We consider the problem of maximizing revenue for a monopolist offering multiple items to multiple heterogeneous buyers. We develop a simple mechanism that obtains a constant factor approximation under the assumption that the buyers' values are additive subject to a matroid feasibility constraint and independent across items. Importantly, different buyers in our setting can have different constraints on the sets of items they desire. Our mechanism is a sequential variant of two-part tariffs. Prior to our work, simple approximation mechanisms for such multi-buyer problems were known only for the special cases of all unit-demand or all additive value buyers.Our work expands upon and unifies long lines of work on unit-demand settings and additive settings. We employ the ex ante relaxation approach developed by Alaei (2011) for reducing a multiple-buyer mechanism design problem with an ex post supply constraint into single-buyer problems with ex ante supply constraints. Solving the single-agent problems requires us to significantly extend a decomposition technique developed in the context of additive values by Li and Yao (2013) and its extension to subadditive values by Rubinstein and Weinberg (2015). This paper belongs to a long line of research on finding simple and approximately optimal mechanisms for multi-parameter settings under various assumptions on the buyers' value functions and type distributions, and on the seller's supply constraint. The first breakthrough along these lines was made by Chawla et al. (2007) who showed that the revenue of an optimal mechanism for a single unit-demand buyer can be approximated within a factor of 3 by an item pricing, 1 a mechanism that allows the buyer to choose any item to buy at fixed per-item prices. More recently, Babaioff et al. (2014) developed a similar result for a single buyer with additive values. 2 They showed that the revenue of an optimal mechanism in this case is approximated within a factor of 6 by one of two simple mechanisms: an item pricing that fixes a price for each item and allows the buyer to choose any subset of items to buy, and a bundle pricing that allows the buyer to buy the grand bundle of all items at a fixed price. Observe that item pricing and bundle pricing are both two-part tariffs (with the entry fee or the per-item prices being zero, respectively).Unit-demand and additive types are two extremes within a broader class of value functions that we call constrained additive values. A constrained additive buyer has a value (drawn independently) for each item under sale; he is interested in buying a set of items that satisfies a certain downward-closed constraint; his value is additive over any such set. We have only recently begun to understand optimal mechanism design for a single agent with constrained additive values. Rubinstein and Weinberg (2015) proved that in this setting, as in the additive case, either item pricing or bundle pricing gives a constant-factor approximation to the optimal revenue. 3 There are many similarities between the t...

Professionals who work with men who self-harm may like to be aware of these relational dynamics.

The central theme of this paper is that men are at one and the same time both damaged and damage-doing. The process of being damaged through the agency of masculinity predisposes men to exploit, dominate and abuse-not only as boys, as partners and fathers, but as priests, teachers, therapists, lawyers, nurses, psychologists and psychiatrists. Mental health services need to see both aspects of this male equation: a focus on male abuse alone leads to punishment, containment and, very likely, the continuation of abuse; a focus on male damagedness alone preserves the ideology of male unaccountability. However, evidence reviewed here suggests that most traditional psychiatric services fail to acknowledge the impact of inequalities on men's mental health as comprehensively as they fail to acknowledge the impact of these inequalities on the mental health of women. This is a significant problem, and we suggest ways that a gendered analysis of masculinity can be used to help address this deficit. This analysis is used to develop a map of men's mental health that not only accommodates traditional categories of mental health difficulty, but other important consequences of the close association between masculinity and sexuality inequality, especially the use of violence and the capacity to do harm. This analytical framework also invites consideration of the invisibility of male distress, the disallowing and desensitizing of 'vulnerability', and their submersion in a kind of psychology of entitlement. Finally, we consider the implications of this mapping exercise for mental health services and for working with men.

We present pricing mechanisms for several online resource allocation problems which obtain tight or nearly tight approximations to social welfare. In our settings, buyers arrive online and purchase bundles of items; buyers' values for the bundles are drawn from known distributions. This problem is closely related to the so-called prophet-inequality of Krengel and Sucheston [22] and its extensions in recent literature. Motivated by applications to cloud economics, we consider two kinds of buyer preferences. In the first, items correspond to different units of time at which a resource is available; the items are arranged in a total order and buyers desire intervals of items. The second corresponds to bandwidth allocation over a tree network; the items are edges in the network and buyers desire paths.Because buyers' preferences have complementarities in the settings we consider, recent constant-factor approximations via item prices do not apply, and indeed strong negative results are known. We develop static, anonymous bundle pricing mechanisms.For the interval preferences setting, we show that static, anonymous bundle pricings achieve a sublogarithmic competitive ratio, which is optimal (within constant factors) over the class of all online allocation algorithms, truthful or not. For the path preferences setting, we obtain a nearlytight logarithmic competitive ratio. Both of these results exhibit an exponential improvement over item pricings for these settings. Our results extend to settings where the seller has multiple copies of each item, with the competitive ratio decreasing linearly with supply. Such a gradual tradeoff between supply and the competitive ratio for welfare was previously known only for the single item prophet inequality.

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 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.We present three main results. First, we characterize optimal mechanisms as menus of two-outcome lotteries. Second, we show that under a reasonable bounded-riskaversion assumption, posted pricing obtains a constant approximation to the optimal revenue. Notably, this result is "risk-robust" in that it does not depend on the details of the buyer's risk attitude. Third, we consider dynamic settings in which the buyer's uncertainty about his future value may allow the seller to extract more revenue. In contrast to the positive result above, here we show it is not possible to achieve any constant-factor approximation to revenue using deterministic mechanisms in a risk-robust manner.