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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.

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.

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 $k$.

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)., Comment: To appear in the Operations Research journal. The abstract appeared in the Proceedings of the 23rd ACM Conference on Economics and Computation (EC 2022)

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.

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.

In the context of auctions for digital goods, an interesting random sampling auction has been proposed by Goldberg, Hartline, and Wright [2001]. This auction has been analyzed by Feige, Flaxman, Hartline, and Kleinberg [2005], who have shown that it is 15-competitive in the worst case {which is substantially better than the previously proven constant bounds but still far from the conjectured competitive ratio of 4. In this paper, we prove that the aforementioned random sampling auction is indeed 4-competitive for a large class of instances where the number of bids above (or equal to) the optimal sale price is at least 6. We also show that it is 4:68-competitive for the small class of remaining instances thus leaving a negligible gap between the lower and upper bound. We employ a mix of probabilistic techniques and dynamic programming to compute these bounds., Comment: A preliminary version appeared in the 10th ACM conference on Electronic Commerce, 2009

We study an abstract optimal auction problem for a single good or service. This problem includes environments where agents have budgets, risk preferences, or multi-dimensional preferences over several possible configurations of the good (furthermore, it allows an agent's budget and risk preference to be known only privately to the agent). These are the main challenge areas for auction theory. A single-agent problem is to optimize a given objective subject to a constraint on the maximum probability with which each type is allocated, a.k.a., an allocation rule. Our approach is a reduction from multi-agent mechanism design problem to collection of single-agent problems. We focus on maximizing revenue, but our results can be applied to other objectives (e.g., welfare). An optimal multi-agent mechanism can be computed by a linear/convex program on interim allocation rules by simultaneously optimizing several single-agent mechanisms subject to joint feasibility of the allocation rules. For single-unit auctions, Border \citeyearpar{B91} showed that the space of all jointly feasible interim allocation rules for $n$ agents is a $\NumTypes$-dimensional convex polytope which can be specified by $2^\NumTypes$ linear constraints, where $\NumTypes$ is the total number of all agents' types. Consequently, efficiently solving the mechanism design problem requires a separation oracle for the feasibility conditions and also an algorithm for ex-post implementation of the interim allocation rules. We show that the polytope of jointly feasible interim allocation rules is the projection of a higher dimensional polytope which can be specified by only $O(\NumTypes^2)$ linear constraints. Furthermore, our proof shows that finding a preimage of the interim allocation rules in the higher dimensional polytope immediately gives an ex-post implementation.

We study Bayesian mechanism design problems in settings where agents have budgets. Specifically, an agent's utility for an outcome is given by his value for the outcome minus any payment he makes to the mechanism, as long as the payment is below his budget, and is negative infinity otherwise. This discontinuity in the utility function presents a significant challenge in the design of good mechanisms, and classical "unconstrained" mechanisms fail to work in settings with budgets. The goal of this paper is to develop general reductions from budget-constrained Bayesian MD to unconstrained Bayesian MD with small loss in performance. We consider this question in the context of the two most well-studied objectives in mechanism design---social welfare and revenue---and present constant factor approximations in a number of settings. Some of our results extend to settings where budgets are private and agents need to be incentivized to reveal them truthfully.

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.

We consider two sided matching markets consisting of agents with non-transferable utilities; agents from the opposite sides form matching pairs (e.g., buyers-sellers) and negotiate the terms of their math which may include a monetary transfer. Competitive equilibria are the elements of the core of this game. We present the first combinatorial characterization of competitive equilibria that relates the utility of each agent at equilibrium to the equilibrium utilities of other agents in a strictly smaller market excluding that agent; thus automatically providing a constructive proof of existence of competitive equilibria in such markets. Our characterization also yields a group strategyproof mechanism for allocating indivisible goods to unit demand buyers with non-quasilinear utilities that highly resembles the Vickrey Clarke Groves (VCG) mechanism. As a direct application of this, we present a group strategyproof welfare maximizing mechanism for Ad-Auctions without requiring the usual assumption that search engine and advertisers have consistent estimates of the clickthrough rates.