Algorithmic Bayesian Persuasion

Persuasion, defined as the act of exploiting an informational advantage in order to effect the decisions of others, is ubiquitous. Indeed, persuasive communication has been estimated to account for almost a third of all economic activity in the US. This paper examines persuasion through a computational lens, focusing on what is perhaps the most basic and fundamental model in this space: the celebrated Bayesian persuasion model of Kamenica and Gentzkow. Here there are two players, a sender and a receiver. The receiver must take one of a number of actions with a-priori unknown payoff, and the sender has access to additional information regarding the payoffs. The sender can commit to revealing a noisy signal regarding the realization of the payoffs of various actions, and would like to do so as to maximize her own payoff assuming a perfectly rational receiver. We examine the sender's optimization task in three of the most natural input models for this problem, and essentially pin down its computational complexity in each. When the payoff distributions of the different actions are i.i.d. and given explicitly, we exhibit a polynomial-time (exact) algorithm, and a "simple" $(1-1/e)$-approximation algorithm. Our optimal scheme for the i.i.d. setting involves an analogy to auction theory, and makes use of Border's characterization of the space of reduced-forms for single-item auctions. When action payoffs are independent but non-identical with marginal distributions given explicitly, we show that it is #P-hard to compute the optimal expected sender utility. Finally, we consider a general (possibly correlated) joint distribution of action payoffs presented by a black box sampling oracle, and exhibit a fully polynomial-time approximation scheme (FPTAS) with a bi-criteria guarantee. We show that this result is the best possible in the black-box model for information-theoretic reasons

Envy Freedom and Prior-free Mechanism Design

We consider the provision of an abstract service to single-dimensional agents. Our model includes position auctions, single-minded combinatorial auctions, and constrained matching markets. When the agents' values are drawn from a distribution, the Bayesian optimal mechanism is given by Myerson (1981) as a virtual-surplus optimizer. We develop a framework for prior-free mechanism design and analysis. A good mechanism in our framework approximates the optimal mechanism for the distribution if there is a distribution; moreover, when there is no distribution this mechanism still performs well. We define and characterize optimal envy-free outcomes in symmetric single-dimensional environments. Our characterization mirrors Myerson's theory. Furthermore, unlike in mechanism design where there is no point-wise optimal mechanism, there is always a point-wise optimal envy-free outcome. Envy-free outcomes and incentive-compatible mechanisms are similar in structure and performance. We therefore use the optimal envy-free revenue as a benchmark for measuring the performance of a prior-free mechanism. A good mechanism is one that approximates the envy free benchmark on any profile of agent values. We show that good mechanisms exist, and in particular, a natural generalization of the random sampling auction of Goldberg et al. (2001) is a constant approximation

Efficiency and Information Aggregation in Auctions with Costly Information

Consider an auction in which $k$ identical objects are sold to $n>k$ bidders who each have a value for one object which can have both private and common components to it. Private information concerning the common component of the object is not exogenously given, but rather endogenous and bidders face a cost to becoming informed. If the cost of information is not prohibitively high, then the equilibrium price in a uniform price auction will not aggregate private information, in contrast to the costless information case. Moreover, for a wide class of auctions if the cost of information is not prohibitively high then the objects can only be allocated in a weakly efficient sense, and then only if the equilibrium proportion of endogenously informed agents is vanishing as the economy grows. In spite of these results, it is shown that there is a mechanism for which there exist equilibria and for which (weak) efficiency is achieved as the economy grows in the face of endogenous information acquisition.Auctions, Efficiency, Information Acquisition, Information Aggregation

Auctions with Severely Bounded Communication

We study auctions with severe bounds on the communication allowed: each bidder may only transmit t bits of information to the auctioneer. We consider both welfare- and profit-maximizing auctions under this communication restriction. For both measures, we determine the optimal auction and show that the loss incurred relative to unconstrained auctions is mild. We prove non-surprising properties of these kinds of auctions, e.g., that in optimal mechanisms bidders simply report the interval in which their valuation lies in, as well as some surprising properties, e.g., that asymmetric auctions are better than symmetric ones and that multi-round auctions reduce the communication complexity only by a linear factor

Allocative and Informational Externalities in Auctions and Related Mechanisms

We study the effects of allocative and informational externalities in (multi-object) auctions and related mechanisms. Such externalities naturally arise in models that embed auctions in larger economic contexts. In particular, they appear when there is downstream interaction among bidders after the auction has closed. The endogeneity of valuations is the main driving force behind many new, specific phenomena with allocative externalities: even in complete information settings, traditional auction formats need not be efficient, and they may give rise to multiple equilibria and strategic non-participation. But, in the absence of informational externalities, welfare maximization can be achieved by Vickrey-Clarke- Groves mechanisms. Welfare-maximizing Bayes-Nash implementation is, however, impossible in multi-object settings with informational externalities, unless the allocation problem is separable across objects (e.g. there are no allocative externalities nor complementarities) or signals are one-dimensional. Moreover, implementation of any choice function via ex-post equilibrium is generically impossible with informational externalities and multidimensional types. A theory of information constraints with multidimensional signals is rather complex, but indispensable for our study

The Value of Knowing Your Enemy

Many auction settings implicitly or explicitly require that bidders are treated equally ex-ante. This may be because discrimination is philosophically or legally impermissible, or because it is practically difficult to implement or impossible to enforce. We study so-called {\em anonymous} auctions to understand the revenue tradeoffs and to develop simple anonymous auctions that are approximately optimal. We consider digital goods settings and show that the optimal anonymous, dominant strategy incentive compatible auction has an intuitive structure --- imagine that bidders are randomly permuted before the auction, then infer a posterior belief about bidder i's valuation from the values of other bidders and set a posted price that maximizes revenue given this posterior. We prove that no anonymous mechanism can guarantee an approximation better than O(n) to the optimal revenue in the worst case (or O(log n) for regular distributions) and that even posted price mechanisms match those guarantees. Understanding that the real power of anonymous mechanisms comes when the auctioneer can infer the bidder identities accurately, we show a tight O(k) approximation guarantee when each bidder can be confused with at most k "higher types". Moreover, we introduce a simple mechanism based on n target prices that is asymptotically optimal and build on this mechanism to extend our results to m-unit auctions and sponsored search

Efficient Design with Interdependent Valuations

We study efficient, Bayes-Nash incentive compatible mechanisms in a social choice setting that allows for informational and allocative externalities. We show that such mechanisms exist only if a congruence condition relating private and social rates of information substitution is satisfied. If signals are multidimensional, the congruence condition is determined by an integrability constraint, and it can hold only in non-generic cases such as the private value case or the symmetric case. If signals are one-dimensional, the congruence condition reduces to a monotonicity constraint and it can be generically satisfied. We apply the results to the study of multi-object auctions, and we discuss why such auctions cannot be reduced to one-dimensional models without loss of generality.