We consider the problem of designing revenue-optimal auctions for selling two items and bidders' valuations are independent among bidders but negatively correlated among items. In this paper, we obtain the closed-form optimal auction for this setting, by directly addressing the two difficulties above. In particular, the first difficulty is that when pointwise maximizing virtual surplus under multi-dimensional feasibility (i.e., the Border feasibility), (1) neither the optimal interim allocation is trivially monotone in the virtual value, (2) nor the virtual value is monotone in the bidder's type. As a result, the optimal interim allocations resulting from virtual surplus maximization no longer guarantees BIC. To address (1), we prove a generalization of Border's theorem and show that optimal interim allocation is indeed monotone in the virtual value. To address (2), we adapt Myerson's ironing procedure to this setting by redefining the (ironed) virtual value as a function of the lowest utility point. The second difficulty, perhaps a more challenging one, is that the lowest utility type in general is no longer at the endpoints of the type interval. To address this difficulty, we show by construction that there exist an allocation rule and an induced lowest utility type such that they form a solution of the virtual surplus maximization and in the meanwhile guarantees IIR. In the single bidder case, the optimal auction consists of a randomized bundle menu and a deterministic bundle menu; while in the multiple bidder case, the optimal auction is a randomization between two extreme mechanisms. The optimal solutions of our setting can be implemented by a Bayesian IC and IR auction, however, perhaps surprisingly, the revenue of this auction cannot be achieved by any (dominant-strategy) IC and IR auction.
Negatively correlated valuations are not uncommon in the literature. For example, consider an instance of the well-known facility location game with two facilities @cite_18 : agents are interested in services from the two facilities, located at the endpoints of a street. Each agent's type is his her location on the street, between the two facilities. The cost (negative utility) to either facility, the same as in @cite_2 , is simply one's distance to that facility, so his overall costs (as well as valuations) sum up to a constant --- the length of the street.
@cite_18 @cite_2
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This paper is devoted to the location of public facilities in a metric space. Selfish agents are located in this metric space, and their aim is to minimize their own cost, which is the distance from their location to the nearest facility. A central authority has to locate the facilities in the space, but she is ignorant of the true locations of the agents. The agents will therefore report their locations, but they may lie if they have an incentive to do it. We consider two social costs in this paper: the sum of the distances of the agents to their nearest facility, or the maximal distance of an agent to her nearest facility. We are interested in designing strategy-proof mechanisms that have a small approximation ratio for the considered social cost. A mechanism is strategy-proof if no agent has an incentive to report false information. In this paper, we design strategyproof mechanisms to locate n - 1 facilities for n agents. We study this problem in the general metric and in the tree metric spaces. We provide lower and upper bounds on the approximation ratio of deterministic and randomized strategy-proof mechanisms. We consider the problem of locating facilities in a metric space to serve a set of selfish agents. The cost of an agent is the distance between her own location and the nearest facility. The social cost is the total cost of the agents. We are interested in designing strategy-proof mechanisms without payment that have a small approximation ratio for social cost. A mechanism is a (possibly randomized) algorithm which maps the locations reported by the agents to the locations of the facilities. A mechanism is strategy-proof if no agent can benefit from misreporting her location in any configuration. This setting was first studied by Procaccia and Tennenholtz [21]. They focused on the facility game where agents and facilities are located on the real line. studied the mechanisms for the facility games in a general metric space [1]. However, they focused on the games with only one facility. In this paper, we study the two-facility game in a general metric space, which extends both previous models. We first prove an Ω(n) lower bound of the social cost approximation ratio for deterministic strategy-proof mechanisms. Our lower bound even holds for the line metric space. This significantly improves the previous constant lower bounds [21, 17]. Notice that there is a matching linear upper bound in the line metric space [21]. Next, we provide the first randomized strategy-proof mechanism with a constant approximation ratio of 4. Our mechanism works in general metric spaces. For randomized strategy-proof mechanisms, the previous best upper bound is O(n) which works only in the line metric space. In the traffic assignment problem, first proposed by Wardrop in 1952, commuters select the shortest available path to travel from their origins to their destinations. We study a generalization of this problem in which competitors, who may control a nonnegligible fraction of the total flow, ship goods across a network. This type of games, usually referred to as atomic games, readily applies to situations in which the competing freight companies have market power. Other applications include intelligent transportation systems, competition among telecommunication network service providers, and scheduling with flexible machines. Our goal is to determine to what extent these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game without centralized control by computing the worst-case inefficiency of Nash equilibria. The main conclusion is that although self-interested competitors will not achieve a fully efficient solution from the system's point of view, the loss is not too severe. We show how to compute several bounds for the worst-case inefficiency that depend on the characteristics of cost functions and on the market structure in the game. In addition, building upon the work of Catoni and Pallotino, we show examples in which market aggregation (or collusion) adversely impacts the aggregated competitors, even though their market power increases. For example, Nash equilibria of atomic network games may be less efficient than the corresponding Wardrop equilibria. When competitors are completely symmetric, we provide a characterization of the Nash equilibrium using a potential function, and prove that this counterintuitive phenomenon does not arise. Finally, we study a pricing mechanism that elicits more coordination from the players by reducing the worst-case inefficiency of Nash equilibria. We consider the problem of fairly dividing a set of items. Much of the fair division literature assumes that the items are goods' i.e., they yield positive utility for the agents. There is also some work where the items are chores' that yield negative utility for the agents. In this paper, we consider a more general scenario where an agent may have negative or positive utility for each item. This framework captures, e.g., fair task assignment, where agents can have both positive and negative utilities for each task. We show that whereas some of the positive axiomatic and computational results extend to this more general setting, others do not. We present several new and efficient algorithms for finding fair allocations in this general setting. We also point out several gaps in the literature regarding the existence of allocations satisfying certain fairness and efficiency properties and further study the complexity of computing such allocations.