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

Leveraging VGI Integrated with 3D Spatial Technology to Support Urban Intensification in Melbourne, Australia

High density residential development in metropolitan Melbourne, where contradictory imperatives of neighbourhood character and urban intensification play important roles, remains an uncertain practice. One key issue for plan implementation is the lack of consistency between authorities, developers and the community in interpreting the standards, design guidelines, and state/local strategies, especially those relating to neighbourhood character. There is currently no mechanism to incorporate community perceptions and place experiences as subjective aspects of neighbourhood character in development assessments. There is also little use of micro-scale and multi-dimensional spatial analysis to integrate these subjective aspects with objective measures (e.g. building volume and height; streetscape) to communicate effectively—and in a limited timeframe—with all stakeholders. This paper explores the potential of two emerging geospatial technologies that can be leveraged to respond to these problems. Evidence in the literature suggests that volunteered geographic information (VGI) can provide community input around subjective aspects of the urban environment. In addition, a deluge of three-dimensional (3D) spatial information (e.g. 3D city models) is increasingly available for micro-level (building- or property-level) assessment of the physical aspects of the urban environment. This paper formulates and discusses a conceptual framework to link these two spatial technological advancements in a virtual geographic environment (VGE) that accounts for micro-scale 3D spatial analysis incorporating both subjective and objective aspects of neighbourhood character relevant in implementing compact city strategies

Multi-outcome and Multidimensional Market Scoring Rules

Hanson's market scoring rules allow us to design a prediction market that still gives useful information even if we have an illiquid market with a limited number of budget-constrained agents. Each agent can "move" the current price of a market towards their prediction. While this movement still occurs in multi-outcome or multidimensional markets we show that no market-scoring rule, under reasonable conditions, always moves the price directly towards beliefs of the agents. We present a modified version of a market scoring rule for budget-limited traders, and show that it does have the property that, from any starting position, optimal trade by a budget-limited trader will result in the market being moved towards the trader's true belief. This mechanism also retains several attractive strategic properties of the market scoring rule

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 pricing mechanism to the buyer with a budget constraint and an indirect mechanism

The present article considers the situation in which the buyer’s taste and budget are his private information. In this multi-dimensional setting, we study the optimal mechanism through a canonical mechanism in the traditional one-dimensional context: a function of one variable, the buyer’s taste. In our multi-dimensional context, however, this is an indirect mechanism. We investigate the effectiveness and limit of this indirect mechanism in the framework of the revelation principle