Inefficiency of equilibria in digital mechanisms with continuous valuations.

A digital mechanism is defined as an iterative procedure in which bidders select an action, from a finite set, in each iteration. When bidders have continuous valuations and make strategic reports, we show that any ex post implementation of the vickrey choice rule via such a mechanism needs infinitely many iterations for almost all realizations of the bidders’ valuations. Thus, when valuations are drawn from a continuous probability distribution, the vickrey choice rule can only be used at the expense of a running time that is infinite with probability one. This infeasibility result even holds in the case of two bidders and the vickrey choice rule only being required to be established with probability one. Establishing the efficient allocation when the nn bidders’ report truthfully contrasts starkly to the previous setting: a bisection procedure has a finite running time almost always, and an expected number of reports are equal to 2n2n. Using a groves payment scheme rather than vickrey’s second price payment scheme somewhat mitigates the problem. We provide an example mechanism with a groves payment scheme, in which the running time of the mechanism in equilibrium is finite with probability 1212

Computational Efficiency Requires Simple Taxation

We characterize the communication complexity of truthful mechanisms. Our departure point is the well known taxation principle. The taxation principle asserts that every truthful mechanism can be interpreted as follows: every player is presented with a menu that consists of a price for each bundle (the prices depend only on the valuations of the other players). Each player is allocated a bundle that maximizes his profit according to this menu. We define the taxation complexity of a truthful mechanism to be the logarithm of the maximum number of menus that may be presented to a player. Our main finding is that in general the taxation complexity essentially equals the communication complexity. The proof consists of two main steps. First, we prove that for rich enough domains the taxation complexity is at most the communication complexity. We then show that the taxation complexity is much smaller than the communication complexity only in "pathological" cases and provide a formal description of these extreme cases. Next, we study mechanisms that access the valuations via value queries only. In this setting we establish that the menu complexity -- a notion that was already studied in several different contexts -- characterizes the number of value queries that the mechanism makes in exactly the same way that the taxation complexity characterizes the communication complexity. Our approach yields several applications, including strengthening the solution concept with low communication overhead, fast computation of prices, and hardness of approximation by computationally efficient truthful mechanisms

Efficiency of scalar-parameterized mechanisms

We consider the problem of allocating a fixed amount of an infinitely divisible resource among multiple competing, fully rational users. We study the efficiency guarantees that are possible when we restrict to mechanisms that satisfy certain scalability constraints motivated by large scale communication networks; in particular, we restrict attention to mechanisms where users are restricted to one-dimensional strategy spaces. We first study the efficiency guarantees possible when the mechanism is not allowed to price differen- tiate. We study the worst-case efficiency loss (ratio of the utility associated with a Nash equilibrium to the maximum possible utility), and show that the proportional allocation mechanism of Kelly (1997) minimizes the efficiency loss when users are price anticipating. We then turn our attention to mechanisms where price differentiation is permitted; using an adaptation of the Vickrey-Clarke-Groves class of mechanisms, we con- struct a class of mechanisms with one-dimensional strategy spaces where Nash equilibria are fully efficient. These mechanisms are shown to be fully efficient even in general convex environments, under reasonable assumptions. Our results highlight a fundamental insight in mechanism design: when the pricing flexibility available to the mechanism designer is limited, restricting the strategic flexibility of bidders may actually improve the efficiency guarantee.National Science FoundationArmy Research OfficeDARPA - Next Generation Internet InitiativeNational Science Foundation Graduate Research Fellowshi

Essays on Microeconomic Theory.

The present work collects three essays on microeconomic theory. In the first essay, I study a model in which a finite number of men and women look for future spouses via random meetings. I ask whether equilibrium marriage outcomes are stable matchings when search frictions are small. The answer is they can but need not be. For any stable matching there is an equilibrium leading to it almost surely. However unstable---even Pareto-dominated---matchings may still arise with positive probability. In addition, inefficiency due to delay may remain significant despite vanishing search frictions. Finally, a condition is identified under which all equilibria are outcome equivalent, stable, and efficient. In the second essay, a joint work Kfir Eliaz, we model a competition between two teams as an all-pay auction with incomplete information. The teams may differ in size and individuals exert effort to increase the performance of one's own team via an additively separable aggregation function. The team with a higher performance wins, and its members enjoy the prize as a public good. The value of the prize is identical to members of the same team but is unknown to the other team. We show that there exists a unique monotone equilibrium in which everyone actively participates, and in this equilibrium a bigger team is more likely to win if the aggregation function is concave, less likely if convex, or equally likely if linear. In the third essay, I study a situation in which a group of people working on a common objective want to share information. Oftentimes information sharing via precise communication is impossible and instead information is aggregated by institutions within which communication is coarse. The paper proposes a unified framework for modeling a general class of such information-aggregating institutions. Within this class, it is shown that institution A outperforms institution B for any common objective if and only if the underlying communication infrastructure of A can be obtained from that of B by a sequence of elementary operations. Each operation either removes redundant communication instruments from B or introduces effective ones to it.PhDEconomicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133250/1/wqg_1.pd

The communication cost of selfishness

We consider how many bits need to be exchanged to implement a given decision rule when the mechanism must be ex post or Bayesian incentive compatible. For ex post incentive compatibility, the communication protocol must reveal enough information to calculate monetary transfers to the agents to motivate them to be truthful (agents' payoffs are assumed to be quasilinear in such transfers). For Bayesian incentive compatibility, the protocol may need to hide some information from the agents to prevent deviations contingent on the information. In both settings with selfish agents, the communication cost can be higher than in the case in which the agents are honest and can be relied upon to report truthfully. The increase is the "communication cost of selfishness." We provide an exponential upper bound on the increase. We show that the bound is tight in the Bayesian setting, but we do not know this in the ex post setting. We describe some cases where the communication cost of selfishness proves to be very low.Communication complexity Algorithmic mechanism design Bayesian incentive compatibility Ex post incentive compatibility Sequential and simultaneous communication protocols Information sets

The Communication Cost of Selfishness: Ex Post Implementation

We consider the communication complexity of implementing a given decision rule when the protocol must also calculate payments to motivate the agents to be honest in an ex post equilibrium (agents payoffs are assumed to be quasi-linear in such payments). We find that the communication cost of selfishness when measured with the average-case communication complexity may be arbitrarily large. For the worst-case communication complexity measure, we provide an exponential upper bound on the communication cost of selfishness. Whether this bound is ever achieved remains an open question. We examine several special cases in which the communication cost of selfishness proves to be very low. These include cases where we want to implement efficiency or where we have only two agents, and the precision of agents' utilities is fixed