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Register a new userOptimal Deterministic Mechanisms for an Additive Buyer
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Added by
Aviad Rubinstein
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We study revenue maximization by deterministic mechanisms for the simplest case for which Myersons characterization does not hold: a single seller selling two items, with independently distributed values, to a single additive buyer. We prove that optimal mechanisms are submodular and hence monotone. Furthermore, we show that in the IID case, optimal mechanisms are symmetric. Our characterizations are surprisingly non-trivial, and we show that they fail to extend in several natural ways, e.g. for correlated distributions or more than two items. In particular, this shows that the optimality of symmetric mechanisms does not follow from the symmetry of the IID distribution.
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A patient seller aims to sell a good to an impatient buyer (i.e., one who discounts utility over time). The buyer will remain in the market for a period of time $T$, and her private value is drawn from a publicly known distribution. What is the revenue-optimal pricing-curve (sequence of (price, time) pairs) for the seller? Is randomization of help here? Is the revenue-optimal pricing-curve computable in polynomial time? We answer these questions in this paper. We give an efficient algorithm for computing the revenue-optimal pricing curve. We show that pricing curves, that post a price at each point of time and let the buyer pick her utility maximizing time to buy, are revenue-optimal among a much broader class of sequential lottery mechanisms: namely, mechanisms that allow the seller to post a menu of lotteries at each point of time cannot get any higher revenue than pricing curves. We also show that the even broader class of mechanisms that allow the menu of lotteries to be adaptively set, can earn strictly higher revenue than that of pricing curves, and the revenue gap can be as big as the support size of the buyers value distribution.
We consider a revenue-maximizing seller with $m$ heterogeneous items and a single buyer whose valuation $v$ for the items may exhibit both substitutes (i.e., for some $S, T$, $v(S cup T) < v(S) + v(T)$) and complements (i.e., for some $S, T$, $v(S cup T) > v(S) + v(T)$). We show that the mechanism first proposed by Babaioff et al. [2014] - the better of selling the items separately and bundling them together - guarantees a $Theta(d)$ fraction of the optimal revenue, where $d$ is a measure on the degree of complementarity. Note that this is the first approximately optimal mechanism for a buyer whose valuation exhibits any kind of complementarity, and extends the work of Rubinstein and Weinberg [2015], which proved that the same simple mechanisms achieve a constant factor approximation when buyer valuations are subadditive, the most general class of complement-free valuations. Our proof is enabled by the recent duality framework developed in Cai et al. [2016], which we use to obtain a bound on the optimal revenue in this setting. Our main technical contributions are specialized to handle the intricacies of settings with complements, and include an algorithm for partitioning edges in a hypergraph. Even nailing down the right model and notion of degree of complementarity to obtain meaningful results is of interest, as the natural extensions of previous definitions provably fail.
Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real agents exhibit risk attitudes which cannot be captured by any expected utility model. We initiate the study of revenue-optimal mechanisms under buyer behavioral models beyond expected utility theory. We adopt a model from prospect theory which arose to explain these discrepancies and incorporates agents under-weighting uncertain outcomes. In our model, an event occurring with probability $x < 1$ is worth strictly less to the agent than $x$ times the value of the event when it occurs with certainty. In contrast to the risk-neutral setting, the optimal mechanism may be randomized and appears challenging to find, even for a single buyer and a single item for sale. Nevertheless, we give a characterization of the optimal mechanism which enables positive approximation results. In particular, we show that under a reasonable bounded-risk-aversion assumption, posted pricing obtains a constant approximation. Notably, this result is risk-robust in that it does not depend on the details of the buyers risk attitude. Finally, we examine a dynamic setting in which the buyer is uncertain about his future value. In contrast to positive results for a risk-neutral buyer, we show that the buyers risk aversion may prevent the seller from approximating the optimal revenue in a risk-robust manner.
The Competition Complexity of an auction setting refers to the number of additional bidders necessary in order for the (deterministic, prior-independent, dominant strategy truthful) Vickrey-Clarke-Groves mechanism to achieve greater revenue than the (randomized, prior-dependent, Bayesian-truthful) optimal mechanism without the additional bidders. We prove that the competition complexity of $n$ bidders with additive valuations over $m$ independent items is at most $n(ln(1+m/n)+2)$, and also at most $9sqrt{nm}$. When $n leq m$, the first bound is optimal up to constant factors, even when the items are i.i.d. and regular. When $n geq m$, the second bound is optimal for the benchmark introduced in [EFFTW17a] up to constant factors, even when the items are i.i.d. and regular. We further show that, while the Eden et al. benchmark is not necessarily tight in the $n geq m$ regime, the competition complexity of $n$ bidders with additive valuations over even $2$ i.i.d. regular items is indeed $omega(1)$. Our main technical contribution is a reduction from analyzing the Eden et al. benchmark to proving stochastic dominance of certain random variables.
In this paper, we design gross product maximization mechanisms which incentivize users to upload high-quality contents on user-generated-content (UGC) websites. We show that, the proportional division mechanism, which is widely used in practice, can perform arbitrarily bad in the worst case. The problem can be formulated using a linear program with bounded and increasing variables. We then present an $O(nlog n)$ algorithm to find the optimal mechanism, where n is the number of players.
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