Implementation in Advised Strategies: Welfare Guarantees from Posted-Price Mechanisms When Demand Queries Are NP-Hard

State-of-the-art posted-price mechanisms for submodular bidders with $m$ items achieve approximation guarantees of $O((\log \log m)^3)$ [Assadi and Singla, 2019]. Their truthfulness, however, requires bidders to compute an NP-hard demand-query. Some computational complexity of this form is unavoidable, as it is NP-hard for truthful mechanisms to guarantee even an $m^{1/2-\varepsilon}$-approximation for any $\varepsilon > 0$ [Dobzinski and Vondr\'ak, 2016]. Together, these establish a stark distinction between computationally-efficient and communication-efficient truthful mechanisms. We show that this distinction disappears with a mild relaxation of truthfulness, which we term implementation in advised strategies, and that has been previously studied in relation to "Implementation in Undominated Strategies" [Babaioff et al, 2009]. Specifically, advice maps a tentative strategy either to that same strategy itself, or one that dominates it. We say that a player follows advice as long as they never play actions which are dominated by advice. A poly-time mechanism guarantees an $\alpha$-approximation in implementation in advised strategies if there exists poly-time advice for each player such that an $\alpha$-approximation is achieved whenever all players follow advice. Using an appropriate bicriterion notion of approximate demand queries (which can be computed in poly-time), we establish that (a slight modification of) the [Assadi and Singla, 2019] mechanism achieves the same $O((\log \log m)^3)$-approximation in implementation in advised strategies

On Simultaneous Two-player Combinatorial Auctions

We consider the following communication problem: Alice and Bob each have some valuation functions $v_1(\cdot)$ and $v_2(\cdot)$ over subsets of $m$ items, and their goal is to partition the items into $S, \bar{S}$ in a way that maximizes the welfare, $v_1(S) + v_2(\bar{S})$. We study both the allocation problem, which asks for a welfare-maximizing partition and the decision problem, which asks whether or not there exists a partition guaranteeing certain welfare, for binary XOS valuations. For interactive protocols with $poly(m)$ communication, a tight 3/4-approximation is known for both [Fei06,DS06]. For interactive protocols, the allocation problem is provably harder than the decision problem: any solution to the allocation problem implies a solution to the decision problem with one additional round and $\log m$ additional bits of communication via a trivial reduction. Surprisingly, the allocation problem is provably easier for simultaneous protocols. Specifically, we show: 1) There exists a simultaneous, randomized protocol with polynomial communication that selects a partition whose expected welfare is at least $3/4$ of the optimum. This matches the guarantee of the best interactive, randomized protocol with polynomial communication. 2) For all $\varepsilon > 0$, any simultaneous, randomized protocol that decides whether the welfare of the optimal partition is $\geq 1$ or $\leq 3/4 - 1/108+\varepsilon$ correctly with probability $> 1/2 + 1/ poly(m)$ requires exponential communication. This provides a separation between the attainable approximation guarantees via interactive ($3/4$) versus simultaneous ($\leq 3/4-1/108$) protocols with polynomial communication. In other words, this trivial reduction from decision to allocation problems provably requires the extra round of communication

Condorcet-Consistent and Approximately Strategyproof Tournament Rules

We consider the manipulability of tournament rules for round-robin tournaments of $n$ competitors. Specifically, $n$ competitors are competing for a prize, and a tournament rule $r$ maps the result of all $\binom{n}{2}$ pairwise matches (called a tournament, $T$) to a distribution over winners. Rule $r$ is Condorcet-consistent if whenever $i$ wins all $n-1$ of her matches, $r$ selects $i$ with probability $1$. We consider strategic manipulation of tournaments where player $j$ might throw their match to player $i$ in order to increase the likelihood that one of them wins the tournament. Regardless of the reason why $j$ chooses to do this, the potential for manipulation exists as long as $\Pr[r(T) = i]$ increases by more than $\Pr[r(T) = j]$ decreases. Unfortunately, it is known that every Condorcet-consistent rule is manipulable (Altman and Kleinberg). In this work, we address the question of how manipulable Condorcet-consistent rules must necessarily be - by trying to minimize the difference between the increase in $\Pr[r(T) = i]$ and decrease in $\Pr[r(T) = j]$ for any potential manipulating pair. We show that every Condorcet-consistent rule is in fact $1/3$-manipulable, and that selecting a winner according to a random single elimination bracket is not $\alpha$-manipulable for any $\alpha > 1/3$. We also show that many previously studied tournament formats are all $1/2$-manipulable, and the popular class of Copeland rules (any rule that selects a player with the most wins) are all in fact $1$-manipulable, the worst possible. Finally, we consider extensions to match-fixing among sets of more than two players.Comment: 20 page

Reducing Revenue to Welfare Maximization: Approximation Algorithms and other Generalizations

It was recently shown in [http://arxiv.org/abs/1207.5518] that revenue optimization can be computationally efficiently reduced to welfare optimization in all multi-dimensional Bayesian auction problems with arbitrary (possibly combinatorial) feasibility constraints and independent additive bidders with arbitrary (possibly combinatorial) demand constraints. This reduction provides a poly-time solution to the optimal mechanism design problem in all auction settings where welfare optimization can be solved efficiently, but it is fragile to approximation and cannot provide solutions to settings where welfare maximization can only be tractably approximated. In this paper, we extend the reduction to accommodate approximation algorithms, providing an approximation preserving reduction from (truthful) revenue maximization to (not necessarily truthful) welfare maximization. The mechanisms output by our reduction choose allocations via black-box calls to welfare approximation on randomly selected inputs, thereby generalizing also our earlier structural results on optimal multi-dimensional mechanisms to approximately optimal mechanisms. Unlike [http://arxiv.org/abs/1207.5518], our results here are obtained through novel uses of the Ellipsoid algorithm and other optimization techniques over {\em non-convex regions}

Matroid prophet inequalities and Bayesian mechanism design

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 42-44).Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of p matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most 0(p), and this factor is also tight. Beyond their interest as theorems about pure online algoritms or optimal stopping rules, these results also have applications to mechanism design. Our results imply improved bounds on the ability of sequential posted-price mechanisms to approximate optimal mechanisms in both single-parameter and multi-parameter Bayesian settings. In particular, our results imply the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings. This work was done in collaboration with Robert Kleinberg.by S. Matthew Weinberg.S.M

Prophet Inequalities with Limited Information

In the classical prophet inequality, a gambler observes a sequence of stochastic rewards $V_1,...,V_n$ and must decide, for each reward $V_i$, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value $V_i$. The gambler's goal is to obtain a constant fraction of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose $k$ items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know the distribution from which the rewards $V_1,...,V_n$ are drawn. The assumption that the gambler knows the distribution from which $V_1,...,V_n$ are drawn is very strong. Instead, we work with the much simpler assumption that the gambler only knows a few samples from this distribution. We construct the first single-sample prophet inequalities for many settings of interest, whose guarantees all match the best possible asymptotically, \emph{even with full knowledge of the distribution}. Specifically, we provide a novel single-sample algorithm when the gambler can choose any $k$ elements whose analysis is based on random walks with limited correlation. In addition, we provide a black-box method for converting specific types of solutions to the related \emph{secretary problem} to single-sample prophet inequalities, and apply it to several existing algorithms. Finally, we provide a constant-sample prophet inequality for constant-degree bipartite matchings. We apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders