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