Nicolas S. Lambert

Complexity of combinatorial market makers

We analyze the computational complexity of market maker pricing algorithms for combinatorial prediction markets. We focus on Hansons popular logarithmic market scoring rule market maker (LMSR). Our goal is to implicitly maintain correct LMSR prices across an exponentially large outcome space. We examine both permutation combinatorics, where outcomes are permutations of objects, and Boolean combinatorics, where outcomes are combinations of binary events. We look at three restrictive languages that limit what traders can bet on. Even with severely limited languages, we find that LMSR pricing is #P-hard, even when the same language admits polynomial-time matching without the market maker. We then propose an approximation technique for pricing permutation markets based on an algorithm for online permutation learning. The connections we draw between LMSR pricing and the literature on online learning with expert advice may be of independent interest.

Eliciting truthful answers to multiple-choice questions

Motivated by the prevalence of online questionnaires in electronic commerce, and of multiple-choice questions in such questionnaires, we consider the problem of eliciting truthful answers to multiple-choice questions from a knowledgeable respondent. Specifically, each question is a statement regarding an uncertain future event, and is multiple-choice -- the responder must select exactly one of the given answers. The principal offers a payment, whose amount is a function of the answer selected and the true outcome (which the principal will eventually observe). This problem significantly generalizes recent work on truthful elicitation of distribution properties, which itself generalized a long line of work in elicitation of complete distributions. We provide necessary and sufficient conditions for the existence of payments that induce truthful answers, and give a characterization of those payments. We also study in greater details the common case of questions with ordinal answers, and illustrate our results with several examples of practical interest.

Self-financed wagering mechanisms for forecasting

We examine a class of wagering mechanisms designed to elicit truthful predictions from a group of people without requiring any outside subsidy. We propose a number of desirable properties for wagering mechanisms, identifying one mechanism - weighted-score wagering - that satisfies all of the properties. Moreover, we show that a single-parameter generalization of weighted-score wagering is the only mechanism that satisfies these properties. We explore some variants of the core mechanism based on practical considerations.

Auctions for social lending: A theoretical analysis☆

Prosper, today the second largest social lending marketplace with nearly 1.5 million members and

Truthful Surveys

380 million in funded loans, employed an auction mechanism amongst lenders to finance each borrowers loan until 2010. Given that a basic premise of social lending is cheap loans for borrowers, how does the Prosper auction do in terms of the borrowers payment, when lenders are strategic agents with private true interest rates? We first analyze the Prosper auction as a game of complete information and fully characterize its Nash equilibria, and show that the uniform-price Prosper mechanism, while simple, can lead to much larger payments for the borrower than the VCG mechanism. We next compare the Prosper mechanism against the borrower-optimal auction in an incomplete information setting, and conclude by examining the Prosper mechanism when modeled as a dynamic auction, and provide tight bounds on the price for a general class of bidding strategies.

An axiomatic characterization of wagering mechanisms

We consider the problem of truthfully sampling opinions of a population for statistical analysis purposes, such as estimating the population distribution of opinions. To obtain accurate results, the surveyor must incentivize individuals to report unbiased opinions. We present a rewarding scheme to elicit opinions that are representative of the population. In contrast with the related literature, we do not assume a specific information structure. In particular, our method does not rely on a common prior assumption.

Asymptotically optimal repeated auctions for sponsored search

We construct a budget-balanced wagering mechanism that flexibly extracts information about event probabilities, as well as the mean, median, and other statistics from a group of individuals whose beliefs are immutable to the actions of others. We show how our mechanism, called the Brier betting mechanism, arises naturally from a modified parimutuel betting market. We prove that it is essentially the unique wagering mechanism that is anonymous, proportional, sybilproof, and homogeneous. While the Brier betting mechanism is designed for individuals with immutable beliefs, we find that it continues to perform well even for Bayesian individuals who learn from the actions of others.

Only valuable experts can be valued

We investigate asymptotically optimal keyword auctions, that is, auctions which maximize revenue as the number of bidders grows. We do so under two alternative behavioral assumptions. The first explicitly models the repeated nature of keyword auctions. It introduces a novel assumption on individual bidding, namely that bidders never overbid their value, and bid their actual value if shut out for long enough. Under these conditions we present a broad class of repeated auctions that are asymptotically optimal among all sequential auctions (a superset of repeated auctions). Those auctions have varying payment schemes but share the ranking method. The Google auction belongs to this class, but not the Yahoo auction, and indeed we show that the latter is not asymptotically optimal. (Nonetheless, with some additional distributional assumptions, the Yahoo auction can be shown to belong to a broad category of auctions that are asymptotically optimal among all auction mechanisms that do not rely on ad relevance.) We then look at the one-shot keyword auction, which can be taken to model repeated auctions in which relatively myopic bidders converge on the equilibrium of the full-information stage game. In this case we show that the Google auction remains asymptotically optimal and the Yahoo auction suboptimal. The distributional assumptions under which our theorems hold are quite general. We do however show that the Google auction is not asymptotically revenue-maximizing for general distributions.

Strategic trading in informationally complex environments

Suppose a principal Alice wishes to reduce her uncertainty regarding some future payoff. Consider a self-proclaimed expert Bob that may either be an informed expert knowing an exact (or approximate) distribution of a future random outcome that may affect Alice’s utility, or an uninformed expert who knows nothing more than Alice does. Alice would like to hire Bob and solicit his signal. Her goal is to incentivize an informed expert to accept the contract and reveal his knowledge while deterring an uninformed expert from accepting the contract altogether. The starting point of this work is a powerful negative result (Olszewski and Sandroni, 2007), which tells us that in the general case for any contract which guarantees an informed expert some positive payoff an uninformed expert (with no extra knowledge) has a strategy which guarantees him a positive payoff as well. At the face of this negative result, we reexamine the notion of an expert and conclude that knowing some hidden variable (i.e., the description of the aforementioned distribution), does not make Bob an expert, or at least not a “valuable expert”. The premise of our paper is that if Alice only tries to incentivize experts which are valuable to her decision making then she can indeed screen them from uninformed experts. On a more technical level, we consider the case where Bob’s signal about the distribution of a future event cannot be an arbitrary distribution but rather comes from some subset P of all possible distributions. We give rather tight conditions on P (which relate to its convexity), under which screening is possible. We formalize our intuition that if these conditions are not met then an expert is not guaranteed to be valuable. We give natural and arguably useful scenarios where indeed such a restriction on the distribution arise.

Collective Choice in Dynamic Public Good Provision

We study trading behavior and the properties of prices in informationally complex markets. Our model is based on the single-period version of the linear-normal framework of [Kyle 1985]. We allow for essentially arbitrary correlations among the random variables involved in the model: the true value of the traded asset, the signals of strategic traders, the signals of competitive market makers, and the demand coming from liquidity traders. We first show that there always exists a unique linear equilibrium, characterize it analytically, and illustrate its properties in a series of examples. We then use this equilibrium characterization to study the informational efficiency of prices as the number of strategic traders becomes large. If the demand from liquidity traders is uncorrelated with the true value of the asset or is positively correlated with it (conditional on other signals), then prices in large markets aggregate all available information. If, however, the demand from liquidity traders is negatively correlated with the true value of the asset, then prices in large markets aggregate all available information except that contained in liquidity demand.