arXiv subjects

Ali Makhdoumi

Publications and source records attributed to Ali Makhdoumi.

15 recordsLinked to original sources

arXiv2025

Deterministic Refund Mechanisms

We consider a mechanism design setting with a single item and a single buyer who is uncertain about the value of the item. Both the buyer and the seller have a common model for the buyer's value, but the buyer discovers her true value only upon receiving the item. Mechanisms in this setting can be interpreted as randomized refund mechanisms, which allocate the item at some price and then offer a (partial and/or randomized) refund to the buyer in exchange for the item if the buyer is unsatisfied with her purchase. Motivated by their practical importance, we study the design of optimal deterministic mechanisms in this setting. We characterize optimal mechanisms as virtual value maximizers for both continuous and discrete type settings. We then use this characterization, along with bounds on the menu size complexity, to develop efficient algorithms for finding optimal and near-optimal deterministic mechanisms.

cs.GT↗

arXiv2024

The Limits of Price Discrimination Under Privacy Constraints

We study a producer's problem of selling a product to a continuum of privacy-conscious consumers, where the producer can implement third-degree price discrimination, offering different prices to different market segments. We consider a privacy mechanism that provides a degree of protection by probabilistically masking each market segment. We establish that the resultant set of all consumer-producer utilities forms a convex polygon, characterized explicitly as a linear mapping of a certain high-dimensional convex polytope into $\mathbb{R}^2$. This characterization enables us to investigate the impact of the privacy mechanism on both producer and consumer utilities. In particular, we establish that the privacy constraint always hurts the producer by reducing both the maximum and minimum utility achievable. From the consumer's perspective, although the privacy mechanism ensures an increase in the minimum utility compared to the non-private scenario, interestingly, it may reduce the maximum utility. Finally, we demonstrate that increasing the privacy level does not necessarily intensify these effects. For instance, the maximum utility for the producer or the minimum utility for the consumer may exhibit nonmonotonic behavior in response to an increase of the privacy level.

econ.TH↗

arXiv2024

On Three-Layer Data Markets

We study a three-layer data market comprising users (data owners), platforms, and a data buyer. Each user benefits from platform services in exchange for data, incurring privacy loss when their data, albeit noisily, is shared with the buyer. The user chooses platforms to share data with, while platforms decide on data noise levels and pricing before selling to the buyer. The buyer selects platforms to purchase data from. We model these interactions via a multi-stage game, focusing on the subgame Nash equilibrium. We find that when the buyer places a high value on user data (and platforms can command high prices), all platforms offer services to the user who joins and shares data with every platform. Conversely, when the buyer's valuation of user data is low, only large platforms with low service costs can afford to serve users. In this scenario, users exclusively join and share data with these low-cost platforms. Interestingly, increased competition benefits the buyer, not the user: as the number of platforms increases, the user utility does not necessarily improve while the buyer utility improves. However, increasing the competition improves the overall utilitarian welfare. Building on our analysis, we then study regulations to improve the user utility. We discover that banning data sharing maximizes user utility only when all platforms are low-cost. In mixed markets of high- and low-cost platforms, users prefer a minimum noise mandate over a sharing ban. Imposing this mandate on high-cost platforms and banning data sharing for low-cost ones further enhances user utility.

econ.TH↗

arXiv2023

Optimal and Differentially Private Data Acquisition: Central and Local Mechanisms

We consider a platform's problem of collecting data from privacy sensitive users to estimate an underlying parameter of interest. We formulate this question as a Bayesian-optimal mechanism design problem, in which an individual can share her (verifiable) data in exchange for a monetary reward or services, but at the same time has a (private) heterogeneous privacy cost which we quantify using differential privacy. We consider two popular differential privacy settings for providing privacy guarantees for the users: central and local. In both settings, we establish minimax lower bounds for the estimation error and derive (near) optimal estimators for given heterogeneous privacy loss levels for users. Building on this characterization, we pose the mechanism design problem as the optimal selection of an estimator and payments that will elicit truthful reporting of users' privacy sensitivities. Under a regularity condition on the distribution of privacy sensitivities we develop efficient algorithmic mechanisms to solve this problem in both privacy settings. Our mechanism in the central setting can be implemented in time $\mathcal{O}(n \log n)$ where $n$ is the number of users and our mechanism in the local setting admits a Polynomial Time Approximation Scheme (PTAS).

cs.GT↗

arXiv2022

Descending Price Auctions with Bounded Number of Price Levels and Batched Prophet Inequality

We consider descending price auctions for selling $m$ units of a good to unit demand i.i.d. buyers where there is an exogenous bound of $k$ on the number of price levels the auction clock can take. The auctioneer's problem is to choose price levels $p_1 > p_2 > \cdots > p_{k}$ for the auction clock such that auction expected revenue is maximized. The prices levels are announced prior to the auction. We reduce this problem to a new variant of prophet inequality, which we call \emph{batched prophet inequality}, where a decision-maker chooses $k$ (decreasing) thresholds and then sequentially collects rewards (up to $m$) that are above the thresholds with ties broken uniformly at random. For the special case of $m=1$ (i.e., selling a single item), we show that the resulting descending auction with $k$ price levels achieves $1- 1/e^k$ of the unrestricted (without the bound of $k$) optimal revenue. That means a descending auction with just 4 price levels can achieve more than 98\% of the optimal revenue. We then extend our results for $m>1$ and provide a closed-form bound on the competitive ratio of our auction as a function of the number of units $m$ and the number of price levels $k$.

cs.GT↗

arXiv2019

Maximizing Sequence-Submodular Functions and its Application to Online Advertising

Motivated by applications in online advertising, we consider a class of maximization problems where the objective is a function of the sequence of actions as well as the running duration of each action. For these problems, we introduce the concepts of \emph{sequence-submodularity} and \emph{sequence-monotonicity} which extend the notions of submodularity and monotonicity from functions defined over sets to functions defined over sequences. We establish that if the objective function is sequence-submodular and sequence-non-decreasing, then there exists a greedy algorithm that achieves $1-1/e$ of the optimal solution. We apply our algorithm and analysis to two applications in online advertising: online ad allocation and query rewriting. We first show that both problems can be formulated as maximizing non-decreasing sequence-submodular functions. We then apply our framework to these two problems, leading to simple greedy approaches with guaranteed performances. In particular, for online ad allocation problem the performance of our algorithm is $1-1/e$, which matches the best known existing performance, and for query rewriting problem the performance of our algorithm is $1- 1/e^{1-1/e}$ which improves upon the best known existing performance in the literature.

cs.DM↗

arXiv2018

Strategic Dynamic Pricing with Network Effects

We study the optimal pricing strategy of a monopolist selling homogeneous goods to customers over multiple periods. The customers choose their time of purchase to maximize their payoff that depends on their valuation of the product, the purchase price, and the utility they derive from past purchases of others, termed the network effect. We first show that the optimal price sequence is non-decreasing. Therefore, by postponing purchase to future rounds, customers trade-off a higher utility from the network effects with a higher price. We then show that a customer's equilibrium strategy can be characterized by a threshold rule in which at each round a customer purchases the product if and only if her valuation exceeds a certain threshold. This implies that customers face an inference problem regarding the valuations of others, i.e., observing that a customer has not yet purchased the product, signals that her valuation is below a threshold. We consider a block model of network interactions, where there are blocks of buyers subject to the same network effect. A natural benchmark, this model allows us to provide an explicit characterization of the optimal price sequence asymptotically as the number of agents goes to infinity, which notably is linearly increasing in time with a slope that depends on the network effect through a scalar given by the sum of entries of the inverse of the network weight matrix. Our characterization shows that increasing the "imbalance" in the network defined as the difference between the in and out degree of the nodes increases the revenue of the monopolist. We further study the effects of price discrimination and show that in earlier periods monopolist offers lower prices to blocks with higher Bonacich centrality to encourage them to purchase, which in turn further incentivizes other customers to buy in subsequent periods.

cs.GT↗

arXiv2017

Guesswork Subject to a Total Entropy Budget

We consider an abstraction of computational security in password protected systems where a user draws a secret string of given length with i.i.d. characters from a finite alphabet, and an adversary would like to identify the secret string by querying, or guessing, the identity of the string. The concept of a "total entropy budget" on the chosen word by the user is natural, otherwise the chosen password would have arbitrary length and complexity. One intuitively expects that a password chosen from the uniform distribution is more secure. This is not the case, however, if we are considering only the average guesswork of the adversary when the user is subject to a total entropy budget. The optimality of the uniform distribution for the user's secret string holds when we have also a budget on the guessing adversary. We suppose that the user is subject to a "total entropy budget" for choosing the secret string, whereas the computational capability of the adversary is determined by his "total guesswork budget." We study the regime where the adversary's chances are exponentially small in guessing the secret string chosen subject to a total entropy budget. We introduce a certain notion of uniformity and show that a more uniform source will provide better protection against the adversary in terms of his chances of success in guessing the secret string. In contrast, the average number of queries that it takes the adversary to identify the secret string is smaller for the more uniform secret string subject to the same total entropy budget.

cs.IT↗

arXiv2017

Informational Braess' Paradox: The Effect of Information on Traffic Congestion

To systematically study the implications of additional information about routes provided to certain users (e.g., via GPS-based route guidance systems), we introduce a new class of congestion games in which users have differing information sets about the available edges and can only use routes consisting of edges in their information set. After defining the notion of Information Constrained Wardrop Equilibrium (ICWE) for this class of congestion games and studying its basic properties, we turn to our main focus: whether additional information can be harmful (in the sense of generating greater equilibrium costs/delays). We formulate this question in the form of Informational Braes' Paradox (IBP), which extends the classic Braess' Paradox in traffic equilibria, and asks whether users receiving additional information can become worse off. We provide a comprehensive answer to this question showing that in any network in the series of linearly independent (SLI) class, which is a strict subset of series-parallel networks, IBP cannot occur, and in any network that is not in the SLI class, there exists a configuration of edge-specific cost functions for which IBP will occur. In the process, we establish several properties of the SLI class of networks, which include the characterization of the complement of the SLI class in terms of embedding a specific set of networks, and also an algorithm which determines whether a graph is SLI in linear time. We further prove that the worst-case inefficiency performance of ICWE is no worse than the standard Wardrop equilibrium.

cs.GT↗

arXiv2017

Principal Inertia Components and Applications

We explore properties and applications of the Principal Inertia Components (PICs) between two discrete random variables $X$ and $Y$. The PICs lie in the intersection of information and estimation theory, and provide a fine-grained decomposition of the dependence between $X$ and $Y$. Moreover, the PICs describe which functions of $X$ can or cannot be reliably inferred (in terms of MMSE) given an observation of $Y$. We demonstrate that the PICs play an important role in information theory, and they can be used to characterize information-theoretic limits of certain estimation problems. In privacy settings, we prove that the PICs are related to fundamental limits of perfect privacy.

cs.IT↗

arXiv2017

Network Maximal Correlation

We introduce Network Maximal Correlation (NMC) as a multivariate measure of nonlinear association among random variables. NMC is defined via an optimization that infers transformations of variables by maximizing aggregate inner products between transformed variables. For finite discrete and jointly Gaussian random variables, we characterize a solution of the NMC optimization using basis expansion of functions over appropriate basis functions. For finite discrete variables, we propose an algorithm based on alternating conditional expectation to determine NMC. Moreover we propose a distributed algorithm to compute an approximation of NMC for large and dense graphs using graph partitioning. For finite discrete variables, we show that the probability of discrepancy greater than any given level between NMC and NMC computed using empirical distributions decays exponentially fast as the sample size grows. For jointly Gaussian variables, we show that under some conditions the NMC optimization is an instance of the Max-Cut problem. We then illustrate an application of NMC in inference of graphical model for bijective functions of jointly Gaussian variables. Finally, we show NMC's utility in a data application of learning nonlinear dependencies among genes in a cancer dataset.

stat.ML↗

arXiv2016

Graph Balancing for Distributed Subgradient Methods over Directed Graphs

We consider a multi agent optimization problem where a set of agents collectively solves a global optimization problem with the objective function given by the sum of locally known convex functions. We focus on the case when information exchange among agents takes place over a directed network and propose a distributed subgradient algorithm in which each agent performs local processing based on information obtained from his incoming neighbors. Our algorithm uses weight balancing to overcome the asymmetries caused by the directed communication network, i.e., agents scale their outgoing information with dynamically updated weights that converge to balancing weights of the graph. We show that both the objective function values and the consensus violation, at the ergodic average of the estimates generated by the algorithm, converge with rate $O(\frac{\log T}{\sqrt{T}})$, where $T$ is the number of iterations. A special case of our algorithm provides a new distributed method to compute average consensus over directed graphs.

math.OC↗

arXiv2016

Convergence Rate of Distributed ADMM over Networks

We propose a distributed algorithm based on Alternating Direction Method of Multipliers (ADMM) to minimize the sum of locally known convex functions using communication over a network. This optimization problem emerges in many applications in distributed machine learning and statistical estimation. We show that when functions are convex, both the objective function values and the feasibility violation converge with rate $O(\frac{1}{T})$, where $T$ is the number of iterations. We then show that if the functions are strongly convex and have Lipschitz continuous gradients, the sequence generated by our algorithm converges linearly to the optimal solution. In particular, an $ε$-optimal solution can be computed with $O(\sqrt{κ_f} \log (1/ε))$ iterations, where $κ_f$ is the condition number of the problem. Our analysis also highlights the effect of network structure on the convergence rate through maximum and minimum degree of nodes as well as the algebraic connectivity of the network.

math.OC↗

arXiv2014

From the Information Bottleneck to the Privacy Funnel

We focus on the privacy-utility trade-off encountered by users who wish to disclose some information to an analyst, that is correlated with their private data, in the hope of receiving some utility. We rely on a general privacy statistical inference framework, under which data is transformed before it is disclosed, according to a probabilistic privacy mapping. We show that when the log-loss is introduced in this framework in both the privacy metric and the distortion metric, the privacy leakage and the utility constraint can be reduced to the mutual information between private data and disclosed data, and between non-private data and disclosed data respectively. We justify the relevance and generality of the privacy metric under the log-loss by proving that the inference threat under any bounded cost function can be upper-bounded by an explicit function of the mutual information between private data and disclosed data. We then show that the privacy-utility tradeoff under the log-loss can be cast as the non-convex Privacy Funnel optimization, and we leverage its connection to the Information Bottleneck, to provide a greedy algorithm that is locally optimal. We evaluate its performance on the US census dataset.

cs.IT↗

arXiv2013

On Locally Decodable Source Coding

Locally decodable channel codes form a special class of error-correcting codes with the property that the decoder is able to reconstruct any bit of the input message from querying only a few bits of a noisy codeword. It is well known that such codes require significantly more redundancy (in particular have vanishing rate) compared to their non-local counterparts. In this paper, we define a dual problem, i.e. locally decodable source codes (LDSC). We consider both almost lossless (block error) and lossy (bit error) cases. In almost lossless case, we show that optimal compression (to entropy) is possible with O(log n) queries to compressed string by the decompressor. We also show the following converse bounds: 1) linear LDSC cannot achieve any rate below one, with a bounded number of queries, 2) rate of any source coding with linear decoder (not necessarily local) in one, 3) for 2 queries, any code construction cannot have a rate below one. In lossy case, we show that any rate above rate distortion is achievable with a bounded number of queries. We also show that, rate distortion is achievable with any scaling number of queries. We provide an achievability bound in the finite block-length regime and compare it with the existing bounds in succinct data structures literature.

cs.IT↗