We study dynamic mechanisms for optimizing revenue in repeated auctions, that are robust to heterogeneous forward-looking and learning behavior of the buyers. Typically it is assumed that the buyers are either all myopic or are all infinite lookahead, and that buyers understand and trust the mechanism. These assumptions raise the following question: is it possible to design approximately revenue optimal mechanisms when the buyer pool is heterogeneous? Facing a heterogeneous population of buyers with an unknown mixture of @math -lookahead buyers, myopic buyers, no-regret-learners and no-policy-regret learners, we design a simple state-based mechanism that achieves a constant fraction of the optimal achievable revenue.
The study of @math -lookahead search can be viewed in the context of bounded rationality , as pioneered by Herb Simon @cite_14 . He argued that, instead of optimizing, agents may apply a class of heuristics, termed satisfying heuristics in decision making. A natural choice of such heuristics is restricting the search space of best-response moves. Lookahead search in decision-making has been motivated and examined in great extent by the artificial intelligence community @cite_7 @cite_31 @cite_35 . Lookahead search is also related to the sequential thinking framework in game theory @cite_21 . More recently, @cite_3 study the quality of equilibrium outcomes for look-ahead search strategies for various classes of games. They observe that the quality of resulting equilibria increases in generalized second-price auctions, and duopoly games, but not in other classes of games. No prior work studies dynamic mechanisms that are robust against various lookahead search strategies.
@cite_35 @cite_14 @cite_7 @cite_21 @cite_3 @cite_31
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Lookahead search is perhaps the most natural and widely used game playing strategy. Given the practical importance of the method, the aim of this paper is to provide a theoretical performance examination of lookahead search in a wide variety of applications. To determine a strategy play using lookahead search, each agent predicts multiple levels of possible re-actions to her move (via the use of a search tree), and then chooses the play that optimizes her future payoff accounting for these re-actions. There are several choices of optimization function the agents can choose, where the most appropriate choice of function will depend on the specifics of the actual game - we illustrate this in our examples. Furthermore, the type of search tree chosen by computationally-constrained agent can vary. We focus on the case where agents can evaluate only a bounded number, k, of moves into the future. That is, we use depth k search trees and call this approach k-lookahead search. We apply our method in five well-known settings: industrial organization (Cournot's model); AdWord auctions; congestion games; valid-utility games and basic-utility games; cost-sharing network design games. We consider two questions. First, what is the expected social quality of outcome when agents apply lookahead search? Second, what interactive behaviours can be exhibited when players use lookahead search? We demonstrate how the answer depends on the game played. We study network design games where @math self-interested agents have to form a network by purchasing links from a given set of edges. We consider Shapley cost sharing mechanisms that split the cost of an edge in a fair manner among the agents using the edge. It is well known that the price of anarchy of these games is as high as @math . Another line of research has focused on evaluating the price of stability, i.e., the cost of the best Nash equilibrium relative to the social optimum. In this paper we investigate to which extent coordination among agents can improve the quality of solutions. We resort to the concept of strong Nash equilibria, which were introduced by Aumann and are resilient to deviations by coalitions of agents. We analyze the price of anarchy of strong Nash equilibria and develop lower and upper bounds for unweighted and weighted games in both directed and undirected graphs. These bounds are tight or nearly tight for many scenarios. It shows that, by using coordination, the price of anarchy drops from linear to logarithmic bounds. We complement these results by also proving the first superconstant lower bound on the price of stability of standard equilibria (without coordination) in undirected graphs. More specifically, we show a lower bound of @math for weighted games, where @math is the total weight of all the agents. This almost matches the known upper bound of @math . Our results imply that, for most settings, the worst-case performance ratios of strong coordinated equilibria are essentially always as good as the performance ratios of the best equilibria achievable without coordination. These settings include unweighted games in directed graphs as well as weighted games in both directed and undirected graphs. In sponsored search auctions, advertisers compete for a number of available advertisement slots of different quality. The auctioneer decides the allocation of advertisers to slots using bids provided by them. Since the advertisers may act strategically and submit their bids in order to maximize their individual objectives, such an auction naturally defines a strategic game among the advertisers. In order to quantify the efficiency of outcomes in generalized second price auctions, we study the corresponding games and present new bounds on their price of anarchy, improving the recent results of Paes Leme and Tardos [16] and Lucier and Paes Leme [13]. For the full information setting, we prove a surprisingly low upper bound of 1.282 on the price of anarchy over pure Nash equilibria. Given the existing lower bounds, this bound denotes that the number of advertisers has almost no impact on the price of anarchy. The proof exploits the equilibrium conditions developed in [16] and follows by a detailed reasoning about the structure of equilibria and a novel relation of the price of anarchy to the objective value of a compact mathematical program. For more general equilibrium classes (i.e., mixed Nash, correlated, and coarse correlated equilibria), we present an upper bound of 2.310 on the price of anarchy. We also consider the setting where advertisers have incomplete information about their competitors and prove a price of anarchy upper bound of 3.037 over Bayes-Nash equilibria. In order to obtain the last two bounds, we adapt techniques of Lucier and Paes Leme [13] and significantly extend them with new arguments. In a combinatorial auction with item bidding, agents participate in multiple single-item second-price auctions at once. As some items might be substitutes, agents need to strategize in order to maximize their utilities. A number of results indicate that high welfare can be achieved this way, giving bounds on the welfare at equilibrium. Recently, however, criticism has been raised that equilibria are hard to compute and therefore unlikely to be attained. In this paper, we take a different perspective. We study simple best-response dynamics. That is, agents are activated one after the other and each activated agent updates his strategy myopically to a best response against the other agents' current strategies. Often these dynamics may take exponentially long before they converge or they may not converge at all. However, as we show, convergence is not even necessary for good welfare guarantees. Given that agents' bid updates are aggressive enough but not too aggressive, the game will remain in states of good welfare after each agent has updated his bid at least once. In more detail, we show that if agents have fractionally subadditive valuations, natural dynamics reach and remain in a state that provides a @math approximation to the optimal welfare after each agent has updated his bid at least once. For subadditive valuations, we can guarantee an @math approximation in case of @math items that applies after each agent has updated his bid at least once and at any point after that. The latter bound is complemented by a negative result, showing that no kind of best-response dynamics can guarantee more than an @math fraction of the optimal social welfare.