A General Theory of Sample Complexity for Multi-Item Profit Maximization

The design of profit-maximizing multi-item mechanisms is a notoriously challenging problem with tremendous real-world impact. The mechanism designer's goal is to field a mechanism with high expected profit on the distribution over buyers' values. Unfortunately, if the set of mechanisms he optimizes over is complex, a mechanism may have high empirical profit over a small set of samples but low expected profit. This raises the question, how many samples are sufficient to ensure that the empirically optimal mechanism is nearly optimal in expectation? We uncover structure shared by a myriad of pricing, auction, and lottery mechanisms that allows us to prove strong sample complexity bounds: for any set of buyers' values, profit is a piecewise linear function of the mechanism's parameters. We prove new bounds for mechanism classes not yet studied in the sample-based mechanism design literature and match or improve over the best known guarantees for many classes. The profit functions we study are significantly different from well-understood functions in machine learning, so our analysis requires a sharp understanding of the interplay between mechanism parameters and buyer values. We strengthen our main results with data-dependent bounds when the distribution over buyers' values is "well-behaved." Finally, we investigate a fundamental tradeoff in sample-based mechanism design: complex mechanisms often have higher profit than simple mechanisms, but more samples are required to ensure that empirical and expected profit are close. We provide techniques for optimizing this tradeoff

Valuation equilibrium

We introduce a new solution concept for games in extensive form with perfect information, valuation equilibrium, which is based on a partition of each player's moves into similarity classes. A valuation of a player'is a real-valued function on the set of her similarity classes. In this equilibrium each player's strategy is optimal in the sense that at each of her nodes, a player chooses a move that belongs to a class with maximum valuation. The valuation of each player is consistent with the strategy profile in the sense that the valuation of a similarity class is the player's expected payoff, given that the path (induced by the strategy profile) intersects the similarity class. The solution concept is applied to decision problems and multi-player extensive form games. It is contrasted with existing solution concepts. The valuation approach is next applied to stopping games, in which non-terminal moves form a single similarity class, and we note that the behaviors obtained echo some biases observed experimentally. Finally, we tentatively suggest a way of endogenizing the similarity partitions in which moves are categorized according to how well they perform relative to the expected equilibrium value, interpreted as the aspiration level

Controlled Information Fusion with Risk-Averse CVaR Social Sensors

Consider a multi-agent network comprised of risk averse social sensors and a controller that jointly seek to estimate an unknown state of nature, given noisy measurements. The network of social sensors perform Bayesian social learning - each sensor fuses the information revealed by previous social sensors along with its private valuation using Bayes' rule - to optimize a local cost function. The controller sequentially modifies the cost function of the sensors by discriminatory pricing (control inputs) to realize long term global objectives. We formulate the stochastic control problem faced by the controller as a Partially Observed Markov Decision Process (POMDP) and derive structural results for the optimal control policy as a function of the risk-aversion factor in the Conditional Value-at-Risk (CVaR) cost function of the sensors. We show that the optimal price sequence when the sensors are risk- averse is a super-martingale; i.e, it decreases on average over time.Comment: IEEE CDC 201