Multi-Armed Bandit Mechanisms for Multi-Slot Sponsored Searth Auctions

In pay-per click sponsored search auctions which are currently extensively used by search engines, the auction for a keyword involves a certain number of advertisers (say k) competing for available slots (say m) to display their ads. This auction is typically conducted for a number of rounds (say T). There are click probabilities mu_ij associated with agent-slot pairs. The search engine's goal is to maximize social welfare, for example, the sum of values of the advertisers. The search engine does not know the true value of an advertiser for a click to her ad and also does not know the click probabilities mu_ij s. A key problem for the search engine therefore is to learn these during the T rounds of the auction and also to ensure that the auction mechanism is truthful. Mechanisms for addressing such learning and incentives issues have recently been introduced and would be referred to as multi-armed-bandit (MAB) mechanisms. When m = 1,characterizations for truthful MAB mechanisms are available in the literature and it has been shown that the regret for such mechanisms will be O(T^{2/3}). In this paper, we seek to derive a characterization in the realistic but nontrivial general case when m > 1 and obtain several interesting results.

Efficient Algorithms for Combinatorial Auctions with Volume Discounts Arising in Web Service Composition

Web services are now a key ingredient of software services offered by software enterprises. Many standardized web services are now available as commodity offerings from web service providers. An important problem for a web service requester is the web service composition problem which involves selecting the right mix of web service offerings to execute an end-to-end business process. Web service offerings are now available in bundled form as composite web services and more recently, volume discounts are also on offer, based on the number of executions of web services requested. In this paper, we develop efficient algorithms for the web service composition problem in the presence of composite web service offerings and volume discounts. We model this problem as a combinatorial auction with volume discounts. We first develop efficient polynomial time algorithms when the end-to-end service involves a linear workflow of web services. Next we develop efficient polynomial time algorithms when the end-to-end service involves a tree workflow of web services.

A Nash Bargaining Approach to Retention Enhancing Bid Optimization in Sponsored Search Auctions with Discrete Bids

Bid optimization is now becoming quite popular in sponsored search auctions on the Web. Given a keyword and the maximum willingness to pay of each advertiser interested in the keyword, the bid optimizer generates a profile of bids for the advertisers with the objective of maximizing customer retention without compromising the revenue of the search engine. In this paper, we present a bid optimization algorithm that is based on a Nash bargaining model where the first player is the search engine and the second player is a virtual agent representing all the bidders. We make the realistic assumption that each bidder specifies a maximum willingness to pay values and a discrete, finite set of bid values. We show that the Nash bargaining solution for this problem always lies on a certain edge of the convex hull such that one end point of the edge is the vector of maximum willingness to pay of all the bidders. We show that the other endpoint of this edge can be computed as a solution of a linear programming problem. We also show how the solution can be transformed to a bid profile of the advertisers.

Bidding Dynamics of Rational Advertisers in Sponsored Search Auctions on the Web

In this paper, we address a key problem faced by advertisers in sponsored search auctions on the web: how much to bid, given the bids of the other advertisers, so as to maximize individual payoffs? Assuming the generalized second price auction as the auction mechanism, we formulate this problem in the framework of an infinite horizon alternative-move game of advertiser bidding behavior. For a sponsored search auction involving two advertisers, we characterize all the pure strategy and mixed strategy Nash equilibria. We also prove that the bid prices will lead to a Nash equilibrium, if the advertisers follow a myopic best response bidding strategy. Following this, we investigate the bidding behavior of the advertisers if they use Q-learning. We discover empirically an interesting trend that the Q-values converge even if both the advertisers learn simultaneously.

Truthful multi-armed bandit mechanisms for multi-slot sponsored search auctions

In pay-per-click sponsored search auctions which are currently extensively used by search engines, the auction for a keyword involves a certain number of advertisers (say k) competing for available slots (say m) to display their advertisements (ads for short). A sponsored search auction for a keyword is typically conducted for a number of rounds (say T). There are click probabilities mu(ij) associated with each agent slot pair (agent i and slot j). The search engine would like to maximize the social welfare of the advertisers, that is, the sum of values of the advertisers for the keyword. However, the search engine does not know the true values advertisers have for a click to their respective advertisements and also does not know the click probabilities. A key problem for the search engine therefore is to learn these click probabilities during the initial rounds of the auction and also to ensure that the auction mechanism is truthful. Mechanisms for addressing such learning and incentives issues have recently been introduced. These mechanisms, due to their connection to the multi-armed bandit problem, are aptly referred to as multi-armed bandit (MAB) mechanisms. When m = 1, exact characterizations for truthful MAB mechanisms are available in the literature. Recent work has focused on the more realistic but non-trivial general case when m > 1 and a few promising results have started appearing. In this article, we consider this general case when m > 1 and prove several interesting results. Our contributions include: (1) When, mu(ij)s are unconstrained, we prove that any truthful mechanism must satisfy strong pointwise monotonicity and show that the regret will be Theta T7) for such mechanisms. (2) When the clicks on the ads follow a certain click precedence property, we show that weak pointwise monotonicity is necessary for MAB mechanisms to be truthful. (3) If the search engine has a certain coarse pre-estimate of mu(ij) values and wishes to update them during the course of the T rounds, we show that weak pointwise monotonicity and type-I separatedness are necessary while weak pointwise monotonicity and type-II separatedness are sufficient conditions for the MAB mechanisms to be truthful. (4) If the click probabilities are separable into agent-specific and slot-specific terms, we provide a characterization of MAB mechanisms that are truthful in expectation.

Mechanism design for green, truthful procurement auctions

Auction based mechanisms have become popular in industrial procurement settings. These mechanisms minimize the cost of procurement and at the same time achieve desirable properties such as truthful bidding by the suppliers. In this paper, we investigate the design of truthful procurement auctions taking into account an additional important issue namely carbon emissions. In particular, we focus on the following procurement problem: A buyer wishes to source multiple units of a homogeneous item from several competing suppliers who offer volume discount bids and who also provide emission curves that specify the cost of emissions as a function of volume of supply. We assume that emission curves are reported truthfully since that information is easily verifiable through standard sources. First we formulate the volume discount procurement auction problem with emission constraints under the assumption that the suppliers are honest (that is they report production costs truthfully). Next we describe a mechanism design formulation for green procurement with strategic suppliers. Our numerical experimentation shows that emission constraints can significantly alter sourcing decisions and affect the procurement costs dramatically. To the best of our knowledge, this is the first effort in explicitly taking into account carbon emissions in planning procurement auctions.