Mechanisms with Unique Learnable Equilibria PAUL DUTTING¨ , Stanford University THOMAS KESSELHEIM, Cornell University EVA´ TARDOS, Cornell University The existence of a unique equilibrium is the classic tool for ensuring predictiveness of game theory. Typical uniqueness results, however, are for Nash and Bayes-Nash equilibria and do not guarantee that natural game playing dynamic converges to this equilibrium. In fact, there are well known examples in which the equilibrium is unique, yet natural learning behavior does not converge to it. Motivated by this, we strive for stronger uniqueness results. We do not only require that there is a unique equilibrium, but also that this equilibrium must be learnable. We adopt correlated equilibrium as our solution concept, as simple and natural learning algorithms guarantee that the empirical distribution of play converges to the space of correlated equilibria. Our main result is to show uniqueness of correlated equilibria in a large class of single-parameter mechanisms with matroid structure. We also show that our uniqueness result extends to problems with polymatroid structure under some conditions. Our model includes a number of special cases interesting on their own right, such as procurement auctions and Bertrand competitions. An interesting feature of our model is that we do not need to assume that the players have quasi-linear utilities, and hence can incorporate models with risk averse players and certain forms of externalities. Categories and Subject Descriptors: F.2 [Theory of Computation]: Analysis of Algorithms and Problem Complexity; J.4 [Computer Applications]: Social and Behavioral Sciences—Economics Additional Key Words and Phrases: Unique Equilibrium, Correlated Equilibrium, Regret Minimization 1. INTRODUCTION A desirable situation in game theory as well as in mechanism design is when the game or mechanism has a unique equilibrium, as the unique equilibrium is typically viewed as the outcome that is likely to emerge. Previous work in showing unique- ness has focused on uniqueness of Nash and Bayes-Nash equilibria. These equilibria, however, may not be learnable. In some cases natural learning dynamic converges to them in polynomial time; in other cases it takes exponentially long to converge or does not converge at all. For example, Kleinberg et al. [2011] give an example where the unique Nash equilibrium of a game has low social welfare, while natural learn- ing behavior in repeated play ensures high average value for all players. Further, no polynomial-time algorithms for computing Nash and Bayes-Nash equilibria are known for these games, and more more generally, computing Nash equilibria is known to be PPAD complete [Daskalakis et al. 2009]. An alternate equilibrium concept that does not share these drawbacks is the concept of a correlated equilibrium. Correlated equi- libria form a convex set, and correlated equilibria can be found in polynomial time Authors’ addresses: P. Dutting,¨ Department of Computer Science, Stanford University, 353 Serra Mall, Stanford, CA 94305, USA; email: [email protected]; T. Kesselheim and E.´ Tardos, Department of Computer Science, Cornell University, Gates Hall, Ithaca, NY 14850, USA; email: fkesselheim,[email protected]. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or repub- lish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. EC’14, June 8–12, 2014, Stanford, CA, USA. ACM 978-1-4503-2565-3/14/06 ...$15.00. Copyright is held by the owner/author(s). Publication rights licensed to ACM. http://dx.doi.org/10.1145/2600057.2602838 even in compactly described games [Papadimitriou and Roughgarden 2008; Jiang and Leyton-Brown 2011]. Maybe more importantly, correlated equilibria emerge quite nat- urally from sequences of repeated play in which the players minimize a form of regret, and simple and natural polynomial-time algorithms are known for minimizing regret (see [Blum and Mansour 2007] for a survey). Motivated by this, we ask: Under which conditions does a mechanism have a unique correlated equilibrium? We thus strive for stronger uniqueness results: We do not only want the equilibrium to be unique, we also want it to be the outcome of natural learning dynamic. We consider single-parameter mechanism design settings. In these settings the pri- vate information held by the players—commonly referred to as their type—can be de- scribed by a single number. We analyze mechanisms that collect reports by the play- ers, and compute an outcome and payments. In the first part of the paper, we focus on a large class of mechanisms that have binary outcomes for each player; they ei- ther are selected or not selected. For such mechanisms, the outcome describes which players win and which players lose, and the payments describe how much the play- ers are charged. In the second part of the paper we extend our results to mechanisms whose outcome for each player is an amount they win. These mechanisms compute the amounts won by the players, and how much they have to pay for it. We use a very general player utility function. We assume that a player’s utility is zero if he loses, and it is given by some continuous function of the bid vector if he wins, that is bounded away from zero if he reports less than its true type. Clearly, first- and second-price mechanisms with quasi-linear utilities satisfy these requirements; but they are also met by other mechanisms and utility functions. — Our model incorporates risk averse players, as we do not assume quasi-linear util- ities. Rather the utility can be any continuous function of the bids (and hence the price). — Our model incorporates certain types of externalities, as the utility of the player can depend on the bid vector, and not only on his own price. For example, the utility of one player can be dependent on the prices paid by others. An interesting special case of our framework is Bertrand competition, where our model includes risk averse firms and price-dependent demand. Another interesting special case of our model are multi-unit auctions with unit-demand bidders where our model—among other things—allows to model fairness by taking the highest-to-lowest price ratio into account. 1.1. Our Results Our main result in Section 3 is that mechanisms selecting an independent set of play- ers maximizing the sum of reported types in a matroid have a unique correlated equi- librium if the player utilities satisfy the mild assumptions above. The unique equilib- rium we exhibit achieves maximum social welfare. Our main result is applicable to a a vast number of practical problems, including — procurement auctions (such as first or second price, or any combination) in which a single buyer buys from multiple sellers, — problems in communication networks in which communication links have to be bought so that the bought communication links form a spanning tree, or — instances of Bertrand competition in which several firms compete to sell their good to a single buyer. When applied to these problems our result shows that a large class of mechanisms has a unique, socially optimal, and learnable equilibrium. In each case, our result applies with very many different pricing rules, and we allow different players to be risk averse to different extents and have externalities on the prices paid by others. We also show that our uniqueness result for matroids is tight by demonstrating that a unilateral relaxation of any of its requirements—mechanism is optimizing, feasibility structure is a matroid, solution concept is that of a correlated equilibrium—can lead to more than one equilibrium. In Section 4, we extend our considerations to polymatroids. Polymatroids are an important generalization of matroids, and arise, for example, in the context of — sponsored search auctions [Goel et al. 2012] in which advertisers seek to be assigned clicks on so-called sponsored search results that are shown along with the organic search results produced by a search engine, or — Bertrand networks [Babaioff et al. 2013] in which firms compete for markets consist- ing of a single buyer and markets are either captive so that only one firm has access to it or shared between two firms. While there can be multiple equilibria in general polymatroid settings, we show condi- tions under which uniqueness carries over. These apply for example in Bertrand net- works without captive markets. Applied to this problem, our result shows that there is a unique, learnable equilibrium with zero payoffs. Together our uniqueness results thus imply stronger uniqueness results for models of Bertrand competition than the ones that were previously known, extending the re- sults to correlated equilibria, to networked markets, and to the general player utilities studied here. 1.2. Proof Technique The technique underlying our uniqueness results has similarities to iterated elimina- tion of strictly dominated strategies, but in our case no strategies are strictly domi- nated. Therefore, we do not argue on a per-player basis considering unilateral devia- tions of a single player only. Instead, we carefully combine effects of unilateral devi- ations by groups of players. As none of the involved players may benefit from these deviations, we argue that certain ranges of the combined strategy space will never be entered when playing a correlated equilibrium. This proof technique may be applicable to settings beyond the settings studied here. 1.3.