2,656 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
Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries
Suppose that an -simplex is partitioned into convex regions having
disjoint interiors and distinct labels, and we may learn the label of any point
by querying it. The learning objective is to know, for any point in the
simplex, a label that occurs within some distance from that point.
We present two algorithms for this task: Constant-Dimension Generalised Binary
Search (CD-GBS), which for constant uses queries, and Constant-Region Generalised Binary
Search (CR-GBS), which uses CD-GBS as a subroutine and for constant uses
queries.
We show via Kakutani's fixed-point theorem that these algorithms provide
bounds on the best-response query complexity of computing approximate
well-supported equilibria of bimatrix games in which one of the players has a
constant number of pure strategies. We also partially extend our results to
games with multiple players, establishing further query complexity bounds for
computing approximate well-supported equilibria in this setting.Comment: 38 pages, 7 figures, second version strengthens lower bound in
Theorem 6, adds footnotes with additional comments and fixes typo
Query Complexity of Approximate Nash Equilibria
We study the query complexity of approximate notions of Nash equilibrium in
games with a large number of players . Our main result states that for
-player binary-action games and for constant , the query
complexity of an -well-supported Nash equilibrium is exponential
in . One of the consequences of this result is an exponential lower bound on
the rate of convergence of adaptive dynamics to approxiamte Nash equilibrium
Model and Reinforcement Learning for Markov Games with Risk Preferences
We motivate and propose a new model for non-cooperative Markov game which
considers the interactions of risk-aware players. This model characterizes the
time-consistent dynamic "risk" from both stochastic state transitions (inherent
to the game) and randomized mixed strategies (due to all other players). An
appropriate risk-aware equilibrium concept is proposed and the existence of
such equilibria is demonstrated in stationary strategies by an application of
Kakutani's fixed point theorem. We further propose a simulation-based
Q-learning type algorithm for risk-aware equilibrium computation. This
algorithm works with a special form of minimax risk measures which can
naturally be written as saddle-point stochastic optimization problems, and
covers many widely investigated risk measures. Finally, the almost sure
convergence of this simulation-based algorithm to an equilibrium is
demonstrated under some mild conditions. Our numerical experiments on a two
player queuing game validate the properties of our model and algorithm, and
demonstrate their worth and applicability in real life competitive
decision-making.Comment: 38 pages, 6 tables, 5 figure
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