Social Cost Function Research Articles

Learning-Augmented Mechanism Design: Leveraging Predictions for Facility Location

In this work, we introduce an alternative model for the design and analysis of strategyproof mechanisms that is motivated by the recent surge of work in “learning-augmented algorithms.” Aiming to complement the traditional worst-case analysis approach in computer science, this line of work has focused on the design and analysis of algorithms that are enhanced with machine-learned predictions. The algorithms can use the predictions as a guide to inform their decisions, aiming to achieve much stronger performance guarantees when these predictions are accurate (consistency), while also maintaining near-optimal worst-case guarantees, even if these predictions are inaccurate (robustness). We initiate the design and analysis of strategyproof mechanisms that are augmented with predictions regarding the private information of the participating agents. To exhibit the important benefits of this approach, we revisit the canonical problem of facility location with strategic agents in the two-dimensional Euclidean space. We study both the egalitarian and utilitarian social cost functions, and we propose new strategyproof mechanisms that leverage predictions to guarantee an optimal trade-off between consistency and robustness. Furthermore, we also prove parameterized approximation results as a function of the prediction error, showing that our mechanisms perform well, even when the predictions are not fully accurate. Funding: The work of E. Balkanski was supported in part by the National Science Foundation [Grants CCF-2210501 and IIS-2147361]. The work of V. Gkatzelis and X. Tan was supported in part by the National Science Foundation [Grant CCF-2210502] and [CAREER Award CCF-2047907]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/moor.2022.0225 .

The economic cost of social distancing during a pandemic: an optimal control approach in the SVIR model

AbstractWe devise a theoretical model for the optimal dynamical control of an infectious disease whose diffusion is described by the SVIR compartmental model. The control is realized through implementing social rules to reduce the disease’s spread, which often implies substantial economic and social costs. We model this trade-off by introducing a functional depending on three terms: a social cost function, the cost supported by the healthcare system for the infected population, and the cost of the vaccination campaign. Using Pontryagin’s Maximum Principle, we are able to characterize the optimal control strategy in three instances of the social cost function, the linear, quadratic, and exponential models, respectively. Finally, we present a set of results on the numerical solution of the optimally controlled system by using Italian data from the recent COVID-19 pandemics for the model calibration.

Large-scale dynamic system optimization using dual decomposition method with approximate dynamic programming

In this paper, multi-agent dynamic optimization with a coupling constraint is studied. The aim is to minimize a strongly convex social cost function, by considering a linear stochastic dynamics for each agent and also coupling constraints among the agents. In order to handle the coupling constraint and also, to avoid high computational cost imposed by a centralized method for large scale systems, the dual decomposition method is used to decompose the problem into multiple individual sub-problems, while the dual variable is adjusted by a coordinator. Nevertheless, since each sub-problem is not a linear–quadratic (LQ) optimal control problem, and hence its closed-form solution does not exist, approximate dynamic programming (ADP) is utilized to solve the sub-problems. The main contribution of the paper is to propose an algorithm by considering the interrelated iterations of dual variable adjustment and ADP, and to prove the convergence of the algorithm to the global optimal solution of the social cost function. Additionally, the implementation of the proposed algorithm using a neural network is presented. Also, the computational advantage of the proposed algorithm in comparison with other bench-marking methods is discussed in simulation results.

On Multidimensional Congestion Games

We introduce multidimensional congestion games, that is, congestion games whose set of players is partitioned into d+1 clusters C0,C1,…,Cd. Players in C0 have full information about all the other participants in the game, while players in Ci, for any 1≤i≤d, have full information only about the members of C0∪Ci and are unaware of all the others. This model has at least two interesting applications: (i) it is a special case of graphical congestion games induced by an undirected social knowledge graph with independence number equal to d, and (ii) it represents scenarios in which players have a type and the level of competition they experience on a resource depends on their type and on the types of the other players using it. We focus on the case in which the cost function associated with each resource is affine and bound the price of anarchy and stability as a function of d with respect to two meaningful social cost functions and for both weighted and unweighted players. We also provide refined bounds for the special case of d=2 in presence of unweighted players.

Decentralized Hierarchical Planning of PEVs Based on Mean-Field Reverse Stackelberg Game

In the reverse Stackelberg mechanism, by considering a decision function for the leader rather than a decision value in the conventional Stackelberg game, the leader can explore a wider decision space. This flexibility can result in realizing the globally optimal solution of the leader’s objective function, while controlling the reaction function of the followers, simultaneously. We consider an aggregator who purchases energy from the wholesale energy market. The aggregator acts as the leader for a group of plugged in electric vehicles (PEVs) and determines the price of energy versus consumption at each hour a day as its decision function. In the followers level, since the optimal charging strategies of the PEVs are coupled through the electricity price, the PEVs in a group are considered to cooperate in finding their Nash–Pareto-optimal charging strategy, by minimizing a social cost function. For a large number of PEVs, the cooperative cost minimization of PEVs can be modeled as a cooperative mean-field (MF) game. We propose a decentralized MF optimal control algorithm and prove that the algorithm converges to leader–follower MF $\varepsilon _{N}$ -Nash equilibrium point of the game. Furthermore, a decentralized reverse Stackelberg algorithm is implemented to achieve the optimal linear price function of the leader. Simulation results and comparison with benchmark methods are performed to demonstrate the advantages of the proposed method. Note to Practitioners —The effect of a large population of PEVs on the power grid such as overload and voltage drop is inevitable. Motivated by this, there are many research articles which propose different centralized and decentralized charging coordination solutions to address this problem. The core idea in the most of literature is: “How to IMPLEMENT a demand response (DR) program appropriately for charging coordination problem to avoid high peak load?” However, none of them propose a practical algorithm on “How to DESIGN a DR optimally by having limited information from the clients?” We propose a bilevel optimization algorithm to both design a DR program (i.e., price function) and also implement DR program to flatten the demand curve, accordingly. Another advantage of the proposed method is that the information structure of the problem is close to reality. There is no information exchange among the clients and also the utility company does not need to know the private information of the clients. The Utility Company only knows the charging profile of the PEVs in each day and broadcasts the price signal to the clients for the next day. The method is illustrated in an IEEE 5-bus system that supports our claims.

Global Social Cost Minimization With Possibly Nonconvex Objective Functions: An Extremum Seeking-Based Approach

A social cost minimization problem is addressed in this article. In the considered problem, a network of agents work collaboratively to minimize the social cost function, which is defined as the sum of the agents’ local objective functions. The engaged agents are supposed to be equipped with an undirected and connected communication graph. Different from most of the existing works, the social cost function in the considered problem is allowed to be nonconvex and possibly admits local extrema. To avoid local extrema and achieve the global minimization of the social cost function, an extremum-seeking-based approach is proposed by introducing a dynamic average consensus protocol to the sinusoidal-dither-signal-based extremum seeking scheme. The dynamic average consensus protocol is leveraged in the proposed extremum-seeker for information sharing and the sinusoidal probing signal is utilized for information extraction. For the avoidance of local extrema, the amplitude of the dither signal is designed to be adaptive. Through Lyapunov stability analysis, it is shown that the proposed method enables the decision variable to converge to a neighborhood of the global minimum point if the conditions on the network connectivity, the existence of unique global minimum and achievability of the global minimum are satisfied. The theoretical result is verified via simulating a numerical example.

Traffic Flow Control in Vehicular Multi-Hop Networks with Data Caching

Control of conventional transportation networks aims at bringing the state of the network (e.g., the traffic flows in the network) to the system optimal (SO) state. This optimum is characterized by the minimality of the social cost function, i.e., the total cost of travel (e.g., travel time) of all drivers. On the other hand, drivers are assumed to be rational and selfish, and make their travel decisions (e.g., route choices) to optimize their own travel costs, bringing the state of the network to a user equilibrium (UE). A classic approach to influence users’ route choice is using congestion tolls. In this paper, we study the SO and UE of future connected vehicular transportation networks, where users consider both the travel cost and the utility from data communication, when making their travel decisions. We leverage the data communication aspect of the decision making to influence the user route choices, driving the UE state to the SO state. We assume the cache-enabled vehicles can communicate with other vehicles via vehicle-to-vehicle (V2V) connections. We propose an algorithm for calculating the values of the data communication utility that drive the UE to the SO. This result provides a guideline on how the system operator can adjust the parameters of the communication network (e.g., data pricing and bandwidth) to achieve the optimal social cost. We discuss the insights that the results shed on a secondary optimization that the operator can conduct to maximize its own utility without deviating the transportation network state from the SO. We validate the proposed communication model via Veins simulation. The simulation results also show that the system cost can be lowered even if the bandwidth allocation does not exactly match the optimal allocation policy under 802.11p protocol.

MinMax Mean-Field Team Approach for a Leader–Follower Network: A Saddle-Point Strategy

This letter investigates a soft-constrained MinMax control problem of a leader-follower network. The network consists of one leader and an arbitrary number of followers that wish to reach consensus with minimum energy consumption in the presence of external disturbances. The leader and followers are coupled in the dynamics and cost function. Two non-classical information structures are considered: 1) mean-field sharing and 2) intermittent mean-field sharing, where the mean-field refers to the aggregate state of the followers. In mean-field sharing, every follower observes its local state, the state of the leader and the mean field while in the intermittent mean-field sharing, the mean-field is only observed at some (possibly no) time instants. A social welfare cost function is defined, and it is shown that a unique saddle-point strategy exists which minimizes the worst-case value of the cost function under mean-field sharing information structure. The solution is obtained by two scalable Riccati equations, which depend on a prescribed attenuation parameter, serving as a robustness factor. For the intermittent mean-field sharing information structure, an approximate saddle-point strategy is proposed, and its converges to the saddle-point is analyzed. Two numerical examples are provided to demonstrate the efficacy of the obtained results.

Economic description of tolerance in a society with asymmetric social cost functions

The evolutionary game dynamics of social tolerance among heterogeneous economic agents have been discussed in an economic interaction model with asymmetric social cost functions, where the individual cost depends only on the share of intolerant people in the opposite group. We show that, very different from the symmetric function case studied previously , economic interactions between individuals in a society with asymmetric social cost functions can be exactly solved in phase plane, and rich behaviours can be revealed by using algebraic approach. Our contribution consists in offering the explicit formula of evolutionary trajectories in the phase plane for the first time. The property of equilibrium is shown to be closely related to the group populations. Based on the explicit formula in the phase plane, the equilibriums of the evolutionary dynamics can be easily identified, and the evolutionary trajectory can be exactly analysed. We also show that the explicit solutions obtained would be especially suited to effective control of the evolutionary dynamics of social tolerance. The necessary and sufficient conditions of the full tolerance equilibrium under asymmetric social cost function are also discussed, which provides guidance and reference to set policies and development strategy of social tolerance.

Tight inefficiency bounds for perception-parameterized affine congestion games

We introduce a new model of congestion games that captures several extensions of the classical congestion game introduced by Rosenthal in 1973. The idea here is to parameterize both the perceived cost of each player and the social cost function of the system designer. Intuitively, each player perceives the load induced by the other players by an extent of ρ≥0, while the system designer estimates that each player perceives the load of all others by an extent of σ≥0. For specific choices of ρ and σ, we obtain extensions such as altruistic player behavior, risk sensitive players and the imposition of taxes on the resources. We derive tight bounds on the price of anarchy and the price of stability for a large range of parameters. Our bounds provide a complete picture of the inefficiency of equilibria for these games. As a result, we obtain tight bounds on the price of anarchy and the price of stability for the above mentioned extensions. Our results also reveal how one should “design” the cost functions of the players in order to reduce the price of anarchy. Somewhat counterintuitively, if each player cares about all other players to the extent of ρ=0.625 (instead of 1 in the standard setting) the price of anarchy reduces from 2.5 to 2.155 and this is best possible.

Neutral agents and social tolerance: a dynamic perspective based on economic interaction model with local social cost functions

ABSTRACTThe influence of neutral agents on the evolutionary dynamics of social tolerance is discussed based on a recently proposed economic interaction model with local social cost functions. We show that the dynamical structure of social tolerance is completely changed even if a few neutral agents are introduced into the society. Especially, the full intolerance steady state, which is stable in previous works, becomes unstable and avoidable due to the economic incentive introduced by the neutral agents. The necessary condition of achieving full tolerance is also reduced compared to previous works without neutral agents.

A framework for parallel and distributed training of neural networks

The aim of this paper is to develop a general framework for training neural networks (NNs) in a distributed environment, where training data is partitioned over a set of agents that communicate with each other through a sparse, possibly time-varying, connectivity pattern. In such distributed scenario, the training problem can be formulated as the (regularized) optimization of a non-convex social cost function, given by the sum of local (non-convex) costs, where each agent contributes with a single error term defined with respect to its local dataset. To devise a flexible and efficient solution, we customize a recently proposed framework for non-convex optimization over networks, which hinges on a (primal) convexification–decomposition technique to handle non-convexity, and a dynamic consensus procedure to diffuse information among the agents. Several typical choices for the training criterion (e.g., squared loss, cross entropy, etc.) and regularization (e.g., ℓ2 norm, sparsity inducing penalties, etc.) are included in the framework and explored along the paper. Convergence to a stationary solution of the social non-convex problem is guaranteed under mild assumptions. Additionally, we show a principled way allowing each agent to exploit a possible multi-core architecture (e.g., a local cloud) in order to parallelize its local optimization step, resulting in strategies that are both distributed (across the agents) and parallel (inside each agent) in nature. A comprehensive set of experimental results validate the proposed approach.

Design and Evaluation of a Mathematical Optimization Model for Traffic Signal Plan Transition Based on Social Cost Function

Signal plan transition is the process of changing from one timing plan to another. It begins when the first intersection starts adjusting signal timing plans and ends when the last intersection completes adjusting signal timing plans. The transition between signal timing plans is required because traffic patterns change during the day. Therefore, it is necessary to modify signal timing parameters offset, phase split, and cycle length for different expectations of traffic volume. This paper presents an alternative and new mathematical model to enhance the performance of traffic signals coordination at intersections during the transition phase. This model is oriented to describe the transition regarding coordination parameters in all intersections of an arterial road for minimizing the social cost during the transition phase expressed in function of costs due to delays, fuel consumption, and air emissions. An ant colony algorithm was designed, coded, and simulated to find the optimal transition parameters using available data. The model was evaluated based on its ability to minimize social costs during the transition period. Results showed that the proposed method performs better than traditional ones.

Vanishing Price of Decentralization in Large Coordinative Nonconvex Optimization

We focus on nonconvex multi-agent optimization where a large number of participants collaboratively optimize some social cost function and their individual preferences. We focus on the semidecentralized multi-agent best response setting where a central planner attempts to coordinate the agents and each participant pursues self-interest. By studying the duality framework, we provide geometric and analytic characterizations of the duality gap and the price of decentralization. We prove that the nonconvex problem becomes increasingly convex as the problem scales up in dimension. Upon appropriate coordination, the price of decentralization asymptotically vanishes to zero as the number of participants grows. We develop a duality-based coordination procedure for the central planner to adapt the price vector and select a particular best response for each participant. The coordination algorithm is able to induce individual best responses to dynamically converge to an approximate global optimum, regardless of the ...

Distributed semi-supervised support vector machines

The semi-supervised support vector machine (S3VM) is a well-known algorithm for performing semi-supervised inference under the large margin principle. In this paper, we are interested in the problem of training a S3VM when the labeled and unlabeled samples are distributed over a network of interconnected agents. In particular, the aim is to design a distributed training protocol over networks, where communication is restricted only to neighboring agents and no coordinating authority is present. Using a standard relaxation of the original S3VM, we formulate the training problem as the distributed minimization of a non-convex social cost function. To find a (stationary) solution in a distributed manner, we employ two different strategies: (i) a distributed gradient descent algorithm; (ii) a recently developed framework for In-Network Nonconvex Optimization (NEXT), which is based on successive convexifications of the original problem, interleaved by state diffusion steps. Our experimental results show that the proposed distributed algorithms have comparable performance with respect to a centralized implementation, while highlighting the pros and cons of the proposed solutions. To the date, this is the first work that paves the way toward the broad field of distributed semi-supervised learning over networks.

Evolutionary dynamics of social tolerance in the economic interaction model with local social cost functions

ABSTRACTThe evolutionary dynamics of social tolerance in a society with local social cost functions have been discussed. We show that, very different from the global function case studied previously, dynamics of social tolerance in a society with local social cost functions is integrable in phase space. The exact solution of the evolutionary dynamics in phase space is obtained, and the evolutionary trajectories are discussed by using the Kolmogorov–Arnold–Moser theorem. We show that the property of steady states is closely related to the group populations. We also discuss the necessary condition of the full tolerance steady state, and it is demonstrated that both global and local social cost functions have the same necessary condition of achieving full tolerance.

Impact of transition between signal timing plans in social cost based in delay, fuel consumption and air emissions

Significant effects of traffic congestion include the cost associated with extra travel time, fuel consumption, and gas emissions. This paper develops a mathematical function to quantify the monetary impact of transition designs between signal timing plans on users and the environment. This function offers an approach to reduce problems such as excessive travel time, pollution emissions and fuel consumption. The proposed social cost function is evaluated for various transition plans to assess the impact of the number of steps required to adjust signal timing. The relationships between delay, fuel consumption and gas emissions and the number of steps needed to achieve the transition are also analysed.

Risky utilities

We develop a theory of “risky utilities,” i.e., private firms that manage an infrastructure for public service and that may be tempted to engage in excessively risky activities, such as reducing maintenance expenditures (at the risk of provoking a breakdown of the system) or in speculation (at the risk of incurring massive losses it cannot bear). These risky utilities include financial utilities like exchanges, clearinghouses or payment systems, as well as standard utilities like electricity transmission networks. Continuation of service is essential, so risky utilities cannot be liquidated. The optimal regulatory contract minimizes the social cost among the contracts that steer the firm away from risky activities. It is simple and implemented with a capital (equity) adequacy requirement and a resolution mechanism when that requirement is breached. The social cost function is explicitly computed, and comparative statics can be simply derived.

The Price of Anarchy for Selfish Ring Routing is Two

We analyze the network congestion game with atomic players, asymmetric strategies, and the maximum latency among all players as social cost. This important social cost function is much less understood than the average latency. We show that the price of anarchy is at most two, when the network is a ring and the link latencies are linear. Our bound is tight. This is the first sharp bound for the maximum latency objective.

Altruism in Atomic Congestion Games

This article studies the effects of altruism, a phenomenon widely observed in practice, in the model of atomic congestion games. Altruistic behavior is modeled by a linear trade-off between selfish and social objectives. Our model can be embedded in the framework of congestion games with player-specific latency functions. Stable states are the pure Nash equilibria of these games, and we examine their existence and the convergence of sequential best-response dynamics. In general, pure Nash equilibria are often absent, and existence is NP-hard to decide. Perhaps surprisingly, if all delay functions are affine, the games remain potential games, even when agents are arbitrarily altruistic. The construction underlying this result can be extended to a class of general potential games and social cost functions, and we study a number of prominent examples. These results give important insights into the robustness of multi-agent systems with heterogeneous altruistic incentives. Furthermore, they yield a general technique to prove that stabilization is robust, even with partly altruistic agents, which is of independent interest. In addition to these results for uncoordinated dynamics, we consider a scenario with a central altruistic institution that can set incentives for the agents. We provide constructive and hardness results for finding the minimum number of altruists to stabilize an optimal congestion profile and more general mechanisms to incentivize agents to adopt favorable behavior. These results are closely related to Stackelberg routing and answer open questions raised recently in the literature.