183 research outputs found
Convex-Concave Min-Max Stackelberg Games
Min-max optimization problems (i.e., min-max games) have been attracting a
great deal of attention because of their applicability to a wide range of
machine learning problems. Although significant progress has been made
recently, the literature to date has focused on games with independent strategy
sets; little is known about solving games with dependent strategy sets, which
can be characterized as min-max Stackelberg games. We introduce two first-order
methods that solve a large class of convex-concave min-max Stackelberg games,
and show that our methods converge in polynomial time. Min-max Stackelberg
games were first studied by Wald, under the posthumous name of Wald's maximin
model, a variant of which is the main paradigm used in robust optimization,
which means that our methods can likewise solve many convex robust optimization
problems. We observe that the computation of competitive equilibria in Fisher
markets also comprises a min-max Stackelberg game. Further, we demonstrate the
efficacy and efficiency of our algorithms in practice by computing competitive
equilibria in Fisher markets with varying utility structures. Our experiments
suggest potential ways to extend our theoretical results, by demonstrating how
different smoothness properties can affect the convergence rate of our
algorithms.Comment: 25 pages, 4 tables, 1 figure, Forthcoming in NeurIPS 202
Zero-Sum Stochastic Stackelberg Games
Zero-sum stochastic games have found important applications in a variety of
fields, from machine learning to economics. Work on this model has primarily
focused on the computation of Nash equilibrium due to its effectiveness in
solving adversarial board and video games. Unfortunately, a Nash equilibrium is
not guaranteed to exist in zero-sum stochastic games when the payoffs at each
state are not convex-concave in the players' actions. A Stackelberg
equilibrium, however, is guaranteed to exist. Consequently, in this paper, we
study zero-sum stochastic Stackelberg games. Going beyond known existence
results for (non-stationary) Stackelberg equilibria, we prove the existence of
recursive (i.e., Markov perfect) Stackelberg equilibria (recSE) in these games,
provide necessary and sufficient conditions for a policy profile to be a recSE,
and show that recSE can be computed in (weakly) polynomial time via value
iteration. Finally, we show that zero-sum stochastic Stackelberg games can
model the problem of pricing and allocating goods across agents and time. More
specifically, we propose a zero-sum stochastic Stackelberg game whose recSE
correspond to the recursive competitive equilibria of a large class of
stochastic Fisher markets. We close with a series of experiments that showcase
how our methodology can be used to solve the consumption-savings problem in
stochastic Fisher markets.Comment: 29 pages 2 figures, Appeared in NeurIPS'2
Fisher Markets with Social Influence
A Fisher market is an economic model of buyer and seller interactions in
which each buyer's utility depends only on the bundle of goods she obtains.
Many people's interests, however, are affected by their social interactions
with others. In this paper, we introduce a generalization of Fisher markets,
namely influence Fisher markets, which captures the impact of social influence
on buyers' utilities. We show that competitive equilibria in influence Fisher
markets correspond to generalized Nash equilibria in an associated pseudo-game,
which implies the existence of competitive equilibria in all influence Fisher
markets with continuous and concave utility functions. We then construct a
monotone pseudo-game, whose variational equilibria and their duals together
characterize competitive equilibria in influence Fisher markets with
continuous, jointly concave, and homogeneous utility functions. This
observation implies that competitive equilibria in these markets can be
computed in polynomial time under standard smoothness assumptions on the
utility functions. The dual of this second pseudo-game enables us to interpret
the competitive equilibria of influence CCH Fisher markets as the solutions to
a system of simultaneous Stackelberg games. Finally, we derive a novel
first-order method that solves this Stackelberg system in polynomial time,
prove that it is equivalent to computing competitive equilibrium prices via
t\^{a}tonnement, and run experiments that confirm our theoretical results
A Computational Approach to Compare Information Revelation Policies
Revelation policies in an electronic marketplace differ in terms of the level of competitive information disseminated to participating sellers. Since sellers who repeatedly compete against one another learn based on the information revealed and alter their future bidding behavior, revelation policies affect welfare parameters—consumer surplus, producer surplus, and social welfare—of the market. Although different revelation policies are adopted in several traditional and Web-based marketplaces, prior work has not studied the implications of these policies on the performance of a market. In this paper, we study and compare a set of revelation policies using a computational marketplace. Specifically, we study this in the context of a reverse-market where each seller’s decision problem of choosing an optimal bid is modeled as an MDP (Markov decision process). Results and analysis presented in this paper are based on market sessions executed using the computational marketplace. The computational model, which employs a machine-learning technique proposed in this paper, ties the simulation results to the model developed using the game-theoretic models. In addition to this, the computational model allows us to relax assumptions of the game-theoretic models and study the problem under a more realistic scenario. Insights gained from this paper will be useful in guiding the buyer in choosing the appropriate policy
Response to the Letter to the Editor, “Depression not related to lower religious involvement in bipolar disorders?”
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/79118/1/j.1399-5618.2010.00842.x.pd
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