1,476 research outputs found
Mixture Selection, Mechanism Design, and Signaling
We pose and study a fundamental algorithmic problem which we term mixture
selection, arising as a building block in a number of game-theoretic
applications: Given a function from the -dimensional hypercube to the
bounded interval , and an matrix with bounded entries,
maximize over in the -dimensional simplex. This problem arises
naturally when one seeks to design a lottery over items for sale in an auction,
or craft the posterior beliefs for agents in a Bayesian game through the
provision of information (a.k.a. signaling).
We present an approximation algorithm for this problem when
simultaneously satisfies two smoothness properties: Lipschitz continuity with
respect to the norm, and noise stability. The latter notion, which
we define and cater to our setting, controls the degree to which
low-probability errors in the inputs of can impact its output. When is
both -Lipschitz continuous and -stable, we obtain an (additive)
PTAS for mixture selection. We also show that neither assumption suffices by
itself for an additive PTAS, and both assumptions together do not suffice for
an additive FPTAS.
We apply our algorithm to different game-theoretic applications from
mechanism design and optimal signaling. We make progress on a number of open
problems suggested in prior work by easily reducing them to mixture selection:
we resolve an important special case of the small-menu lottery design problem
posed by Dughmi, Han, and Nisan; we resolve the problem of revenue-maximizing
signaling in Bayesian second-price auctions posed by Emek et al. and Miltersen
and Sheffet; we design a quasipolynomial-time approximation scheme for the
optimal signaling problem in normal form games suggested by Dughmi; and we
design an approximation algorithm for the optimal signaling problem in the
voting model of Alonso and C\^{a}mara
Signaling in Quasipolynomial time
Strategic interactions often take place in an environment rife with
uncertainty. As a result, the equilibrium of a game is intimately related to
the information available to its players. The \emph{signaling problem}
abstracts the task faced by an informed "market maker", who must choose how to
reveal information in order to effect a desirable equilibrium.
In this paper, we consider two fundamental signaling problems: one for
abstract normal form games, and the other for single item auctions. For the
former, we consider an abstract class of objective functions which includes the
social welfare and weighted combinations of players' utilities, and for the
latter we restrict our attention to the social welfare objective and to
signaling schemes which are constrained in the number of signals used. For both
problems, we design approximation algorithms for the signaling problem which
run in quasi-polynomial time under various conditions, extending and
complementing the results of various recent works on the topic.
Underlying each of our results is a "meshing scheme" which effectively
overcomes the "curse of dimensionality" and discretizes the space of
"essentially different" posterior beliefs -- in the sense of inducing
"essentially different" equilibria. This is combined with an algorithm for
optimally assembling a signaling scheme as a convex combination of such
beliefs. For the normal form game setting, the meshing scheme leads to a convex
partition of the space of posterior beliefs and this assembly procedure is
reduced to a linear program, and in the auction setting the assembly procedure
is reduced to submodular function maximization
Algorithmic Bayesian Persuasion
Persuasion, defined as the act of exploiting an informational advantage in
order to effect the decisions of others, is ubiquitous. Indeed, persuasive
communication has been estimated to account for almost a third of all economic
activity in the US. This paper examines persuasion through a computational
lens, focusing on what is perhaps the most basic and fundamental model in this
space: the celebrated Bayesian persuasion model of Kamenica and Gentzkow. Here
there are two players, a sender and a receiver. The receiver must take one of a
number of actions with a-priori unknown payoff, and the sender has access to
additional information regarding the payoffs. The sender can commit to
revealing a noisy signal regarding the realization of the payoffs of various
actions, and would like to do so as to maximize her own payoff assuming a
perfectly rational receiver.
We examine the sender's optimization task in three of the most natural input
models for this problem, and essentially pin down its computational complexity
in each. When the payoff distributions of the different actions are i.i.d. and
given explicitly, we exhibit a polynomial-time (exact) algorithm, and a
"simple" -approximation algorithm. Our optimal scheme for the i.i.d.
setting involves an analogy to auction theory, and makes use of Border's
characterization of the space of reduced-forms for single-item auctions. When
action payoffs are independent but non-identical with marginal distributions
given explicitly, we show that it is #P-hard to compute the optimal expected
sender utility. Finally, we consider a general (possibly correlated) joint
distribution of action payoffs presented by a black box sampling oracle, and
exhibit a fully polynomial-time approximation scheme (FPTAS) with a bi-criteria
guarantee. We show that this result is the best possible in the black-box model
for information-theoretic reasons
On the Hardness of Signaling
There has been a recent surge of interest in the role of information in
strategic interactions. Much of this work seeks to understand how the realized
equilibrium of a game is influenced by uncertainty in the environment and the
information available to players in the game. Lurking beneath this literature
is a fundamental, yet largely unexplored, algorithmic question: how should a
"market maker" who is privy to additional information, and equipped with a
specified objective, inform the players in the game? This is an informational
analogue of the mechanism design question, and views the information structure
of a game as a mathematical object to be designed, rather than an exogenous
variable.
We initiate a complexity-theoretic examination of the design of optimal
information structures in general Bayesian games, a task often referred to as
signaling. We focus on one of the simplest instantiations of the signaling
question: Bayesian zero-sum games, and a principal who must choose an
information structure maximizing the equilibrium payoff of one of the players.
In this setting, we show that optimal signaling is computationally intractable,
and in some cases hard to approximate, assuming that it is hard to recover a
planted clique from an Erdos-Renyi random graph. This is despite the fact that
equilibria in these games are computable in polynomial time, and therefore
suggests that the hardness of optimal signaling is a distinct phenomenon from
the hardness of equilibrium computation. Necessitated by the non-local nature
of information structures, en-route to our results we prove an "amplification
lemma" for the planted clique problem which may be of independent interest
When Are Welfare Guarantees Robust?
Computational and economic results suggest that social welfare maximization and combinatorial auction design are much easier when bidders\u27 valuations satisfy the "gross substitutes" condition. The goal of this paper is to evaluate rigorously the folklore belief that the main take-aways from these results remain valid in settings where the gross substitutes condition holds only approximately. We show that for valuations that pointwise approximate a gross substitutes valuation (in fact even a linear valuation), optimal social welfare cannot be approximated to within a subpolynomial factor and demand oracles cannot be simulated using a subexponential number of value queries. We then provide several positive results by imposing additional structure on the valuations (beyond gross substitutes), using a more stringent notion of approximation, and/or using more powerful oracle access to the valuations. For example, we prove that the performance of the greedy algorithm degrades gracefully for near-linear valuations with approximately decreasing marginal values; that with demand queries, approximate welfare guarantees for XOS valuations degrade gracefully for valuations that are pointwise close to XOS; and that the performance of the Kelso-Crawford auction degrades gracefully for valuations that are close to various subclasses of gross substitutes valuations
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