Abstract. Sponsored search involves running an auction among advertisers who bid in order to have their ad shown next to search results for specific keywords. The most popular auction for sponsored search is the “Generalized Second Price ” (GSP) auction where advertisers are assigned to slots in the decreasing order of their score, which is defined as the product of their bid and click-through rate. One of the main advantages of this simple ranking is that bidding strategy is intuitive: to move up to a more prominent slot on the results page, bid more. This makes it simple for advertisers to strategize. However this ranking only maximizes efficiency under the assumption that the probability of a user clicking on an ad is independent of the other ads shown on the page. We study a Markovian user model that does not make this assumption. Under this model, the most efficient assignment is no longer a simple ranking function as in GSP. We show that the optimal assignment can be found efficiently (even in near-linear time). As a result of the more sophisticated structure of the optimal assignment, bidding dynamics become more complex: indeed it is no longer clear that bidding more moves one higher on the page. Our main technical result is that despite the added complexity of the bidding dynamics, the optimal assignment has the property that ad position is still monotone in bid. Thus even in this richer user model, our mechanism retains the core bidding dynamics of the GSP auction that make it useful for advertisers. 1

We study multi-unit auctions where the bidders have a budget constraint, a situation very common in practice that has received very little attention in the auction theory literature. Our main result is an impossibility: there are no incentive-compatible auctions that always produce a Pareto-optimal allocation. We also obtain some surprising positive results for certain special cases. 1

Position auctions such as the Generalized Second Price (GSP) are commonly used for sponsored search, e.g., by Yahoo! and Google. We now have an understanding of the equilibria of these auctions, via game-theoretic concepts like Generalized English Auctions and the “locally envy-free ” property, as well as through a relationship to the well-known, truthful Vickrey-Clarke-Groves (VCG) mechanism. In practice, however, position auctions are implemented with additional constraints, in particular, bidder-specific minimum prices. Such minimum prices are used to control the quality of the ads that appear on the page. We study the effect of bidder-specific minimum prices in position auctions. Naïvely enforcing minimum prices in the VCG mechanism breaks the truthfulness of the auction; we describe two variants of VCG for which revealing the truth is a dominant strategy. The implications of bidder-specific minimum prices are more intricate for the GSP auction. Some properties proved for standard GSP no longer hold in this setting. For example, we show that the GSP allocation is now not always efficient (in terms of advertiser value). Also, the property of“envy-locality”enjoyed by GSP—which is essential in the prior analysis of strategies and equilibria— no longer holds. Our main result is to show that despite losing envy locality, GSP with bidder-specific minimum prices still has an envy-free equilibrium. 1.

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Motivated by sponsored search auctions with hard budget constraints given by the advertisers, we study multi-unit auctions of a single item. An important example is a sponsored result slot for a keyword, with many units representing its inventory in a month, say. In this single-item multi-unit auction, each bidder has a private value for each unit, and a private budget which is the total amount of money she can spend in the auction. A recent impossibility result [Dobzinski et al., FOCS’08] precludes the existence of a truthful mechanism with Pareto-optimal allocations in this important setting. We propose Sort-Cut, a mechanism which does the next best thing from the auctioneer’s point of view, that we term semitruthful. In our mechanism, it is a weakly dominant strategy for all agents to state their true budgets and to not understate their values.

Recently, there has been a surge of interest in algorithms that allocate advertisement space in an online revenue-competitive manner. Most such algorithms, however, assume a pay-as-you-bid pricing scheme. In this paper, we study the query allocation problem where the ad space is priced using the well-known and widely-used generalized second price (GSP) scheme. We observe that the previous algorithms fail to achieve a bounded competitive ratio under the GSP scheme. On the positive side, we present online constant-competitive algorithms for the problem. 1

We study multi unit auctions in presence of budget constraints. This situation is very common in real world but has not received proper attention in theoretical literature. We will study the impact of budget on the different aspects of multi unit auctions. These results try to define a frame work in which multi unit auctions with budget constraints are working and problem we are facing in designing mechanisms. 1

We consider a resource allocation problem where providers have general convex constraints on the allocations that they can distribute to users. This type of problems has many applications in computer science, from sharing network bandwidth to scheduling jobs in data centers and distributing click-through resources in sponsored search. We introduce the weighted proportional sharing mechanism, where providers can apply different weights to users. This is an extension of the traditional proportional sharing scheme that has been well studied and applied in many real-world applications. However, to the best of our knowledge, this line of research has not considered strategic providers. Our paper investigates the problem where providers act strategically to maximize their revenues in both monopolies and oligopolies. In the application of sponsored search, we give a new approach for the general second price auction currently used. Our general convex constraints on the resources can capture complex externalities among click-through rates and discrimination rates among users are chosen in a similar way that search engines use different weights for different advertisers Our results show that providers can obtain a revenue that is at least k/(k+1) times the maximum revenue under standard third-degree price discrimination with a set of k users excluded. Moreover, when users ’ utilities are linear, the social welfare of a Nash equilibrium is at least 2 √ 3 − 3 (approx. 46%) of the maximum social welfare, and this bound is tight. We extend this result to an oligopoly consisting of multiple competing providers and to a broad class of utility functions, which includes most of the utility functions found in literature. ∗ The work by T. Nguyen was performed in part while an intern with Microsoft Research Cambridge. 1 1

Recently, there has been a surge of interest in algorithms that allocate advertisement space in an online revenuecompetitive manner. Most such algorithms, however, assume a pay-as-you-bid pricing scheme. In this paper, we study the query allocation problem where the ad space is priced using the well-known and widely-used generalized second price (GSP) scheme. We observe that the previous algorithms fail to achieve a bounded competitive ratio under the GSP scheme. On the positive side, we present online constant-competitive algorithms for the problem. Key words: sponsored search, generalized second-pricing scheme, throttling policies, greedy algorithms 1.