ACM Transactions on Economics and Computation
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2021 ◽  
Vol 9 (4) ◽  
pp. 1-41
Author(s):  
Nawal Benabbou ◽  
Mithun Chakraborty ◽  
Ayumi Igarashi ◽  
Yair Zick

In this article, we present new results on the fair and efficient allocation of indivisible goods to agents whose preferences correspond to matroid rank functions . This is a versatile valuation class with several desirable properties (such as monotonicity and submodularity), which naturally lends itself to a number of real-world domains. We use these properties to our advantage; first, we show that when agent valuations are matroid rank functions, a socially optimal (i.e., utilitarian social welfare-maximizing) allocation that achieves envy-freeness up to one item (EF1) exists and is computationally tractable. We also prove that the Nash welfare-maximizing and the leximin allocations both exhibit this fairness/efficiency combination by showing that they can be achieved by minimizing any symmetric strictly convex function over utilitarian optimal outcomes. To the best of our knowledge, this is the first valuation function class not subsumed by additive valuations for which it has been established that an allocation maximizing Nash welfare is EF1. Moreover, for a subclass of these valuation functions based on maximum (unweighted) bipartite matching, we show that a leximin allocation can be computed in polynomial time. Additionally, we explore possible extensions of our results to fairness criteria other than EF1 as well as to generalizations of the above valuation classes.


2021 ◽  
Vol 9 (4) ◽  
pp. 1-27
Author(s):  
Anat Ganor ◽  
Karthik C. S. ◽  
Dömötör Pálvölgyi

Brouwer’s fixed point theorem states that any continuous function from a compact convex space to itself has a fixed point. Roughgarden and Weinstein (FOCS 2016) initiated the study of fixed point computation in the two-player communication model, where each player gets a function from [0,1]^n to [0,1]^n , and their goal is to find an approximate fixed point of the composition of the two functions. They left it as an open question to show a lower bound of 2^{\Omega (n)} for the (randomized) communication complexity of this problem, in the range of parameters which make it a total search problem. We answer this question affirmatively. Additionally, we introduce two natural fixed point problems in the two-player communication model. Each player is given a function from [0,1]^n to [0,1]^{n/2} , and their goal is to find an approximate fixed point of the concatenation of the functions. Each player is given a function from [0,1]^n to [0,1]^{n} , and their goal is to find an approximate fixed point of the mean of the functions. We show a randomized communication complexity lower bound of 2^{\Omega (n)} for these problems (for some constant approximation factor). Finally, we initiate the study of finding a panchromatic simplex in a Sperner-coloring of a triangulation (guaranteed by Sperner’s lemma) in the two-player communication model: A triangulation T of the d -simplex is publicly known and one player is given a set S_A\subset T and a coloring function from S_A to \lbrace 0,\ldots ,d/2\rbrace , and the other player is given a set S_B\subset T and a coloring function from S_B to \lbrace d/2+1,\ldots ,d\rbrace , such that S_A\dot{\cup }S_B=T , and their goal is to find a panchromatic simplex. We show a randomized communication complexity lower bound of |T|^{\Omega (1)} for the aforementioned problem as well (when d is large). On the positive side, we show that if d\le 4 then there is a deterministic protocol for the Sperner problem with O((\log |T|)^2) bits of communication.


2021 ◽  
Vol 9 (4) ◽  
pp. 1-39
Author(s):  
Paul GÖlz ◽  
Anson Kahng ◽  
Simon Mackenzie ◽  
Ariel D. Procaccia

Liquid democracy is the principle of making collective decisions by letting agents transitively delegate their votes. Despite its significant appeal, it has become apparent that a weakness of liquid democracy is that a small subset of agents may gain massive influence. To address this, we propose to change the current practice by allowing agents to specify multiple delegation options instead of just one. Much like in nature, where—fluid mechanics teaches us—liquid maintains an equal level in connected vessels, we seek to control the flow of votes in a way that balances influence as much as possible. Specifically, we analyze the problem of choosing delegations to approximately minimize the maximum number of votes entrusted to any agent by drawing connections to the literature on confluent flow. We also introduce a random graph model for liquid democracy and use it to demonstrate the benefits of our approach both theoretically and empirically.


2021 ◽  
Vol 9 (4) ◽  
pp. 1-55
Author(s):  
Jiehua Chen ◽  
Piotr Skowron ◽  
Manuel Sorge

We propose two solution concepts for matchings under preferences: robustness and near stability . The former strengthens while the latter relaxes the classical definition of stability by Gale and Shapley (1962). Informally speaking, robustness requires that a matching must be stable in the classical sense, even if the agents slightly change their preferences. Near stability, however, imposes that a matching must become stable (again, in the classical sense) provided the agents are willing to adjust their preferences a bit. Both of our concepts are quantitative; together they provide means for a fine-grained analysis of the stability of matchings. Moreover, our concepts allow the exploration of tradeoffs between stability and other criteria of social optimality, such as the egalitarian cost and the number of unmatched agents. We investigate the computational complexity of finding matchings that implement certain predefined tradeoffs. We provide a polynomial-time algorithm that, given agent preferences, returns a socially optimal robust matching (if it exists), and we prove that finding a socially optimal and nearly stable matching is computationally hard.


2021 ◽  
Vol 9 (4) ◽  
pp. 1-14
Author(s):  
Simon Mauras

Stable matching in a community consisting of N men and N women is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal work by Gale and Shapley. When the input preference profile is generated from a distribution, we study the output distribution of two stable matching procedures: women-proposing-deferred-acceptance and men-proposing-deferred-acceptance. We show that the two procedures are ex-ante equivalent—that is, under certain conditions on the input distribution, their output distributions are identical. In terms of technical contributions, we generalize (to the non-uniform case) an integral formula, due to Knuth and Pittel, which gives the probability that a fixed matching is stable. Using an inclusion-exclusion principle on the set of rotations, we give a new formula that gives the probability that a fixed matching is the women/men-optimal stable matching.


2021 ◽  
Vol 9 (3) ◽  
pp. 1-31
Author(s):  
Khaled Elbassioni

We consider the problem of pricing edges of a line graph so as to maximize the profit made from selling intervals to single-minded customers. An instance is given by a set E of n edges with a limited supply for each edge, and a set of m clients, where each client specifies one interval of E she is interested in and a budget B j which is the maximum price she is willing to pay for that interval. An envy-free pricing is one in which every customer is allocated an (possibly empty) interval maximizing her utility. Grandoni and Rothvoss (SIAM J. Comput. 2016) proposed a polynomial-time approximation scheme ( PTAS ) for the unlimited supply case with running time ( nm ) O ((1/ɛ) 1/ɛ ) , which was extended to the limited supply case by Grandoni and Wiese (ESA 2019). By utilizing the known hierarchical decomposition of doubling metrics , we give a PTAS with running time ( nm ) O (1/ ɛ 2 ) for the unlimited supply case. We then consider the limited supply case, and the notion of ɛ-envy-free pricing in which a customer gets an allocation maximizing her utility within an additive error of ɛ. For this case, we develop an approximation scheme with running time ( nm ) O (log 5/2 max e H e /ɛ 3 ) , where H e = B max ( e )/ B min ( e ) is the maximum ratio of the budgets of any two customers demanding edge e . This yields a PTAS in the uniform budget case, and a quasi-PTAS for the general case. The best approximation known, in both cases, for the exact envy-free pricing version is O (log c max ), where c max is the maximum item supply. Our method is based on the known hierarchical decomposition of doubling metrics, and can be applied to other problems, such as the maximum feasible subsystem problem with interval matrices.


2021 ◽  
Vol 9 (3) ◽  
pp. 1-33
Author(s):  
Dario Paccagnan ◽  
Rahul Chandan ◽  
Bryce L. Ferguson ◽  
Jason R. Marden

How can we design mechanisms to promote efficient use of shared resources? Here, we answer this question in relation to the well-studied class of atomic congestion games, used to model a variety of problems, including traffic routing. Within this context, a methodology for designing tolling mechanisms that minimize the system inefficiency (price of anarchy) exploiting solely local information is so far missing in spite of the scientific interest. In this article, we resolve this problem through a tractable linear programming formulation that applies to and beyond polynomial congestion games. When specializing our approach to the polynomial case, we obtain tight values for the optimal price of anarchy and corresponding tolls, uncovering an unexpected link with load balancing games. We also derive optimal tolling mechanisms that are constant with the congestion level, generalizing the results of Caragiannis et al. [8] to polynomial congestion games and beyond. Finally, we apply our techniques to compute the efficiency of the marginal cost mechanism. Surprisingly, optimal tolling mechanism using only local information perform closely to existing mechanism that utilize global information, e.g., Bilò and Vinci [6], while the marginal cost mechanism, known to be optimal in the continuous-flow model, has lower efficiency than that encountered levying no toll. All results are tight for pure Nash equilibria and extend to coarse correlated equilibria.


2021 ◽  
Vol 9 (3) ◽  
pp. 1-39
Author(s):  
Mithun Chakraborty ◽  
Ayumi Igarashi ◽  
Warut Suksompong ◽  
Yair Zick

We introduce and analyze new envy-based fairness concepts for agents with weights that quantify their entitlements in the allocation of indivisible items. We propose two variants of weighted envy-freeness up to one item (WEF1): strong , where envy can be eliminated by removing an item from the envied agent’s bundle, and weak , where envy can be eliminated either by removing an item (as in the strong version) or by replicating an item from the envied agent’s bundle in the envying agent’s bundle. We show that for additive valuations, an allocation that is both Pareto optimal and strongly WEF1 always exists and can be computed in pseudo-polynomial time; moreover, an allocation that maximizes the weighted Nash social welfare may not be strongly WEF1, but it always satisfies the weak version of the property. Moreover, we establish that a generalization of the round-robin picking sequence algorithm produces in polynomial time a strongly WEF1 allocation for an arbitrary number of agents; for two agents, we can efficiently achieve both strong WEF1 and Pareto optimality by adapting the adjusted winner procedure. Our work highlights several aspects in which weighted fair division is richer and more challenging than its unweighted counterpart.


2021 ◽  
Vol 9 (2) ◽  
pp. 1-22
Author(s):  
Shuchi Chawla ◽  
Joseph (Seffi) Naor ◽  
Debmalya Panigrahi ◽  
Mohit Singh ◽  
Seeun William Umboh

This article studies the equilibrium states that can be reached in a network design game via natural game dynamics. First, we show that an arbitrarily interleaved sequence of arrivals and departures of players can lead to a polynomially inefficient solution at equilibrium. This implies that the central controller must have some control over the timing of agent arrivals and departures to ensure efficiency of the system at equilibrium. Indeed, we give a complementary result showing that if the central controller is allowed to restore equilibrium after every set of arrivals/departures via improving moves , then the eventual equilibrium states reached have exponentially better efficiency.


2021 ◽  
Vol 9 (3) ◽  
pp. 1-18
Author(s):  
Susanne Albers ◽  
Dennis Kraft

People tend to behave inconsistently over time due to an inherent present bias. As this may impair performance, social and economic settings need to be adapted accordingly. Common tools to reduce the impact of time-inconsistent behavior are penalties and prohibition. Such tools are called commitment devices. In recent work Kleinberg and Oren [6, 7] connect the design of prohibition-based commitment devices to a combinatorial problem in which edges are removed from a task graph G with n nodes. However, this problem is NP-hard to approximate within a ratio less than √n/3 [2]. To address this issue, we propose a penalty-based commitment device that does not delete edges but raises their cost. The benefits of our approach are twofold. On the conceptual side, we show that penalties are up to 1/β times more efficient than prohibition, where β ϵ (0,1] parameterizes the present bias. On the computational side, we significantly improve approximability by presenting a 2-approximation algorithm for allocating the penalties. To complement this result, we prove that optimal penalties are NP-hard to approximate within a ratio of 1.08192.


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