game theoretic (integral) allocation model from the (fractional) flow model by demonstrating that even a very small or negligible amount of integrality can lead to a dramatic performance degradation. 4. We unify recent results on selfish routing under different objectives by showing that an unbounded price of anarchy under the min-max objective implies an unbounded price of anarchy under the average cost objective and vice versa. Our special focus lies on cost functions describing the behavior of Web servers that can open only a limited number of Transmission Control Protocol (TCP) connections. In particular, we compare the performance of queueing systems that serve all incoming requests with servers that reject requests in case of overload. Our analysis indicates that all queueing systems without rejection cannot give any reasonable guarantee on the expected delay of requests under selfish routing even when the injected load is far away from the capacity of the system. In contrast, Web server farms that are allowed to reject requests can guarantee a high quality of service for every individual request stream even under relatively high injection rates.