Efficient Decentralized Learning Dynamics for Extensive-Form Coarse Correlated Equilibrium: No Expensive Computation of Stationary Distributions Required

While in two-player zero-sum games the Nash equilibrium is a well-established prescriptive notion of optimal play, its applicability as a prescriptive tool beyond that setting is limited. Consequently, the study of decentralized learning dynamics that guarantee convergence to correlated solution concepts in multiplayer, general-sum extensive-form (i.e., tree-form) games has become an important topic of active research. The per-iteration complexity of the currently known learning dynamics depends on the specific correlated solution concept considered. For example, in the case of extensive-form correlated equilibrium (EFCE), all known dynamics require, as an intermediate step at each iteration, to compute the stationary distribution of multiple Markov chains, an expensive operation in practice. Oppositely, in the case of normal-form coarse correlated equilibrium (NFCCE), simple no-external-regret learning dynamics that amount to a linear-time traversal of the tree-form decision space of each agent suffice to guarantee convergence. This paper focuses on extensive-form coarse correlated equilibrium (EFCCE), an intermediate solution concept that is a subset of NFCCE and a superset of EFCE. Being a superset of EFCE, any learning dynamics for EFCE automatically guarantees convergence to EFCCE. However, since EFCCE is a simpler solution concept, this begs the question: do learning dynamics for EFCCE that avoid the expensive computation of stationary distributions exist? This paper answers the previous question in the positive. Our learning dynamics only require the orchestration of no-external-regret minimizers, thus showing that EFCCE is more akin to NFCCE than to EFCE from a learning perspective. Our dynamics guarantees that the empirical frequency of play after $T$ iteration is a $O(1/sqrt{T})$-approximate EFCCE with high probability, and an EFCCE almost surely in the limit.

Learning Fair Equilibria in Sponsored Search Auctions

In this work we investigate the strategic learning implications of the deployment of sponsored search auction mechanisms that obey to fairness criteria. We introduce a new class of mechanisms composing a traditional Generalized Second Price auction (GSP) with different fair division schemes to achieve some desired level of fairness between two groups of Bayesian strategic advertisers. We propose two mechanisms, $beta$-Fair GSP and GSP-EFX, that compose GSP with, respectively, an envy-free up to one item (EF1), and an envy-free up to any item (EFX) fair division scheme. The payments of GSP are adjusted in order to compensate the advertisers that suffer a loss of efficiency due the fair division stage. We prove that, for both mechanisms, if bidders play so as to minimize their external regret they are guaranteed to reach an equilibrium with good social welfare. We also prove that the mechanisms are budget balanced, so that the payments charged by the traditional GSP mechanism are a good proxy of the total compensation offered to the advertisers. Finally, we evaluate the quality of the allocations of the two mechanisms through experiments on real-world data.

The Parity Ray Regularizer for Pacing in Auction Markets

Budget-management systems are one of the key components of modern auction markets. Internet advertising platforms typically offer advertisers the possibility to pace the rate at which their budget is depleted, through budget-pacing mechanisms. We focus on multiplicative pacing mechanisms in an online setting in which a bidder is repeatedly confronted with a series of advertising opportunities. After collecting bids, each item is then allocated through a single-item, second-price auction. If there were no budgetary constraints, bidding truthfully would be an optimal choice for the advertiser. However, since their budget is limited, the advertiser may want to shade their bid downwards in order to preserve their budget for future opportunities, and to spread expenditures evenly over time. The literature on online pacing problems mostly focuses on the setting in which the bidder optimizes an additive separable objective, such as the total click-through rate or the revenue of the allocation. In many settings, however, bidders may also care about other objectives which oftentimes are non-separable, and therefore not amenable to traditional online learning techniques. Building on recent work, we study the frequent case in which advertisers seek to reach a certain distribution of impressions over a target population of users. We introduce a novel regularizer to achieve this desideratum, and show how to integrate it into an online mirror descent scheme attaining the optimal order of sub-linear regret compared to the optimal allocation in hindsight when inputs are drawn independently, from an unknown distribution. Moreover, we show that our approach can easily be incorporated in standard existing pacing systems that are not usually built for this objective. The effectiveness of our algorithm in internet advertising applications is confirmed by numerical experiments on real-world data.

No-Regret Learning Dynamics for Extensive-Form Correlated Equilibrium

The existence of simple, uncoupled no-regret dynamics that converge to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their internal regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium. Extensive-form (that is, tree-form) games generalize normal-form games by modeling both sequential and simultaneous moves, as well as private information. Because of the sequential nature and presence of partial information in the game, extensive-form correlation has significantly different properties than the normal-form counterpart, many of which are still open research directions. Extensive-form correlated equilibrium (EFCE) has been proposed as the natural extensive-form counterpart to normal-form correlated equilibrium. However, it was currently unknown whether EFCE emerges as the result of uncoupled agent dynamics. In this paper, we give the first uncoupled no-regret dynamics that converge to the set of EFCEs in $n$-player general-sum extensive-form games with perfect recall. First, we introduce a notion of trigger regret in extensive-form games, which extends that of internal regret in normal-form games. When each player has low trigger regret, the empirical frequency of play is close to an EFCE. Then, we give an efficient no-trigger-regret algorithm. Our algorithm decomposes trigger regret into local subproblems at each decision point for the player, and constructs a global strategy of the player from the local solutions at each decision point.

Signaling in Bayesian Network Congestion Games: the Subtle Power of Symmetry

Network congestion games are a well-understood model of multi-agent strategic interactions. Despite their ubiquitous applications, it is not clear whether it is possible to design information structures to ameliorate the overall experience of the network users. We focus on Bayesian games with atomic players, where network vagaries are modeled via a (random) state of nature which determines the costs incurred by the players. A third-party entity---the sender---can observe the realized state of the network and exploit this additional information to send a signal to each player. A natural question is the following: is it possible for an informed sender to reduce the overall social cost via the strategic provision of information to players who update their beliefs rationally? The paper focuses on the problem of computing optimal ex ante persuasive signaling schemes, showing that symmetry is a crucial property for its solution. Indeed, we show that an optimal ex ante persuasive signaling scheme can be computed in polynomial time when players are symmetric and have affine cost functions. Moreover, the problem becomes NP-hard when players are asymmetric, even in non-Bayesian settings.

Persuading Voters: Its Easy to Whisper, Its Hard to Speak Loud

We focus on the following natural question: is it possible to influence the outcome of a voting process through the strategic provision of information to voters who update their beliefs rationally? We investigate whether it is computationally tractable to design a signaling scheme maximizing the probability with which the senders preferred candidate is elected. We focus on the model recently introduced by Arieli and Babichenko (2019) (i.e., without inter-agent externalities), and consider, as explanatory examples, $k$-voting rule and plurality voting. There is a sharp contrast between the case in which private signals are allowed and the more restrictive setting in which only public signals are allowed. In the former, we show that an optimal signaling scheme can be computed efficiently both under a $k$-voting rule and plurality voting. In establishing these results, we provide two general (i.e., applicable to settings beyond voting) contributions. Specifically, we extend a well known result by Dughmi and Xu (2017) to more general settings, and prove that, when the senders utility function is anonymous, computing an optimal signaling scheme is fixed parameter tractable w.r.t. the number of receivers actions. In the public signaling case, we show that the senders optimal expected return cannot be approximated to within any factor under a $k$-voting rule. This negative result easily extends to plurality voting and problems where utility functions are anonymous.

Bayesian Persuasion with Sequential Games

We study an information-structure design problem (a.k.a. persuasion) with a single sender and multiple receivers with actions of a priori unknown types, independently drawn from action-specific marginal distributions. As in the standard Bayesian persuasion model, the sender has access to additional information regarding the action types, which she can exploit when committing to a (noisy) signaling scheme through which she sends a private signal to each receiver. The novelty of our model is in considering the case where the receivers interact in a sequential game with imperfect information, with utilities depending on the game outcome and the realized action types. After formalizing the notions of ex ante and ex interim persuasiveness (which differ in the time at which the receivers commit to following the senders signaling scheme), we investigate the continuous optimization problem of computing a signaling scheme which maximizes the senders expected revenue. We show that computing an optimal ex ante persuasive signaling scheme is NP-hard when there are three or more receivers. In contrast with previous hardness results for ex interim persuasion, we show that, for games with two receivers, an optimal ex ante persuasive signaling scheme can be computed in polynomial time thanks to a novel algorithm based on the ellipsoid method which we propose.

Computing Optimal Coarse Correlated Equilibria in Sequential Games

We investigate the computation of equilibria in extensive-form games where ex ante correlation is possible, focusing on correlated equilibria requiring the least amount of communication between the players and the mediator. Motivated by the hardness results on the computation of normal-form correlated equilibria, we introduce the notion of normal-form coarse correlated equilibrium, extending the definition of coarse correlated equilibrium to sequential games. We show that, in two-player games without chance moves, an optimal (e.g., social welfare maximizing) normal-form coarse correlated equilibrium can be computed in polynomial time, and that in general multi-player games (including two-player games with Chance), the problem is NP-hard. For the former case, we provide a polynomial-time algorithm based on the ellipsoid method and also propose a more practical one, which can be efficiently applied to problems of considerable size. Then, we discuss how our algorithm can be extended to games with Chance and games with more than two players.

Computational Results for Extensive-Form Adversarial Team Games

We provide, to the best of our knowledge, the first computational study of extensive-form adversarial team games. These games are sequential, zero-sum games in which a team of players, sharing the same utility function, faces an adversary. We define three different scenarios according to the communication capabilities of the team. In the first, the teammates can communicate and correlate their actions both before and during the play. In the second, they can only communicate before the play. In the third, no communication is possible at all. We define the most suitable solution concepts, and we study the inefficiency caused by partial or null communication, showing that the inefficiency can be arbitrarily large in the size of the game tree. Furthermore, we study the computational complexity of the equilibrium-finding problem in the three scenarios mentioned above, and we provide, for each of the three scenarios, an exact algorithm. Finally, we empirically evaluate the scalability of the algorithms in random games and the inefficiency caused by partial or null communication.

Team-maxmin equilibrium: efficiency bounds and algorithms

The Team-maxmin equilibrium prescribes the optimal strategies for a team of rational players sharing the same goal and without the capability of correlating their strategies in strategic games against an adversary. This solution concept can capture situations in which an agent controls multiple resources-corresponding to the team members-that cannot communicate. It is known that such equilibrium always exists and it is unique (unless degeneracy) and these properties make it a credible solution concept to be used in real-world applications, especially in security scenarios. Nevertheless, to the best of our knowledge, the Team-maxmin equilibrium is almost completely unexplored in the literature. In this paper, we investigate bounds of (in)efficiency of the Team-maxmin equilibrium w.r.t. the Nash equilibria and w.r.t. the Maxmin equilibrium when the team members can play correlated strategies. Furthermore, we study a number of algorithms to find and/or approximate an equilibrium, discussing their theoretical guarantees and evaluating their performance by using a standard testbed of game instances.