The constraints of the valuation distribution for solving a class of games by using a best response algorithm

Infinite games with incomplete information are common in practice. First-price, sealed-bid auctions are a prototypical example. To solve this kind of infinite game, a heuristic approach is to discretise the strategy spaces and enumerate to approximate the equilibrium strategies. However, an approximate algorithm might not be guaranteed to converge. This paper discusses the utilisation of a best response algorithm in solving infinite games with incomplete information. We show the constraints of the valuation distributions define the necessary conditions of the convergence of the best response algorithm for several classes of infinite games, including auctions. A salient feature of the necessary convergence conditions lies in that they can be employed to compute the exact Nash equilibria without discretising the strategy space if the best response algorithm converges

Robust Dynamic Pricing with Strategic Customers

We consider the canonical revenue management (RM) problem wherein a seller must sell an inventory of some product over a finite horizon via an anonymous, posted price mechanism. Unlike typical models in RM, we assume that customers are forward looking. In particular, customers arrive randomly over time and strategize about their times of purchases. The private valuations of these customers decay over time and the customers incur monitoring costs; both the rates of decay and these monitoring costs are private information. This setting has resisted the design of optimal dynamic mechanisms heretofore. Optimal pricing schemes-an almost necessary mechanism format for practical RM considerations-have been similarly elusive. The present paper proposes a mechanism we dub robust pricing. Robust pricing is guaranteed to achieve expected revenues that are at least within 29% of those under an optimal (not necessarily posted price) dynamic mechanism. We thus provide the first approximation algorithm for this problem. The robust pricing mechanism is practical, since it is an anonymous posted price mechanism and since the seller can compute the robust pricing policy for a problem without any knowledge of the distribution of customer discount factors and monitoring costs. The robust pricing mechanism also enjoys the simple interpretation of solving a dynamic pricing problem for myopic customers with the additional requirement of a novel “restricted sub-martingale constraint” on prices that discourages rapid discounting. We believe this interpretation is attractive to practitioners. Finally, numerical experiments suggest that the robust pricing mechanism is, for all intents, near optimal

Dynamic Online-Advertising Auctions as Stochastic Scheduling

We study dynamic models of online-advertising auctions in the Internet: advertisers compete for space on a web page over multiple time periods, and the web page displays ads in differentiated slots based on their bids and other considerations. The complex interactions between the advertisers and the website (which owns the web page) is modeled as a dynamic game. Our goal is to derive ad-slot placement and pricing strategies which maximize the expected revenue of the website. We show that the problem can be transformed into a scheduling problem familiar to queueing theorists. When only one advertising slot is available on a webpage, we derive the optimal revenue-maximizing solution by making connections to the familiar cμ rule used in queueing theory. More generally, we show that a cμ-like rule can serve as a good suboptimal solution, while the optimal solution itself may be computed using dynamic programming techniques

The probability of sale and price premiums in withdrawn auctioned properties

This paper examines the impact of the auction process of residential properties that whilst unsuccessful at auction sold subsequently. The empirical analysis considers both the probability of sale and the premium of the subsequent sale price over the guide price, reserve and opening bid. The findings highlight that the final achieved sale price is influenced by key price variables revealed both prior to and during the auction itself. Factors such as auction participation, the number of individual bidders and the number of bids are significant in a number of the alternative specifications

Online Auction and List Price Revenue Management

We analyze a revenue management problem in which a seller facing a Poisson arriving stream of customers operates an online multiunit auction. Customers have an alternative list price channel where to get the product from. We consider two variants of this problem: In the first one, the list price is an external channel run by another firm. In the second variant, the seller manages simultaneously both the auction and the list price channels. Each consumer, trying to maximize his own surplus, must decide either to buy at the posted price and get the item at no risk, or to join the auction and wait until its end, where the winners are revealed and the auction price is disclosed. Our approach consists of two parts. First, we study structural properties of the problem, and show that the equilibrium strategy for both versions of this game is of the threshold type, meaning that a consumer will join the auction only if his arrival time is above a function of his own valuation. This consumerâs strategy can be computed using an iterative algorithm in a function space, provably convergent under some conditions. Unfortunately, this procedure is computationally intensive. To overcome this, we formulate an asymptotic version of the problem, in which the demand rate and the initial number of units grow proportionally large. We get a simple closed form for the equilibrium strategy in this regime, which is then used as an approximated solution for the original problem. Numerical computations show that this heuristic is very accurate. The asymptotic solution culminates then in simple and precise recipes for how bidders should behave, and how the seller should structure the auction, and price the product in the dual channel case.Information Systems Working Papers Serie

Optimal dynamic mechanism design with deadlines

A seller maximizes revenue from selling an object in a dynamic environment, with buyers that differ in their patience: Each buyer has a privately known deadline for buying and a privately known valuation. First, we derive the optimal mechanism, neglecting the incentive constraint for the deadline. The deadline of the winner determines the time of the allocation and therefore also the amount of information available to the seller when he decides whether to allocate to a buyer. Depending on the shape of the markup that the seller uses, this can lead to a violation of the neglected incentive constraint. We give sufficient conditions on the type distribution under which the neglected constraint is fulfilled or violated. Second, for the case that the constraint cannot be neglected, we consider a model with two periods and two buyers. Here, the optimal mechanism is implemented by a fixed price in period one and an asymmetric auction in period two. The asymmetry, which is introduced to prevent the patient type of the first buyer from buying in period one leads to pooling of deadlines at the top of the type space