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.

Multi-Receiver Online Bayesian Persuasion

Bayesian persuasion studies how an informed sender should partially disclose information to influence the behavior of a self-interested receiver. Classical models make the stringent assumption that the sender knows the receivers utility. This can be relaxed by considering an online learning framework in which the sender repeatedly faces a receiver of an unknown, adversarially selected type. We study, for the first time, an online Bayesian persuasion setting with multiple receivers. We focus on the case with no externalities and binary actions, as customary in offline models. Our goal is to design no-regret algorithms for the sender with polynomial per-iteration running time. First, we prove a negative result: for any $0 < alpha leq 1$, there is no polynomial-time no-$alpha$-regret algorithm when the senders utility function is supermodular or anonymous. Then, we focus on the case of submodular senders utility functions and we show that, in this case, it is possible to design a polynomial-time no-$(1 - frac{1}{e})$-regret algorithm. To do so, we introduce a general online gradient descent scheme to handle online learning problems with a finite number of possible loss functions. This requires the existence of an approximate projection oracle. We show that, in our setting, there exists one such projection oracle which can be implemented in polynomial time.

Simple Uncoupled No-Regret Learning Dynamics for Extensive-Form Correlated Equilibrium

The existence of simple uncoupled no-regret learning 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 games generalize normal-form games by modeling both sequential and simultaneous moves, as well as imperfect information. Because of the sequential nature and presence of private information in the game, correlation in extensive-form games possesses significantly different properties than its counterpart in normal-form games, many of which are still open research directions. Extensive-form correlated equilibrium (EFCE) has been proposed as the natural extensive-form counterpart to the classical notion of correlated equilibrium in normal-form games. Compared to the latter, the constraints that define the set of EFCEs are significantly more complex, as the correlation device must keep into account the evolution of beliefs of each player as they make observations throughout the game. Due to that significant added complexity, the existence of uncoupled learning dynamics leading to an EFCE has remained a challenging open research question for a long time. In this article, we settle that question by giving the first uncoupled no-regret dynamics that converge to the set of EFCEs in n-player general-sum extensive-form games with perfect recall. We show that each iterate can be computed in time polynomial in the size of the game tree, and that, when all players play repeatedly according to our learning dynamics, the empirical frequency of play is proven to be a O(T^-0.5)-approximate EFCE with high probability after T game repetitions, and an EFCE almost surely in the limit.

Multi-Agent Coordination in Adversarial Environments through Signal Mediated Strategies

Many real-world scenarios involve teams of agents that have to coordinate their actions to reach a shared goal. We focus on the setting in which a team of agents faces an opponent in a zero-sum, imperfect-information game. Team members can coordinate their strategies before the beginning of the game, but are unable to communicate during the playing phase of the game. This is the case, for example, in Bridge, collusion in poker, and collusion in bidding. In this setting, model-free RL methods are oftentimes unable to capture coordination because agents policies are executed in a decentralized fashion. Our first contribution is a game-theoretic centralized training regimen to effectively perform trajectory sampling so as to foster team coordination. When team members can observe each other actions, we show that this approach provably yields equilibrium strategies. Then, we introduce a signaling-based framework to represent team coordinated strategies given a buffer of past experiences. Each team members policy is parametrized as a neural network whose output is conditioned on a suitable exogenous signal, drawn from a learned probability distribution. By combining these two elements, we empirically show convergence to coordinated equilibria in cases where previous state-of-the-art multi-agent RL algorithms did not.

Faster Algorithms for Optimal Ex-Ante Coordinated Collusive Strategies in Extensive-Form Zero-Sum Games

We focus on the problem of finding an optimal strategy for a team of two players that faces an opponent in an imperfect-information zero-sum extensive-form game. Team members are not allowed to communicate during play but can coordinate before the game. In that setting, it is known that the best the team can do is sample a profile of potentially randomized strategies (one per player) from a joint (a.k.a. correlated) probability distribution at the beginning of the game. In this paper, we first provide new modeling results about computing such an optimal distribution by drawing a connection to a different literature on extensive-form correlation. Second, we provide an algorithm that computes such an optimal distribution by only using profiles where only one of the team members gets to randomize in each profile. We can also cap the number of such profiles we allow in the solution. This begets an anytime algorithm by increasing the cap. We find that often a handful of well-chosen such profiles suffices to reach optimal utility for the team. This enables team members to reach coordination through a relatively simple and understandable plan. Finally, inspired by this observation and leveraging theoretical concepts that we introduce, we develop an efficient column-generation algorithm for finding an optimal distribution for the team. We evaluate it on a suite of common benchmark games. It is three orders of magnitude faster than the prior state of the art on games that the latter can solve and it can also solve several games that were previously unsolvable.

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.

Public Bayesian Persuasion: Being Almost Optimal and Almost Persuasive

Persuasion studies how an informed principal may influence the behavior of agents by the strategic provision of payoff-relevant information. We focus on the fundamental multi-receiver model by Arieli and Babichenko (2019), in which there are no inter-agent externalities. Unlike prior works on this problem, we study the public persuasion problem in the general setting with: (i) arbitrary state spaces; (ii) arbitrary action spaces; (iii) arbitrary senders utility functions. We fully characterize the computational complexity of computing a bi-criteria approximation of an optimal public signaling scheme. In particular, we show, in a voting setting of independent interest, that solving this problem requires at least a quasi-polynomial number of steps even in settings with a binary action space, assuming the Exponential Time Hypothesis. In doing so, we prove that a relaxed version of the Maximum Feasible Subsystem of Linear Inequalities problem requires at least quasi-polynomial time to be solved. Finally, we close the gap by providing a quasi-polynomial time bi-criteria approximation algorithm for arbitrary public persuasion problems that, in specific settings, yields a QPTAS.