Group Strategy-proof Research Articles

Robust group strategy-proof rules in the object allocation problem with money: The role of tie-breaking rules

We study the object allocation problem with money. The owner owns identical objects. Each agent receives at most one unit of the object, and has a preference that is not necessarily quasi-linear. Recently, Kivinen and Tumennasan (2021) propose a group incentive property that they call robust group strategy-proofness. It takes into account a coalitional manipulation with which a coalition agrees without knowing the other agents’ preferences. We propose a generalized Vickrey rule with lowest priority agents that is associated with a tie-breaking rule such that for each coalitional report of preferences, there is an agent (a lowest priority agent) who has a chance to receive the object only after all the other members of the coalition receive the object. We show that the generalized Vickrey rules with lowest priority agents are the only rules satisfying robust group strategy-proofness, efficiency, individual rationality, and no subsidy for losers. Our result highlights the importance of the tie-breaking rules for robust group strategy-proofness, which contrasts with group strategy-proofness that a generalized Vickrey rule satisfies regardless of the tie-breaking rule.

Group incentive compatibility in a market with indivisible goods: A comment

We note that the proofs of Bird (1984), the first to show group strategy-proofness of top trading cycles (TTC), require correction. We provide a counterexample to a critical claim and present corrected proofs in the spirit of the originals. We also present a novel proof of strong group strategy-proofness using the corrected results.

Characterizing the typewise top-trading-cycles mechanism for multiple-type housing markets

We consider the generalization of the classical Shapley and Scarf housing market model (Shapley and Scarf, 1974) to so-called multiple-type housing markets (Moulin, 1995). Throughout the paper, we focus on strict preferences. When preferences are separable, the prominent solution for these markets is the typewise top-trading-cycles (tTTC) mechanism.We first show that for lexicographic preferences, a mechanism is unanimous (or onto), individually rational, strategy-proof, and non-bossy if and only if it is the tTTC mechanism. Second, we obtain a corresponding characterization for separable preferences. We obtain additional characterizations when replacing [strategy-proofness and non-bossiness] with self-enforcing group (or pairwise) strategy-proofness. Finally, we show that for strict preferences, there is no mechanism satisfying unanimity, individual rationality, and strategy-proofness.Our characterizations of the tTTC mechanism constitute the first characterizations of an extension of the prominent top-trading-cycles (TTC) mechanism to multiple-type housing markets.

Group strategy-proof rules in multidimensional binary domains

We consider a setting in which the alternatives are binary vectors and the preferences of the agents are determined by the Hamming distance from their most preferred alternatives. We consider only rules that are unanimous, anonymous, and component-neutral, and focus on strategy-proofness, weak group strategy-proofness, and strong group strategy-proofness. We show that component-wise majority rules are strategy-proof, and for three agents or two components also weakly group strategy-proof, but not otherwise. These rules are even strongly group strategy-proof if there are two or three agents. Our main result is an impossibility result: if there are at least four agents and at least three components, then no rule is strongly group strategy-proof.

A Strongly Group Strategyproof and Shill Resistant Bargaining Mechanism for Fog Resource Pricing

A Strongly Group Strategyproof and Shill Resistant Bargaining Mechanism for Fog Resource Pricing

On the manipulability of equitable voting rules

We consider the equity-manipulability trade-off when voting over two alternatives. Pairwise equity is a weakening of anonymity requiring certain permutations of the voters' identities to preserve the outcome of the election. ℓ-robust group strategy-proofness (ℓ-RGSP) and fine robust group strategy-proofness (FRGSP) are notions of group strategy-proofness that allow voters to have different beliefs about players' types (Kivinen and Tumennasan, 2021). Our main results show that, under mild conditions, for pairwise equitable voting rules anonymity is equivalent to 2-RGSP, 3-RGSP, and FRGSP. In this way, anonymity is special in its equity and non-manipulability. These results follow from several preliminary results, one of which equates 2-RGSP to comonotonicity and swap-robustness. Finally, we consider the manipulability of k-unanimous rules, and 2-RGSP when agents can be indifferent and abstain.

New axioms for top trading cycles

AbstractThe school choice problem is of great importance both in theory and practice. This paper studies the (student‐optimal) top trading cycles mechanism (TTCM) in an axiomatic way. We introduce two new axioms: MBG (mutual best group)‐quota‐rationality and MBG‐robust efficiency. While stability implies MBG‐quota rationality, MBG‐robust efficiency is weaker than robust efficiency, which is stronger than the combination of efficiency and group strategy proofness. The TTCM is characterized by MBG‐quota rationality and MBG‐robust efficiency. Our results construct a new basis to compare the TTCM with the other school choice mechanisms, in terms of both fairness and incentive‐related axioms.

Collusion-proof mechanisms for multi-unit procurement

A principal wants to procure multiple homogeneous units from finitely many agents. Each agent has an increasing and convex cost function, whose exact shape is unknown to the principal. Utility is quasilinear in money. We study which mechanisms are strategy-proof and robust to collusion, both when the agents can exchange money and physical units (reallocation-proofness) and when they cannot (group strategy-proofness). To achieve reallocation-proofness, the principal must offer the agents a fixed price per unit. While group-strategy-proof mechanisms can be more complex, they are inefficient and run the risk of procuring no units at all. We characterize the set of group-strategy-proof and anonymous mechanisms with a uniform price. A standout feature is that the number of potential prices is bounded above by the number of agents.

Group strategy-proof probabilistic voting with single-peaked preferences

We study probabilistic voting rules in the case where agents have single-peaked preferences over alternatives. A probabilistic rule decides a probability distribution over the set of alternatives for each profile of agents’ preferences. In this paper, we characterize the class of group strategy-proof and peak-only probabilistic rules.

Fair and Truthful Giveaway Lotteries

We consider a setting where a large number of agents are all interested in attending some public resource of limited capacity. Attendance is thus allotted by lottery. If agents arrive individually, then randomly choosing the agents – one by one - is a natural, fair and efficient solution. We consider the case where agents are organized in groups (e.g. families, friends), the members of each of which must all be admitted together. We study the question of how best to design such lotteries. We first establish the desired properties of such lotteries, in terms of fairness and efficiency, and define the appropriate notions of strategy proofness (providing that agents cannot gain by misrepresenting the true groups, e.g. joining or splitting groups). We establish inter-relationships between the different properties, proving properties that cannot be fulfilled simultaneously (e.g. leximin optimality and strong group stratagy proofness). Our main contribution is a polynomial mechanism for the problem, which guarantees many of the desired properties, including: leximin optimality, Pareto-optimality, anonymity, group strategy proofness, and adjunctive strategy proofness (which provides that no benefit can be obtained by registering additional - uninterested or bogus - individuals). The mechanism approximates the utilitarian optimum to within a factor of 2, which, we prove, is optimal for any mechanism that guarantees any one of the following properties: egalitarian welfare optimality, leximin optimality, envyfreeness, and adjunctive strategy proofness.

Truthful and Fair Mechanisms for Matroid-Rank Valuations

We study the problem of allocating indivisible goods among strategic agents. We focus on settings wherein monetary transfers are not available and each agent's private valuation is a submodular function with binary marginals, i.e., the agents' valuations are matroid-rank functions. In this setup, we establish a notable dichotomy between two of the most well-studied fairness notions in discrete fair division; specifically, between envy-freeness up to one good (EF1) and maximin shares (MMS). First, we show that a known Pareto-efficient mechanism is group strategy-proof for finding EF1 allocations, under matroid-rank valuations. The group strategy-proofness guarantee strengthens an existing result that establishes truthfulness (individually for each agent) in the same context. Our result also generalizes prior work from binary additive valuations to the matroid-rank case. Next, we establish that an analogous positive result cannot be achieved for MMS, even when considering truthfulness on an individual level. Specifically, we prove that, for matroid-rank valuations, there does not exist a truthful mechanism that is index oblivious, Pareto efficient, and maximin fair. For establishing our results, we develop a characterization of truthful mechanisms for matroid-rank functions. This characterization in fact holds for a broader class of valuations (specifically, holds for binary XOS functions) and might be of independent interest.

Restricted environments and incentive compatibility in interdependent values models

We study mechanisms that operate in interdependent values environments. We show that when defined on knit and strict environments, only constant mechanisms can be ex post incentive compatible. Knitness is also necessary for this result to hold for mechanisms with two alternatives in the range. For partially knit and strict environments, we prove that ex post incentive compatibility extends to groups, and that strategy-proofness implies strong group strategy-proofness in the special case of private values. The results extend to mechanisms operating on non-strict domains under an additional requirement of respectfulness. We discuss examples of environments where our theorems apply.

Local global equivalence for unanimous social choice functions

We identify a condition on preference domains that ensures that every locally strategy-proof and unanimous random social choice function is also strategy-proof. Furthermore every unanimous, locally strategy-proof deterministic social choice function is also group strategy-proof. The condition identified is significantly weaker than the characterization condition for local-global equivalence without unanimity in Kumar et al. (2020). The condition is not necessary for equivalence with unanimous random/deterministic social choice functions. However, we show the weaker condition of connectedness remains necessary.

On the relation between preference reversal and strategy-proofness

We analyze the relation between strategy-proofness and preference reversal in the case that agents may declare indifference. Interestingly, Berga and Moreno (2020) have recently derived preference reversal from group strategy-proofness of social choice functions on strict preferences domains if the range has no more than three elements. We extend this result and, at the same time, present a simple argument for proving it. Our analysis points out the role of individual strategy-proofness in deriving the preference reversal property, giving back to the latter its original individual nature (cfr. Eliaz, 2004). Finally, we show that group strategy-proofness is needed to obtain preference reversal if there are infinitely many voters.

Necessary and sufficient conditions for pairwise majority decisions on path-connected domains

In this paper, we consider choice functions that are unanimous, anonymous, symmetric, and group strategy-proof and consider domains that are single-peaked on some tree. We prove the following three results in this setting. First, there exists a unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. Second, a choice function is unanimous, anonymous, symmetric, and group strategy-proof on a single-peaked domain on a tree if and only if it is the pairwise majority rule (also known as the tree-median rule) and the number of agents is odd. Third, there exists a unanimous, anonymous, symmetric, and strategy-proof choice function on a strongly path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. As a corollary of these results, we obtain that there exists no unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if the number of agents is even.

Group incentive compatibility and welfare for matching with contracts

In the matching with contracts framework, under the assumption that each hospital has a choice function that satisfies the observable substitutability condition, we show that the strategy-proof cumulative offer mechanism is no longer guaranteed to be group strategy-proof, yet it nevertheless outputs a weakly Pareto efficient outcome for any input. We also discuss why, unlike the previous observations in the literature, the equivalence of strategy-proofness and group strategy-proofness does not carry over to our context.

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.

Social Choice in Large Populations with Single-Peaked Preferences

Anonymous social choice function for a large atomless population maps cross-section distributions of preferences into outcomes. Because any one individual is too insignificant to a¤ect these distributions, every anonymous social choice function is individually strategy-proof. However, not every anonymous social choice function is group strategy-proof. If the set of outcomes is linearly ordered and participants have single-peaked preferences, an anonymous social choice function is group strategy-proof if and only if it can be implemented by a mechanism involving binary votes between neighbouring outcomes with nondecreasing thresholds for moving higher up. Such a mechanism can be interpreted as a version of Moulin's (1980) generalized median-voter mechanism for a large population.