In sponsored search, a number of advertising slots is available on a search results page, and have to be allocated among a set of advertisers competing to display an ad on the page. This gives rise to a bipartite matching market that is typically cleared by the way of an automated auction. Several auction mechanisms have been proposed, with variants of the Generalized Second Price (GSP) being widely used in practice. A rich body of work on bipartite matching markets builds upon the stable marriage model of Gale and Shapley and the assignment model of Shapley and Shubik. We apply insights from this line of research into the structure of stable outcomes and their incentive properties to advertising auctions. We model advertising auctions in terms of an assignment model with linear utilities, extended with bidder and item specific maximum and minimum prices. Auction mechanisms like the commonly used GSP or the well-known Vickrey-Clarke-Groves (VCG) are interpreted as simply computing a in this model, for a suitably defined set of bidder preferences. In our model, the existence of a stable matching is guaranteed, and under a non-degeneracy assumption a bidder-optimal stable matching exists as well. We give an algorithm to find such matching in polynomial time, and use it to design truthful mechanism that generalizes GSP, is truthful for profit-maximizing bidders, implements features like bidder-specific minimum prices and position-specific bids, and works for rich mixtures of bidders and preferences.
The assignment model @cite_25 , (see also @cite_20 @cite_6 ) differs in that each player derives a certain value from being matched to each person of the opposite sex, and side payments between partners are allowed. The goal of each player is to maximize his or her payoff which is the sum of partner's value and monetary payment (positive or negative negative) from the partner. The set of stable outcomes is non-empty by a linear programming argument. In fact, each stable outcome corresponds to a maximum-weight matching, and player payoffs correpond to dual variables of the maximum matching LP. A man-optimal outcome is guaranteed to exist, and its allocation and prices are identical to the VCG mechanism for maximum weight matchings @cite_19 @cite_13 .
@cite_6 @cite_19 @cite_13 @cite_25 @cite_20
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Payoff-driven adjustment dynamics lead to stable and optimal outcomes in decentralized two-sided assignment markets. Pairs of agents from both sides of the market randomly encounter each other and match if profitable'. Very little information is available, in particular agents have no knowledge of others' preferences, their past actions and payoffs or the value of the different matches. This process implements optimal and stable - i.e. core - allocations even though agents interact asynchronously and randomly, and there is no central authority enforcing matchings or sharing rules. We consider a problem known as the restricted assignment version of the max-min allocation problem with indivisible goods. There are n items of various nonnegative values and m players. Every player is interested only in some of the items and has zero value for the other items. One has to distribute the items among the players in a way that maximizes a certain notion of fairness, namely, maximizes the minimum of the sum of values of items given to any player. Bansal and Sviridenko [STOC 2006] describe a linear programming relaxation for this problem, and present a rounding technique that recovers an allocation of value at least Ω(log log log m log log m) of the optimum. We show that the value of this LP relaxation in fact approximates the optimum value to within a constant factor. Our proof is not constructive and does not by itself provide an efficient algorithm for finding an allocation that is within constant factors of optimal. The set of optimal matchings in the assignment matrix allows to define a reflexive and symmetric binary relation on each side of the market, the equal-partner binary relation. The number of equivalence classes of the transitive closure of the equal-partner binary relation determines the dimension of the core of the assignment game. This result provides an easy procedure to determine the dimension of the core directly from the entries of the assignment matrix and shows that the dimension of the core is not as much determined by the number of optimal matchings as by their relative position. In many two-sided markets, each side has incomplete information about the other but has an opportunity to learn (some) relevant information before final matches are made. For instance, clients seeking workers to perform tasks often conduct interviews that require the workers to perform some tasks and thereby provide information to both sides. The performance of a worker in such an interview assessment - and hence the information revealed - depends both on the inherent characteristics of the worker and the task and also on the actions taken by the worker (e.g. the effort expended); thus there is both adverse selection (on both sides) and moral hazard (on one side). When interactions are ongoing, incentives for workers to expend effort in the current assessment can be provided by the payment rule used and also by the matching rule that assesses and determines the tasks to which the worker is assigned in the future; thus workers have career concerns. We derive mechanisms - payment, assessment and matching rules - that lead to final matchings that are stable in the long run and achieve close to the optimal performance (profit or social welfare maximizing) in equilibrium (unique) thus mitigating both adverse selection and moral hazard (in many settings).