48 research outputs found
On the Efficiency of the Proportional Allocation Mechanism for Divisible Resources
We study the efficiency of the proportional allocation mechanism, that is
widely used to allocate divisible resources. Each agent submits a bid for each
divisible resource and receives a fraction proportional to her bids. We
quantify the inefficiency of Nash equilibria by studying the Price of Anarchy
(PoA) of the induced game under complete and incomplete information. When
agents' valuations are concave, we show that the Bayesian Nash equilibria can
be arbitrarily inefficient, in contrast to the well-known 4/3 bound for pure
equilibria. Next, we upper bound the PoA over Bayesian equilibria by 2 when
agents' valuations are subadditive, generalizing and strengthening previous
bounds on lattice submodular valuations. Furthermore, we show that this bound
is tight and cannot be improved by any simple or scale-free mechanism. Then we
switch to settings with budget constraints, and we show an improved upper bound
on the PoA over coarse-correlated equilibria. Finally, we prove that the PoA is
exactly 2 for pure equilibria in the polyhedral environment.Comment: To appear in SAGT 201
Valuation Compressions in VCG-Based Combinatorial Auctions
The focus of classic mechanism design has been on truthful direct-revelation
mechanisms. In the context of combinatorial auctions the truthful
direct-revelation mechanism that maximizes social welfare is the VCG mechanism.
For many valuation spaces computing the allocation and payments of the VCG
mechanism, however, is a computationally hard problem. We thus study the
performance of the VCG mechanism when bidders are forced to choose bids from a
subspace of the valuation space for which the VCG outcome can be computed
efficiently. We prove improved upper bounds on the welfare loss for
restrictions to additive bids and upper and lower bounds for restrictions to
non-additive bids. These bounds show that the welfare loss increases in
expressiveness. All our bounds apply to equilibrium concepts that can be
computed in polynomial time as well as to learning outcomes
Expressiveness and Robustness of First-Price Position Auctions
Since economic mechanisms are often applied to very different instances of
the same problem, it is desirable to identify mechanisms that work well in a
wide range of circumstances. We pursue this goal for a position auction setting
and specifically seek mechanisms that guarantee good outcomes under both
complete and incomplete information. A variant of the generalized first-price
mechanism with multi-dimensional bids turns out to be the only standard
mechanism able to achieve this goal, even when types are one-dimensional. The
fact that expressiveness beyond the type space is both necessary and sufficient
for this kind of robustness provides an interesting counterpoint to previous
work on position auctions that has highlighted the benefits of simplicity. From
a technical perspective our results are interesting because they establish
equilibrium existence for a multi-dimensional bid space, where standard
techniques break down. The structure of the equilibrium bids moreover provides
an intuitive explanation for why first-price payments may be able to support
equilibria in a wider range of circumstances than second-price payments
On the Efficiency of the Walrasian Mechanism
Central results in economics guarantee the existence of efficient equilibria
for various classes of markets. An underlying assumption in early work is that
agents are price-takers, i.e., agents honestly report their true demand in
response to prices. A line of research in economics, initiated by Hurwicz
(1972), is devoted to understanding how such markets perform when agents are
strategic about their demands. This is captured by the \emph{Walrasian
Mechanism} that proceeds by collecting reported demands, finding clearing
prices in the \emph{reported} market via an ascending price t\^{a}tonnement
procedure, and returns the resulting allocation. Similar mechanisms are used,
for example, in the daily opening of the New York Stock Exchange and the call
market for copper and gold in London.
In practice, it is commonly observed that agents in such markets reduce their
demand leading to behaviors resembling bargaining and to inefficient outcomes.
We ask how inefficient the equilibria can be. Our main result is that the
welfare of every pure Nash equilibrium of the Walrasian mechanism is at least
one quarter of the optimal welfare, when players have gross substitute
valuations and do not overbid. Previous analysis of the Walrasian mechanism
have resorted to large market assumptions to show convergence to efficiency in
the limit. Our result shows that approximate efficiency is guaranteed
regardless of the size of the market
Stability and auctions in labor markets with job security
Fu et al. (2016) introduced a stability concept for labor markets with job security. We show that their proposed outcomes form Nash equilibria of an auction where firms compete for workers. This parallels literature on stable outcomes and similar auctions, and yields new price of anarchy bounds