57 research outputs found
Strategy-Proof Facility Location for Concave Cost Functions
We consider k-Facility Location games, where n strategic agents report their
locations on the real line, and a mechanism maps them to k facilities. Each
agent seeks to minimize his connection cost, given by a nonnegative increasing
function of his distance to the nearest facility. Departing from previous work,
that mostly considers the identity cost function, we are interested in
mechanisms without payments that are (group) strategyproof for any given cost
function, and achieve a good approximation ratio for the social cost and/or the
maximum cost of the agents.
We present a randomized mechanism, called Equal Cost, which is group
strategyproof and achieves a bounded approximation ratio for all k and n, for
any given concave cost function. The approximation ratio is at most 2 for Max
Cost and at most n for Social Cost. To the best of our knowledge, this is the
first mechanism with a bounded approximation ratio for instances with k > 2
facilities and any number of agents. Our result implies an interesting
separation between deterministic mechanisms, whose approximation ratio for Max
Cost jumps from 2 to unbounded when k increases from 2 to 3, and randomized
mechanisms, whose approximation ratio remains at most 2 for all k. On the
negative side, we exclude the possibility of a mechanism with the properties of
Equal Cost for strictly convex cost functions. We also present a randomized
mechanism, called Pick the Loser, which applies to instances with k facilities
and n = k+1 agents, and for any given concave cost function, is strongly group
strategyproof and achieves an approximation ratio of 2 for Social Cost
The Value of Knowing Your Enemy
Many auction settings implicitly or explicitly require that bidders are
treated equally ex-ante. This may be because discrimination is philosophically
or legally impermissible, or because it is practically difficult to implement
or impossible to enforce. We study so-called {\em anonymous} auctions to
understand the revenue tradeoffs and to develop simple anonymous auctions that
are approximately optimal.
We consider digital goods settings and show that the optimal anonymous,
dominant strategy incentive compatible auction has an intuitive structure ---
imagine that bidders are randomly permuted before the auction, then infer a
posterior belief about bidder i's valuation from the values of other bidders
and set a posted price that maximizes revenue given this posterior.
We prove that no anonymous mechanism can guarantee an approximation better
than O(n) to the optimal revenue in the worst case (or O(log n) for regular
distributions) and that even posted price mechanisms match those guarantees.
Understanding that the real power of anonymous mechanisms comes when the
auctioneer can infer the bidder identities accurately, we show a tight O(k)
approximation guarantee when each bidder can be confused with at most k "higher
types". Moreover, we introduce a simple mechanism based on n target prices that
is asymptotically optimal and build on this mechanism to extend our results to
m-unit auctions and sponsored search
Strong Duality for a Multiple-Good Monopolist
We characterize optimal mechanisms for the multiple-good monopoly problem and
provide a framework to find them. We show that a mechanism is optimal if and
only if a measure derived from the buyer's type distribution satisfies
certain stochastic dominance conditions. This measure expresses the marginal
change in the seller's revenue under marginal changes in the rent paid to
subsets of buyer types. As a corollary, we characterize the optimality of
grand-bundling mechanisms, strengthening several results in the literature,
where only sufficient optimality conditions have been derived. As an
application, we show that the optimal mechanism for independent uniform
items each supported on is a grand-bundling mechanism, as long as
is sufficiently large, extending Pavlov's result for items [Pavlov'11]. At
the same time, our characterization also implies that, for all and for all
sufficiently large , the optimal mechanism for independent uniform items
supported on is not a grand bundling mechanism
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