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A powerful feature in mechanism design is the ability to irrevocably commit to the rules of a mechanism. Commitment is achieved by public declaration, which enables players to verify incentive properties in advance and the outcome in retrospect. However, public declaration can reveal superfluous information that the mechanism designer might prefer not to disclose, such as her target function or private costs. Avoiding this may be possible via a trusted mediator; however, the availability of a trusted mediator, especially if mechanism secrecy must be maintained for years, might be unrealistic. We propose a new approach to commitment, and show how to commit to, and run, any given mechanism without disclosing it, while enabling the verification of incentive properties and the outcome -- all without the need for any mediators. Our framework is based on zero-knowledge proofs -- a cornerstone of modern cryptographic theory. Applications include non-mediated bargaining with hidden yet binding offers.

We study sequential bilateral trade where sellers and buyers valuations are completely arbitrary (i.e., determined by an adversary). Sellers and buyers are strategic agents with private valuations for the good and the goal is to design a mechanism that maximizes efficiency (or gain from trade) while being incentive compatible, individually rational and budget balanced. In this paper we consider gain from trade which is harder to approximate than social welfare. We consider a variety of feedback scenarios and distinguish the cases where the mechanism posts one price and when it can post different prices for buyer and seller. We show several surprising results about the separation between the different scenarios. In particular we show that (a) it is impossible to achieve sublinear $\alpha$-regret for any $\alpha<2$, (b) but with full feedback sublinear $2$-regret is achievable (c) with a single price and partial feedback one cannot get sublinear $\alpha$ regret for any constant $\alpha$ (d) nevertheless, posting two prices even with one-bit feedback achieves sublinear $2$-regret, and (e) there is a provable separation in the $2$-regret bounds between full and partial feedback., Comment: The conference version of this paper appeared in NeurIPS 22, while a journal version was published in the Artificial Intelligence Journal. With respect to the previous arXiv version, the current one contains a revised proof of Theorem 6

We consider a market setting of agents with additive valuations over heterogeneous divisible resources. Agents are assigned a budget of tokens (possibly unequal budgets) they can use to obtain resources; leftover tokens are worthless. We show how to support any Pareto efficient allocation in equilibrium, using anonymous resource prices and agent specific budgets. We also give computationally efficient algorithms for those tasks. In particular, this allows us to support the Rawlsian max-min allocation.

The existence of EFX allocations is a major open problem in fair division, even for additive valuations. The current state of the art is that no setting where EFX allocations are impossible is known, and EFX is known to exist for ($i$) agents with identical valuations, ($ii$) 2 agents, ($iii$) 3 agents with additive valuations, ($iv$) agents with one of two additive valuations and ($v$) agents with two-valued instances. It is also known that EFX exists if one can leave $n-1$ items unallocated, where $n$ is the number of agents. We develop new techniques that allow us to push the boundaries of the enigmatic EFX problem beyond these known results, and, arguably, to simplify proofs of earlier results. Our main results are ($i$) every setting with 4 additive agents admits an EFX allocation that leaves at most a single item unallocated, ($ii$) every setting with $n$ additive valuations has an EFX allocation with at most $n-2$ unallocated items. Moreover, all of our results extend beyond additive valuations to all nice cancelable valuations (a new class, including additive, unit-demand, budget-additive and multiplicative valuations, among others). Furthermore, using our new techniques, we show that previous results for additive valuations extend to nice cancelable valuations.

A distance-approximation algorithm for a graph property $\mathcal{P}$ in the adjacency-matrix model is given an approximation parameter $\epsilon \in (0,1)$ and query access to the adjacency matrix of a graph $G=(V,E)$. It is required to output an estimate of the \emph{distance} between $G$ and the closest graph $G'=(V,E')$ that satisfies $\mathcal{P}$, where the distance between graphs is the size of the symmetric difference between their edge sets, normalized by $|V|^2$. In this work we introduce property covers, as a framework for using distance-approximation algorithms for "simple" properties to design distance-approximation. Applying this framework we present distance-approximation algorithms with $poly(1/\epsilon)$ query complexity for induced $P_3$-freeness, induced $P_4$-freeness, and Chordality. For induced $C_4$-freeness our algorithm has query complexity $exp(poly(1/\epsilon))$. These complexities essentially match the corresponding known results for testing these properties and provide an exponential improvement on previously known results.

ReLU neural-networks have been in the focus of many recent theoretical works, trying to explain their empirical success. Nonetheless, there is still a gap between current theoretical results and empirical observations, even in the case of shallow (one hidden-layer) networks. For example, in the task of memorizing a random sample of size $m$ and dimension $d$, the best theoretical result requires the size of the network to be $\tilde{\Omega}(\frac{m^2}{d})$, while empirically a network of size slightly larger than $\frac{m}{d}$ is sufficient. To bridge this gap, we turn to study a simplified model for ReLU networks. We observe that a ReLU neuron is a product of a linear function with a gate (the latter determines whether the neuron is active or not), where both share a jointly trained weight vector. In this spirit, we introduce the Gated Linear Unit (GaLU), which simply decouples the linearity from the gating by assigning different vectors for each role. We show that GaLU networks allow us to get optimization and generalization results that are much stronger than those available for ReLU networks. Specifically, we show a memorization result for networks of size $\tilde{\Omega}(\frac{m}{d})$, and improved generalization bounds. Finally, we show that in some scenarios, GaLU networks behave similarly to ReLU networks, hence proving to be a good choice of a simplified model.

We study combinatorial auctions with interdependent valuations. In such settings, each agent $i$ has a private signal $s_i$ that captures her private information, and the valuation function of every agent depends on the entire signal profile, ${\bf s}=(s_1,\ldots,s_n)$. The literature in economics shows that the interdependent model gives rise to strong impossibility results, and identifies assumptions under which optimal solutions can be attained. The computer science literature provides approximation results for simple single-parameter settings (mostly single item auctions, or matroid feasibility constraints). Both bodies of literature focus largely on valuations satisfying a technical condition termed {\em single crossing} (or variants thereof). We consider the class of {\em submodular over signals} (SOS) valuations (without imposing any single-crossing type assumption), and provide the first welfare approximation guarantees for multi-dimensional combinatorial auctions, achieved by universally ex-post IC-IR mechanisms. Our main results are: $(i)$ 4-approximation for any single-parameter downward-closed setting with single-dimensional signals and SOS valuations; $(ii)$ 4-approximation for any combinatorial auction with multi-dimensional signals and {\em separable}-SOS valuations; and $(iii)$ $(k+3)$- and $(2\log(k)+4)$-approximation for any combinatorial auction with single-dimensional signals, with $k$-sized signal space, for SOS and strong-SOS valuations, respectively. All of our results extend to a parameterized version of SOS, $d$-SOS, while losing a factor that depends on $d$., Comment: To appear in the 20th ACM conference on Economics and Computation (ACM EC '19)

We consider a setting where an auctioneer sells a single item to $n$ potential agents with {\em interdependent values}. That is, each agent has her own private signal, and the valuation of each agent is a known function of all $n$ private signals. This captures settings such as valuations for artwork, oil drilling rights, broadcast rights, and many more. In the interdependent value setting, all previous work has assumed a so-called {\sl single-crossing condition}. Single-crossing means that the impact of agent $i$'s private signal, $s_i$, on her own valuation is greater than the impact of $s_i$ on the valuation of any other agent. It is known that without the single-crossing condition an efficient outcome cannot be obtained. We study welfare maximization for interdependent valuations through the lens of approximation. We show that, in general, without the single-crossing condition, one cannot hope to approximate the optimal social welfare any better than the approximation given by assigning the item to a random bidder. Consequently, we introduce a relaxed version of single-crossing, {\sl $c$-single-crossing}, parameterized by $c\geq 1$, which means that the impact of $s_i$ on the valuation of agent $i$ is at least $1/c$ times the impact of $s_i$ on the valuation of any other agent ($c=1$ is single-crossing). Using this parameterized notion, we obtain a host of positive results. We propose a prior-free deterministic mechanism that gives an $(n-1)c$-approximation guarantee to welfare. We then show that a random version of the proposed mechanism gives a prior-free universally truthful $2c$-approximation to the optimal welfare for any concave $c$-single crossing setting (and a $2\sqrt{n}c^{3/2}$-approximation in the absence of concavity). We extend this mechanism to a universally truthful mechanism that gives $O(c^2)$-approximation to the optimal revenue.

We explore issues of dynamic supply and demand in ride sharing services such as Lyft and Uber, where demand fluctuates over time and geographic location. We seek to maximize social welfare which depends on taxicab and passenger locations, passenger valuations for service, and the distances between taxicabs and passengers. Our only means of control is to set surge prices, then taxicabs and passengers maximize their utilities subject to these prices. We study two related models: a continuous passenger-taxicab setting, similar to the Wardrop model, and a discrete passenger-taxicab setting. In the continuous setting, every location is occupied by a set of infinitesimal strategic taxicabs and a set of infinitesimal non-strategic passengers. In the discrete setting every location is occupied by a set of strategic agents, taxicabs and passengers, passengers have differing values for service. We expand the continuous model to a time-dependent setting and study the corresponding online environment. Surge prices are in passenger-taxicab equilibrium if there exists a min cost flow that moves taxicabs about such that (a) every taxicab follows a best response, (b) all strategic passengers at $v$ with value above the surge price $r_v$ for $v$, are served and (c) no strategic passengers with value below $r_v$ are served (non-strategic infinitesimal passengers are always served). This paper computes surge prices such that resulting passenger-taxicab equilibrium maximizes social welfare, and the computation of such surge prices is in poly time. Moreover, it is a dominant strategy for passengers to reveal their true values. We seek to maximize social welfare in the online environment, and derive tight competitive ratio bounds to this end. Our online algorithms make use of the surge prices computed over time and geographic location, inducing successive passenger-taxicab equilibria.

In their seminal paper, Karp, Vazirani and Vazirani (STOC'90) introduce the online bipartite matching problem, and the RANKING algorithm, which admits a tight $1-\frac{1}{e}$ competitive ratio. Since its publication, the problem has received considerable attention, including a sequence of simplified proofs. In this paper we present a new proof that gives an economic interpretation of the RANKING algorithm -- further simplifying the proof and avoiding arguments such as duality. The new proof gives a new perspective on previous proofs.

We give a prompt online mechanism for minimizing the sum of [weighted] completion times. This is the first prompt online algorithm for the problem. When such jobs are strategic agents, delaying scheduling decisions makes little sense. Moreover, the mechanism has a particularly simple form of an anonymous menu of options.

We consider job scheduling settings, with multiple machines, where jobs arrive online and choose a machine selfishly so as to minimize their cost. Our objective is the classic makespan minimization objective, which corresponds to the completion time of the last job to complete. The incentives of the selfish jobs may lead to poor performance. To reconcile the differing objectives, we introduce posted machine prices. The selfish job seeks to minimize the sum of its completion time on the machine and the posted price for the machine. Prices may be static (i.e., set once and for all before any arrival) or dynamic (i.e., change over time), but they are determined only by the past, assuming nothing about upcoming events. Obviously, such schemes are inherently truthful. We consider the competitive ratio: the ratio between the makespan achievable by the pricing scheme and that of the optimal algorithm. We give tight bounds on the competitive ratio for both dynamic and static pricing schemes for identical, restricted, related, and unrelated machine settings. Our main result is a dynamic pricing scheme for related machines that gives a constant competitive ratio, essentially matching the competitive ratio of online algorithms for this setting. In contrast, dynamic pricing gives poor performance for unrelated machines. This lower bound also exhibits a gap between what can be achieved by pricing versus what can be achieved by online algorithms.

We study mechanisms for candidate selection that seek to minimize the social cost, where voters and candidates are associated with points in some underlying metric space. The social cost of a candidate is the sum of its distances to each voter. Some of our work assumes that these points can be modeled on a real line, but other results of ours are more general. A question closely related to candidate selection is that of minimizing the sum of distances for facility location. The difference is that in our setting there is a fixed set of candidates, whereas the large body of work on facility location seems to consider every point in the metric space to be a possible candidate. This gives rise to three types of mechanisms which differ in the granularity of their input space (voting, ranking and location mechanisms). We study the relationships between these three classes of mechanisms. While it may seem that Black's 1948 median algorithm is optimal for candidate selection on the line, this is not the case. We give matching upper and lower bounds for a variety of settings. In particular, when candidates and voters are on the line, our universally truthful spike mechanism gives a [tight] approximation of two. When assessing candidate selection mechanisms, we seek several desirable properties: (a) efficiency (minimizing the social cost) (b) truthfulness (dominant strategy incentive compatibility) and (c) simplicity (a smaller input space). We quantify the effect that truthfulness and simplicity impose on the efficiency.

Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, this assumes carefully breaking ties amongst different bundles in the buyer demand set. Presumably, the shopkeeper cleverly convinces the buyer to break ties in a manner consistent with maximizing social welfare. Lacking such a shopkeeper, if buyers arrive sequentially and simply choose some arbitrary bundle in their demand set, the social welfare may be arbitrarily bad. In the context of matching markets, we show how to compute dynamic prices, based upon the current inventory, that guarantee that social welfare is maximized. Such prices are set without knowing the identity of the next buyer to arrive. We also show that this is impossible in general (e.g., for coverage valuations), but consider other scenarios where this can be done. We further extend our results to Bayesian and bounded rationality models.

We consider a generalization of the secretary problem where contracts are temporary, and for a fixed duration. This models online hiring of temporary employees, or online auctions for re-usable resources. The problem is related to the question of Finding a large independent set in a random unit interval graph.

We consider dynamic pricing schemes in online settings where selfish agents generate online events. Previous work on online mechanisms has dealt almost entirely with the goal of maximizing social welfare or revenue in an auction settings. This paper deals with quite general settings and minimizing social costs. We show that appropriately computed posted prices allow one to achieve essentially the same performance as the best online algorithm. This holds in a wide variety of settings. Unlike online algorithms that learn about the event, and then make enforceable decisions, prices are posted without knowing the future events or even the current event, and are thus inherently dominant strategy incentive compatible. In particular we show that one can give efficient posted price mechanisms for metrical task systems, some instances of the $k$-server problem, and metrical matching problems. We give both deterministic and randomized algorithms. Such posted price mechanisms decrease the social cost dramatically over selfish behavior where no decision incurs a charge. One alluring application of this is reducing the social cost of free parking exponentially., Comment: Appeared in the ACM-SIAM Symposium on Discrete Algorithms (SODA), 2015