Sponsored Search Auctions (SSAs) arguably represent the problem at the intersection of computer science and economics with the deepest applications in real life. Within the realm of SSAs, the study of the effects that showing one ad has on the other ads, a.k.a. externalities in economics, is of utmost importance and has so far attracted the attention of much research. However, even the basic question of modeling the problem has so far escaped a definitive answer. The popular cascade model is arguably too idealized to really describe the phenomenon yet it allows a good comprehension of the problem. Other models, instead, describe the setting more adequately but are too complex to permit a satisfactory theoretical analysis. In this work, we attempt to get the best of both approaches: firstly, we define a number of general mathematical formulations for the problem in the attempt to have a rich description of externalities in SSAs and, secondly, prove a host of results drawing a nearly complete picture about the computational complexity of the problem. We complement these approximability results with some considerations about mechanism design in our context.

In spite of the considerable research effort devoted to studying externalities in Sponsored Search Auctions (SSAs), even the basic question of modeling the problem has so far escaped a definitive answer. The popular cascade model appears too idealized to really describe the phenomenon yet it allows a good comprehension of the problem. Other models, instead, arguably describe the real setting more adequately but are too complex to permit a satisfactory theoretical analysis. In this work, we attempt to get the best of both approaches: (i) we generalize the cascade model along a number of directions in the attempt to have mathematical formulations that are close to SSAs in the real world and (ii) prove a host of results drawing a nearly complete picture about the computational complexity of the problem. We complement these approximability results with some considerations about mechanism design in our context.

In recent years, effective sponsored search auctions (SSAs) have been designed to incentivize advertisers (advs) to bid their truthful valuations and, at the same time, to assure both the advs and the auctioneer a non--negative utility. Nonetheless, when the click--through--rates (CTRs) of the advs are unknown to the auction, these mechanisms must be paired with a learning algorithm for the estimation of the CTRs. This introduces the critical problem of designing a learning mechanism able to estimate the CTRs as the same time as implementing a truthful mechanism with a revenue loss as small as possible. In this paper, we extend previous results [2, 3] to the general case of multi--slot auctions with position-- and ad--dependent externalities with particular attention on the dependency of the regret on the number of slots K and the number of advertisements n.

Sponsored Search Auctions (SSAs) constitute one of the most successful applications of microeconomic mechanisms. In mechanism design, auctions are usually designed to incentivize advertisers to bid their truthful valuations and, at the same time, to guarantee both the advertisers and the auctioneer a non-negative utility. Nonetheless, in sponsored search auctions, the Click–Through–Rates (CTRs) of the advertisers are often unknown to the auctioneer and thus standard truthful mechanisms cannot be directly applied and must be paired with an effective learning algorithm for the estimation of the CTRs. This introduces the critical problem of designing a learning mechanism able to estimate the CTRs at the same time as implementing a truthful mechanism with a revenue loss as small as possible compared to the mechanism that can exploit the true CTRs. Previous work showed that, when dominant-strategy truthfulness is adopted, in single-slot auctions the problem can be solved using suitable exploration–exploitation mechanisms able to achieve a cumulative regret (on the auctioneer's revenue) of order O˜(T23), where T is the number of times the auction is repeated. It is also known that, when truthfulness in expectation is adopted, a cumulative regret (over the social welfare) of order O˜(T12) can be obtained. In this paper we extend the results available in the literature to the more realistic case of multi-slot auctions. In this case, a model of the user is needed to characterize how the CTR of an ad changes as its position in the allocation changes. In particular, we adopt the cascade model, one of the most popular models for sponsored search auctions, and we prove a number of novel upper bounds and lower bounds on both auctioneer's revenue loss and social welfare w.r.t. the Vickrey–Clarke–Groves (VCG) auction. Furthermore, we report numerical simulations investigating the accuracy of the bounds in predicting the dependency of the regret on the auction parameters.

Sponsored search auctions constitute one of the most successful applications of microeconomic mechanisms. In mechanism design, auctions are usually designed to incentivize advertisers to bid their truthful valuations and to assure both the advertisers and the auctioneer a non-negative utility. Nonetheless, in sponsored search auctions, the click-through-rates (CTRs) of the advertisers are often unknown to the auctioneer and thus standard truthful mechanisms cannot be directly applied and must be paired with an effective learning algorithm for the estimation of the CTRs. This introduces the critical problem of designing a learning mechanism able to estimate the CTRs at the same time as implementing a truthful mechanism with a revenue loss as small as possible compared to an optimal mechanism designed with the true CTRs. Previous work showed that, when dominant-strategy truthfulness is adopted, in single-slot auctions the problem can be solved using suitable exploration-exploitation mechanisms able to achieve a per-step regret (over the auctioneer's revenue) of order O(T^-1/3) (where T is the number of times the auction is repeated). It is also known that, when truthfulness in expectation is adopted, a per-step regret (over the social welfare) of order O(T^-1/2) can be obtained. In this paper we extend the results known in the literature to the case of multi-slot auctions. In this case, a model of the user is needed to characterize how the advertisers' valuations change over the slots. We adopt the cascade model that is the most famous model in the literature for sponsored search auctions. We prove a number of novel upper bounds and lower bounds both on the auctioneer's revenue loss and social welfare w.r.t. to the VCG auction and we report numerical simulations investigating the accuracy of the bounds in predicting the dependency of the regret on the auction parameters.

Mobile geo-location advertising, where mobile ads are targeted based on a user's location, has been identified as a key growth factor for the mobile market. As with online advertising, a crucial ingredient for their success is the development of effective economic mechanisms. An important difference is that mobile ads are shown sequentially over time and information about the user can be learned based on their movements. Furthermore, ads need to be shown selectively to prevent ad fatigue. To this end, we introduce, for the first time, a user model and suitable economic mechanisms which take these factors into account. Specifically, we design two truthful mechanisms which produce an advertisement plan based on the user's movements. One mechanism is allocatively efficient, but requires exponential compute time in the worst case. The other requires polynomial time, but is not allocatively efficient. Finally, we experimentally evaluate the tradeoff between compute time and efficiency of our mechanisms.

Strong Nash equilibrium (SNE) is an appealing solution concept when rational agents can form coalitions. A strategy profile is an SNE if no coalition of agents can benefit by deviating. We present the first general-purpose algorithms for SNE finding in games with more than two agents. An SNE must simultaneously be a Nash equilibrium (NE) and the optimal solution of multiple non-convex optimization problems. This makes even the derivation of necessary and sufficient mathematical equilibrium constraints difficult. We show that forcing an SNE to be resilient only to pure-strategy deviations by coalitions, unlike for NEs, is only a necessary condition here. Second, we show that the application of Karush-Kuhn-Tucker conditions leads to another set of necessary conditions that are not sufficient. Third, we show that forcing the Pareto efficiency of an SNE for each coalition with respect to coalition correlated strategies is sufficient but not necessary. We then develop a tree search algorithm for SNE finding. At each node, it calls an oracle to suggest a candidate SNE and then verifies the candidate. We show that our new necessary conditions can be leveraged to make the oracle more powerful. Experiments validate the overall approach and show that the new conditions significantly reduce search tree size compared to using NE conditions alone.

Sponsored search is one of the most successful applications of economic mechanisms in real life. A crucial issue is the modeling of the user behavior to provide the best targeting of ads to each user. Experimental studies show that the click through rate of an ad is dramatically affected by both its position and the other displayed ads. However, these externalities rise severe currently open computational issues in the determination of the best allocation and of the payments, preventing their adoption in practice so far. In the present paper, we provide a number of results when the most famous externality model, the cascade model, is adopted: we design the first exact algorithm for computing the efficient allocation, we show that the previously presented constant--approximation algorithm does not lead to any incentive compatible mechanism, we design a monotonic constant--approximation algorithm for finding the allocation and two different polynomial--time algorithms for the payments, each with different properties, leading to incentive compatible mechanisms. Finally, we provide a thorough experimental evaluation of the presented algorithms with Yahoo! Webscope A3 dataset to identify which mechanism should be adopted in concrete applications.

The computational characterization of game-theoretic solution concepts is a prominent topic in computer science. The central solution concept is Nash equilibrium (NE). However, it fails to capture the possibility that agents can form coalitions. Strong Nash equilibrium (SNE) refines NE to this setting. It is known that finding an SNE is NP-complete when the number of agents is constant. This hardness is solely due to the existence of mixed-strategy SNEs, given that the problem of enumerating all pure-strategy SNEs is trivially in P. Our central result is that, in order for an n-agent game to have at least one non-pure-strategy SNE, the agents' payoffs restricted to the agents' supports must lie on an (n - 1)-dimensional space. Small perturbations make the payoffs fall outside such a space and thus, unlike NE, finding an SNE is in smoothed polynomial time.

My Ph.D. thesis is focused on the field of mechanism design, a branch of game theory that aims to study interaction mechanisms for rational agents. The basic goal is the design of direct--revelation mechanisms that are stable. Given the valuations reported by the agents, a mechanism determines the outcome of the interaction by means of an objective function. In some contexts, such objective function cannot lead to a stable mechanism or finding an optimal solution is NP-hard. The goal of my studies is the development of techniques to deal with these situations. The basic idea is to design a new objective function that approximates at best the original one, but, at the same time can be used to build a stable mechanism computable in polynomial time. During my Ph.D., I will extend the techniques currently available in the state of the art to more general situations and I will apply such techniques to the important field of ad auctions.

Computing equilibria of games is a central task in computer science. A large number of results are known for Nash equilibrium (NE). However, these can be adopted only when coalitions are not an issue. When instead agents can form coalitions, NE is inadequate and an appropriate solution concept is strong Nash equilibrium (SNE). Few computational results are known about SNE. In this paper, we first study the problem of verifying whether a strategy profile is an SNE, showing that the problem is in P. We then design a spatial branch--and--bound algorithm to find an SNE, and we experimentally evaluate the algorithm.

The computational characterization of game-theoretic solution concepts is a central topic in artificial intelligence, with the aim of developing computationally efficient tools for finding optimal ways to behave in strategic interactions. The central solution concept in game theory is Nash equilibrium (NE). However, it fails to capture the possibility that agents can form coalitions (even in the 2-agent case). Strong Nash equilibrium (SNE) refines NE to this setting. It is known that finding an SNE is NP-complete when the number of agents is constant. This hardness is solely due to the existence of mixed-strategy SNEs, given that the problem of enumerating all pure-strategy SNEs is trivially in P. Our central result is that, in order for a game to have at least one non-pure-strategy SNE, the agents' payoffs restricted to the agents' supports must, in the case of 2 agents, lie on the same line, and, in the case of n agents, lie on an (n - 1)-dimensional hyperplane. Leveraging this result, we provide two contributions. First, we develop worst-case instances for support-enumeration algorithms. These instances have only one SNE and the support size can be chosen to be of any size-in particular, arbitrarily large. Second, we prove that, unlike NE, finding an SNE is in smoothed polynomial time: generic game instances (i.e., all instances except knife-edge cases) have only pure-strategy SNEs.

Computing a Nash equilibrium (NE) is a central task in computer science. An NE is a particularly appropriate solution concept for two-agent settings because coalitional deviations are not an issue. However, even in this case, finding an NE is PPAD-complete. In this paper, we combine path following algorithms with local search techniques to design new algorithms for finding exact and approximate NEs. We show that our algorithms largely outperform the state of the art and that almost all the known benchmark game classes are easily solvable or approximable (except for the GAMUT CovariantGame-Rand class).

The paper has a double aim. On the one hand, it studies some monotone extensions to the bidual of monotone operators which are not of type (D), paying particular attention to the issue of representability of these extensions by means of convex functions. On the other hand, it derives a density property for maximal monotone operators that, not only are of type (D), but have their unique maximal monotone extensions to the bidual which are of type (D) as well, as it is the case for subdifferentials of lower semicontinuous convex functions.

In this paper, we develop a theory of monotone operators in the framework of abstract convexity. First, we provide a surjectivity result for a broad class of abstract monotone operators. Then, by using an additivity constraint qualification, we prove a generalization of Fenchel’s duality theorem in the framework of abstract convexity and give some criteria for maximal abstract monotonicity. Finally, we present necessary and sufficient conditions for maximality of abstract monotone operators.

In the framework of real Banach spaces, the present paper provides a necessary and sufficient condition for the Fitzpatrick function of the subdifferential of a proper lower semicontinuous convex function to be autoconjugate. This enables us to: obtain a new proof of the fact that subdifferentials of indicator and sublinear functions have autoconjugate Fitzpatrick functions; characterize those classes of functions whose subdifferentials fulfill the condition under study in the same special way as indicator and sublinear functions do; prove that, in the one-dimensional case, the functions of these classes are the only ones whose subdifferentials have autoconjugate Fitzpatrick functions, while this is not true in higher dimensions.

A generalization of Rockafellar's surjectivity theorem was provided in [14], replacing the duality mapping by any maximal monotone operator having finite-valued Fitzpatrick function. The present paper extends this result to the nonreflexive setting for maximal monotone operators of type (D) and refines the finite-valuedness condition on the Fitzpatrick function. Moreover, a characterization of surjectivity properties for the sum of two maximal monotone operators of type (D) in terms of Fenchel duality is given.

The computation of a Nash equilibrium of a game is a challenging problem in artificial intelligence. This is because the computational time of the algorithms provided by the literature is, in the worst case, exponential in the size of the game. In this paper, we present, to the best of our knowledge, the first anytime algorithm based on the combination of support enumeration methods and local search techniques to find a Nash equilibrium in two-player general-sum games. The algorithm searches for a Nash equilibrium and, if it is stopped before it has found an equilibrium, it returns the best approximate equilibrium found so far. We design some dimensions for our algorithm and we experimentally evaluate them. Our algorithm solves with high probability games that are unsolvable with the algorithms known in the literature within a reasonable time and provides good anytime performance.