Group Strategy-proof Mechanisms Research Articles

Exchange of indivisible goods under matroid constraints

We study the problem of reallocating indivisible goods among a set of strategic agents by generalizing the original Sharpley-Scarf market to the setting where an object can be allocated to multiple agents but subject to an associated matroid constraint. We refer to this modified market as the matroid Sharpley-Scarf market. For this general market, we present a Pareto-optimal, individually rational and group strategy-proof mechanism. Our mechanism is simple and natural generalization of the Top Trading Cycle mechanism and the Serial Dictator mechanism. In our analysis, we demonstrate that the weak core may be absent in the matroid Sharpley-Scarf market, while a relaxed version called the constrained core does exist. Furthermore, we extend the concept called competitive equilibrium to the matroid Sharpley-Scarf market and show that an allocation is a competitive equilibrium if and only if it can be produced by our mechanism.

Multi-Stage Facility Location Problems with Transient Agents

We study various models for the one-dimensional multi-stage facility location problems with transient agents, where a transient agent arrives in some stage and stays for a number of consecutive stages. In the problems, we need to serve each agent in one of their stages by determining the location of the facility at each stage. In the first model, we assume there is no cost for moving the facility across the stages. We focus on optimal algorithms to minimize both the social cost objective, defined as the total distance of all agents to the facility over all stages, and the maximum cost objective, defined as the max distance of any agent to the facility over all stages. For each objective, we give a slice-wise polynomial (XP) algorithm (i.e., solvable in m^f(k) for some fixed parameter k and computable function f, where m is the input size) and show that there is a polynomial-time algorithm when a natural first-come-first-serve (FCFS) order of agent serving is enforced. We then consider the mechanism design problem, where the agents' locations and arrival stages are private, and design a group strategy-proof mechanism that achieves good approximation ratios for both objectives and settings with and without FCFS ordering. In the second model, we consider the facility's moving cost between adjacent stages under the social cost objective, which accounts for the total moving distance of the facility. Correspondingly, we design XP (and polynomial time) algorithms and a group strategy-proof mechanism for settings with or without the FCFS ordering.

The obnoxious facility location game with dichotomous preferences

We consider a facility location game in which n agents reside at known locations on a path, and k heterogeneous facilities are to be constructed on the path. Each agent is adversely affected by some subset of the facilities, and is unaffected by the others. We design two classes of mechanisms for choosing the facility locations given the reported agent preferences: utilitarian mechanisms that strive to maximize social welfare (i.e., to be efficient), and egalitarian mechanisms that strive to maximize the minimum welfare. For the utilitarian objective, we present a weakly group-strategyproof efficient mechanism for up to three facilities, we give strongly group-strategyproof mechanisms that achieve approximation ratios of 5/3 and 2 for k=1 and k>1, respectively, and we prove that no strongly group-strategyproof mechanism achieves an approximation ratio less than 5/3 for the case of a single facility. For the egalitarian objective, we present a strategyproof egalitarian mechanism for arbitrary k, and we prove that no weakly group-strategyproof mechanism achieves a o(n) approximation ratio for two facilities. We extend our egalitarian results to the case where the agents are located on a cycle, and we extend our first egalitarian result to the case where the agents are located in the unit square.

Strategyproof mechanisms for 2-facility location games with minimax envy

We study a fairness-based model for 2-facility location games on the real line where the social objective is to minimize the maximum envy over all agents. All the agents seek to minimize their personal costs, and the envy between any two of them is the difference in their personal costs. We consider two cases of personal costs, called min-dist cost and sum-dist cost. We are interested in pursuing strategyproof mechanisms for 2-facility location games in both cases. For the min-dist personal cost, we first show that a lower bound of the additive approximation for any deterministic strategyproof mechanism is 1/4, then devise a deterministic group strategyproof mechanism with additive approximation of 1/2 and two randomized strategyproof mechanisms with additive approximation of 1/4. For the sum-dist personal cost, we devise a group strategyproof deterministic mechanism which is also optimal.

Two-facility Location Games with Minimum Distance Requirement

We study the mechanism design problem of a social planner for locating two facilities on a line interval [0, 1], where a set of n strategic agents report their locations and a mechanism determines the locations of the two facilities. We consider the requirement of a minimum distance 0 ≤ d ≤ 1 between the two facilities. Given the two facilities are heterogeneous, we model the cost/utility of an agent as the sum of his distances to both facilities. In the heterogeneous two-facility location game to minimize the social cost, we show that the optimal solution can be computed in polynomial time and prove that carefully choosing one optimal solution as output is strategyproof. We also design a strategyproof mechanism minimizing the maximum cost. Given the two facilities are homogeneous, we model the cost/utility of an agent as his distance to the closer facility. In the homogeneous two-facility location game for minimizing the social cost, we show that any deterministic strategyproof mechanism has unbounded approximation ratio. Moreover, in the obnoxious heterogeneous two-facility location game for maximizing the social utility, we propose new deterministic group strategyproof mechanisms with provable approximation ratios and establish a lower bound (7 − d)/6 for any deterministic strategyproof mechanism. We also design a strategyproof mechanism maximizing the minimum utility. In the obnoxious homogeneous two-facility location game for maximizing the social utility, we propose deterministic group strategyproof mechanisms with provable approximation ratios and establish a lower bound 4/3. Besides, in the two-facility location game with triple-preference, where each facility may be favorable, obnoxious, indifferent for any agent, we further motivate agents to report both their locations and preferences towards the two facilities truthfully, and design a deterministic group strategyproof mechanism with an approximation ratio 4.

Multiple facility location games with envy ratio

We study deterministic mechanism design without money for k-facility location games with envy ratio on a real line segment, where a set of strategic agents report their locations and a social planner locates k facilities for minimizing the envy ratio. The objective of envy ratio, which is defined as the maximum over the ratios between any two agents' utilities, is derived from fair division to measure the fairness with respect to a certain facility location profile.The problem is studied in two settings. In the homogeneous k-facility location game where k facilities serve the same purpose, we propose a 2k2k−1-approximate deterministic group strategyproof mechanism which is also the best deterministic strategyproof mechanism. In the heterogeneous k-facility location game where each facility serves a different purpose, when k is even, we devise an optimal and group strategyproof mechanism; when k is odd, we provide a k+1k−1-approximate deterministic group strategyproof mechanism.

Stable Mechanisms in Controlled School Choice

We consider a controlled school choice model where students are divided into several types such as socioeconomically advantaged and disadvantaged. The priority for a school is dependent on the type distribution of the assignment. Our model is a generalization of several models in previous studies such as those in which the priorities are based on type quotas and/or reserves. We consider the case where the priority for each school is represented by a weak order and introduce a stable and group strategy-proof mechanism. Moreover, if the priority order for each school is restricted to be a strict order, then the mechanism is also a student optimal stable one. We generalize the model allowing that the priority orders for some schools are not weak, but partial. In this case, a stable mechanism is also introduced.

Incentive Compatible Cost Sharing of a Coalition Initiative with Probabilistic Inspection and Penalties for Misrepresentation

This research proposes cost sharing mechanisms such that payments for a coalition initiative are allocated among players based on their honest valuations of the initiative, probabilistic inspection efforts, and deception penalties. Specifically, we develop a set of multiobjective, nonlinear optimization problem formulations that alternatively impose Bayesian incentive compatible, strategyproof, or group strategyproof mechanisms with generalized cost sharing and penalty functions that can be tailored to specific applications. Any feasible solution to these problems corresponds to a Bayesian game with stochastic payoffs wherein a collectively honest declaration is a Bayes–Nash equilibrium, a Nash equilibrium in dominant strategies, or a collusion resistant Nash equilibrium, respectively, and wherein an optimal solution considers the central authority’s relative priorities between inspection and penalization. In addition to this general framework, we introduce special cases having specific cost sharing and penalty functions such that the set of mechanisms are budget-balanced-in-equilibrium and proportional by design. The convexity of the resulting mathematical programs are examined, and formulation size reductions due to constraint redundancy analyses are presented. The Pareto fronts associated with each multiobjective optimization problem are assessed, as are computer memory limitations. Finally, an experiment considers the clustering of available valuations and the player probability distributions over them to examine their effects.

Facility location game with envy ratio

We study the one-facility location game on a real line with a new objective called envy ratio. The envy ratio, which is adopted from fair division and represents the egalitarianism, is defined as the maximum over the ratios between any two agents’ utilities. We are interested in designing strategyproof or group strategyproof mechanisms that can minimize the envy ratio objective.We consider the model in two settings that can capture natural scenarios: the facility location and all the agents’ locations are restricted on a fixed interval; every agent’s location can be any point on the real line but the facility location is restricted on a relative interval. In both settings, we obtain the optimal solutions and the best deterministic strategyproof mechanisms which are also group strategyproof. In the first setting, we provide a lower bound for randomized strategyproof mechanisms. In the second setting, we give a lower bound and two upper bounds for randomized strategyproof mechanisms.

On group strategyproof and optimal object allocation

We consider a two-agent, single indivisible object allocation problem. We focus on continuous mechanisms that satisfy agent sovereignty, and investigate implications of group strategyproofness. In particular, we provide an explicit characterization of the strategyproof mechanisms and show that there are non-affine maximizer mechanisms that do not belong to the class characterized by Roberts (North-Holland, 1979). Further, we show that there are no budget-balanced strategyproof mechanisms. Also, we obtain an impossibility for existence of strong group strategyproof mechanism. We find that this impossibility goes away upon relaxing our notion of group strategyproofness, and consequently, present a class of weak group strategyproof mechanisms. Finally, we completely characterize the class of feasible strategyproof mechanisms satisfying individual rationality, and show that there are no optimal strategyproof expected revenue maximizing mechanisms under a general class of well behaved type distributions.

Efficient and Envy Minimal Matching

In priority-based matching problems, there is a trade-off between efficiency and reducing justified-envy. We show that finding an efficient and justified-envy minimal matching is an NP-hard problem when objects have weak priority rankings. Consequently, we focus on resolving the trade-off in a subclass of efficient mechanisms. First, we consider the class of efficient and strategyproof mechanisms and show that no group-strategyproof mechanism minimizes justified-envy in that class. Then, we restrict attention to the class of hierarchical exchanges and sequential dictatorship mechanisms, which are widely studied in the literature and applied in real-life resource allocation problems. We propose justified-envy minimal mechanisms for each of those classes.

Group Strategy-proof Mechanisms for Shuttle Facility Games

Group Strategy-proof Mechanisms for Shuttle Facility Games

Facility location games with distinct desires

In facility location games, one aims at designing a mechanism to decide the facility location based on the addresses reported by all agents. In the facility location game, each agent wants to minimize his/her distance from the facility, while in the obnoxious facility game, each agent prefers to be as far away from the facility as possible. In this paper we revisit the two games on a line network by finely defining another reasonable agent utility function in terms of their satisfaction degree with respect to the facility location. Namely, a happiness factor within [0,1] is introduced to measure the difference between the best facility location for an agent and the one given by the mechanism. An agent wants to gain a largest possible happiness factor while the social satisfaction is to maximize the sum of the factors. For the facility location game, we first show that the median mechanism (Procaccia and Tennenholtz, 2009) is 32-approximation. We then devise a 54-approximate group strategy-proof mechanism. Subsequently, we investigate the variant of maximizing the smallest happiness factor over all agents for this setting. For the obnoxious facility game, we show that the majority mechanism (Cheng et al., 2011) remains the best possible with approximation ratio of two. Finally, we devise a 43 -approximate randomized mechanism.

Approximate efficiency and strategy-proofness for moneyless mechanisms on single-dipped policy domain

The single-dipped domain can be used to model any allocation problem where a single output must be chosen in an interval with the assumption that agents’ preferences have a single most loathful point (the dip) in the interval, and the preferences are increasing as one moves away from that dip. Practical domains like this appear in political voting system where each voter has his most-hated candidate and alternative candidates are evaluated by their proximity to this candidate or in obnoxious location problem, where each agent prefers to have the obnoxious location to be distant from his own location, among others. We first characterize deterministic and anonymous strategy-proof and group strategy-proof mechanisms on single-dipped public policy domain, complementing the well-known results on single-peaked policy domain first investigated by Moulin (Pub. Choice 35:437–455, 1980). Then we consider the tradeoff between strategy-proofness and efficiency by applying our characterization. As concrete applications of our results, we extend existing models and results, and resolve several open questions related to the obnoxious facility location game from the algorithmic mechanism design literature.

Approximation Randomized Strategy-Proof Mechanisms in Obnoxious Facility Game with Weighted Agents

In this paper, we investigate the obnoxious facility location game with weighted agents. First, we design a randomized group strategy-proof mechanism with approximation ratio \(\frac{3W_\mathrm{max}}{2W_\mathrm{min}}\) when the weighted agents are located on a line; then, on the cycle metric, we also discuss the strategy-proofness and the approximation ratios of a class of group strategy-proof deterministic mechanisms.

Group strategyproof cost sharing: The role of indifferences

Every agent reports his willingness to pay for one unit of a good. A mechanism allocates goods and cost shares to some agents. We characterize the group strategyproof (GSP) mechanisms under two alternative continuity conditions interpreted as tie-breaking rules. With the maximalist rule (MAX) an indifferent agent is always served. With the minimalist rule (MIN) an indifferent agent does not get a unit of the good.GSP and MAX characterize the cross-monotonic mechanisms. These mechanisms are appropriate whenever symmetry is required. On the other hand, GSP and MIN characterize the sequential mechanisms. These mechanisms are appropriate whenever there is scarcity of the good.Our results are independent of an underlying cost function; they unify and strengthen earlier results for particular classes of cost functions.

Fair and group strategy-proof good allocation with money

We completely characterize the class of fair and group strategy-proof mechanisms. We consider two notions of fairness, anonymity in welfare and no-envy. Both fairness axioms, when applied with strategy-proofness, imply decision efficiency, and lead to the same class of group strategy-proof mechanisms (where the group size is restricted to two). We find that the only feasible mechanism satisfying a mild zero transfer axiom, in this class, is the Pivotal mechanism.

Optimal collusion-resistant mechanisms with verification

We present the first general positive result on the construction of collusion-resistant mechanisms, that is, mechanisms that guarantee dominant strategies even when agents can form arbitrary coalitions and exchange compensations (sometimes referred to as transferable utilities or side payments). This is a much stronger solution concept as compared to truthful or even group strategyproof mechanisms, and only impossibility results were known for this type of mechanisms in the “classical” model.We describe collusion-resistant mechanisms with verification that return optimal solutions for a wide class of mechanism design problems (which includes utilitarian ones as a special case). Note that every collusion-resistant mechanism without verification must have an unbounded approximation factor and, in general, optimal solutions cannot be obtained even if we content ourselves with truthful (“non-collusion-resistant”) mechanisms. All these results apply to problems that have been extensively studied in the algorithmic mechanism design literature like, for instance, task scheduling and inter-domain routing.

A Complete Characterization of Group-Strategyproof Mechanisms of Cost-Sharing

We study the problem of designing group-strategyproof cost-sharing mechanisms. The players report their bids for getting serviced and the mechanism decides a set of players that are going to be serviced and how much each one of them is going to pay. We determine three conditions: Fence Monotonicity, Stability of the allocation and Validity of the tie-breaking rule that are necessary and sufficient for group-strategyproofness, regardless of the cost function. Consequently, Fence Monotonicity characterizes group-strategyproof cost-sharing schemes closing an important open problem. Finally, we use our results to prove that there exist families of cost functions, where any group-strategyproof mechanism has arbitrarily poor budget balance.

Limitations of cross-monotonic cost-sharing schemes

A cost-sharing scheme is a set of rules defining how to share the cost of a service (often computed by solving a combinatorial optimization problem) amongs serviced customers. A cost-sharing scheme is cross-monotonic if it satisfies the property that everyone is better off when the set of people who receive the service expands. In this article, we develop a novel technique for proving upper bounds on the budget-balance factor of cross-monotonic cost-sharing schemes or the worst-case ratio of recovered cost to total cost. We apply this technique to games defined, based on several combinatorial optimization problems, including the problems of edge cover, vertex cover, set cover, and metric facility location and, in each case, derive tight or nearly-tight bounds. In particular, we show that for the facility location game, there is no cross-monotonic cost-sharing scheme that recovers more than a third of the total cost. This result, together with a recent 1/3-budget-balanced cross-monotonic cost-sharing scheme of Pál and Tardos [2003] closes the gap for the facility location game. For the vertex cover and set cover games, we show that no cross-monotonic cost-sharing scheme can recover more than a O ( n −1/3 ) and O (1/ n ) fraction of the total cost, respectively. Finally, we study the implications of our results on the existence of group-strategyproof mechanisms. We show that every group-strategyproof mechanism corresponds to a cost-sharing scheme that satisfies a condition weaker than cross-monotonicity. Using this, we prove that group-strategyproof mechanisms satisfying additional properties give rise to cross-monotonic cost-sharing schemes and therefore our upper bounds hold.