5 research outputs found
Efficient Stackelberg Strategies for Finitely Repeated Games
We study the problem of efficiently computing optimal strategies in
asymmetric leader-follower games repeated a finite number of times, which
presents a different set of technical challenges than the infinite-horizon
setting. More precisely, we give efficient algorithms for finding approximate
Stackelberg equilibria in finite-horizon repeated two-player games, along with
rates of convergence depending on the horizon . We give two algorithms, one
computing strategies with an optimal rate at the expense of an
exponential dependence on the number of actions, and another (randomized)
approach computing strategies with no dependence on the number of actions but a
worse dependence on of . Both algorithms build upon a
linear program to produce simple automata leader strategies and induce
corresponding automata best-responses for the follower. We complement these
results by showing that approximating the Stackelberg value in three-player
finite-horizon repeated games is a computationally hard problem via a reduction
from the balanced vertex cover problem.Comment: An earlier version of this paper used incorrect asymptotic notation
in the statement of the main hardness result as well as in the description of
related hardness results in a table (in the related work section). The proofs
and implication of the result remain unchanged, but a correction has been
made to the statement of the resul
Fair Division with a Secretive Agent
We study classic fair-division problems in a partial information setting.
This paper respectively addresses fair division of rent, cake, and indivisible
goods among agents with cardinal preferences. We will show that, for all of
these settings and under appropriate valuations, a fair (or an approximately
fair) division among n agents can be efficiently computed using only the
valuations of n-1 agents. The nth (secretive) agent can make an arbitrary
selection after the division has been proposed and, irrespective of her choice,
the computed division will admit an overall fair allocation.
For the rent-division setting we prove that the (well-behaved) utilities of
n-1 agents suffice to find a rent division among n rooms such that, for every
possible room selection of the secretive agent, there exists an allocation (of
the remaining n-1 rooms among the n-1 agents) which ensures overall envy
freeness (fairness). We complement this existential result by developing a
polynomial-time algorithm that finds such a fair rent division under
quasilinear utilities.
In this partial information setting, we also develop efficient algorithms to
compute allocations that are envy-free up to one good (EF1) and
epsilon-approximate envy free. These two notions of fairness are applicable in
the context of indivisible goods and divisible goods (cake cutting),
respectively. This work also addresses fairness in terms of proportionality and
maximin shares. Our key result here is an efficient algorithm that, even with a
secretive agent, finds a 1/19-approximate maximin fair allocation (of
indivisible goods) under submodular valuations of the non-secretive agents.
One of the main technical contributions of this paper is the development of
novel connections between different fair-division paradigms, e.g., we use our
existential results for envy-free rent-division to develop an efficient EF1
algorithm.Comment: 27 page
Pipeline Interventions
We introduce the \emph{pipeline intervention} problem, defined by a layered
directed acyclic graph and a set of stochastic matrices governing transitions
between successive layers. The graph is a stylized model for how people from
different populations are presented opportunities, eventually leading to some
reward. In our model, individuals are born into an initial position (i.e. some
node in the first layer of the graph) according to a fixed probability
distribution, and then stochastically progress through the graph according to
the transition matrices, until they reach a node in the final layer of the
graph; each node in the final layer has a \emph{reward} associated with it. The
pipeline intervention problem asks how to best make costly changes to the
transition matrices governing people's stochastic transitions through the
graph, subject to a budget constraint. We consider two objectives: social
welfare maximization, and a fairness-motivated maximin objective that seeks to
maximize the value to the population (starting node) with the \emph{least}
expected value. We consider two variants of the maximin objective that turn out
to be distinct, depending on whether we demand a deterministic solution or
allow randomization. For each objective, we give an efficient approximation
algorithm (an additive FPTAS) for constant width networks. We also tightly
characterize the "price of fairness" in our setting: the ratio between the
highest achievable social welfare and the highest social welfare consistent
with a maximin optimal solution. Finally we show that for polynomial width
networks, even approximating the maximin objective to any constant factor is NP
hard, even for networks with constant depth. This shows that the restriction on
the width in our positive results is essential