Routers in networking use simple learning algorithms to find the best way to deliver packets to their desired destination. This simple, myopic and distributed decision system makes large queuing systems simple to operate, but at the same time, the system needs more capacity than would be required if all traffic were centrally coordinated. In a recent paper, Gaitonde and Tardos (EC 2020 and JACM 2023) initiate the study of such systems, modeling them as an infinitely repeated game in which routers compete for servers and the system maintains a state (number of packets held by each queue) resulting from outcomes of previous rounds. Queues get to send a packet at each step to one of the servers, and servers attempt to process only one of the arriving packets, modeling routers. However, their model assumes that servers have no buffers at all, so queues have to resend all packets that were not served successfully. They show that, even with hugely increased server capacity relative to what is needed in the centrally-coordinated case, ensuring that the system is stable requires using timestamps and priority for older packets. We consider a system with two important changes, which make the model more realistic: first we add a very small buffer to each server, allowing it to hold on to a single packet to be served later (even if it fails to serve it); and second, we do not require timestamps or priority for older packets. Our main result is to show that when queues are learning, a small constant factor increase in server capacity, compared to what would be needed if centrally coordinating, suffices to keep the system stable, even if servers select randomly among packets arriving simultaneously. This work contributes to the growing literature on the impact of selfish learning in systems with carryover effects between rounds: when outcomes in the present round affect the game in the future.

We analyze the performance of heterogeneous learning agents in asset markets with stochastic payoffs. Our agents aim to maximize the expected growth rate of their wealth but have different theories on how to learn this best. We focus on comparing Bayesian and no-regret learners in market dynamics. Bayesian learners with a prior over a finite set of models that assign positive prior probability to the correct model have posterior probabilities that converge exponentially to the correct model. Consequently, they survive even in the presence of agents who invest according to the correct model of the stochastic process. Bayesians with a continuum prior converge to the correct model at a rate of $O((\log T)/T)$. Online learning theory provides no-regret algorithms for maximizing the log of wealth in this setting, achieving a worst-case regret bound of $O(\log T)$ without assuming a steady underlying stochastic process but comparing to the best fixed investment rule. This regret, as we observe, is of the same order of magnitude as that of a Bayesian learner with a continuum prior. However, we show that even such low regret may not be sufficient for survival in asset markets: an agent can have regret as low as $O(\log T)$, but still vanish in market dynamics when competing against agents who invest according to the correct model or even against a perfect Bayesian with a finite prior. On the other hand, we show that Bayesian learning is fragile, while no-regret learning requires less knowledge of the environment and is therefore more robust. Any no-regret learner will drive out of the market an imperfect Bayesian whose finite prior or update rule has even small errors. We formally establish the relationship between notions of survival, vanishing, and market domination studied in economics and the framework of regret minimization, thus bridging these theories.

In many repeated auction settings, participants care not only about how frequently they win but also how their winnings are distributed over time. This problem arises in various practical domains where avoiding congested demand is crucial, such as online retail sales and compute services, as well as in advertising campaigns that require sustained visibility over time. We introduce a simple model of this phenomenon, modeling it as a budgeted auction where the value of a win is a concave function of the time since the last win. This implies that for a given number of wins, even spacing over time is optimal. We also extend our model and results to the case when not all wins result in "conversions" (realization of actual gains), and the probability of conversion depends on a context. The goal is to maximize and evenly space conversions rather than just wins. We study the optimal policies for this setting in second-price auctions and offer learning algorithms for the bidders that achieve low regret against the optimal bidding policy in a Bayesian online setting. Our main result is a computationally efficient online learning algorithm that achieves $\tilde O(\sqrt T)$ regret. We achieve this by showing that an infinite-horizon Markov decision process (MDP) with the budget constraint in expectation is essentially equivalent to our problem, even when limiting that MDP to a very small number of states. The algorithm achieves low regret by learning a bidding policy that chooses bids as a function of the context and the system's state, which will be the time elapsed since the last win (or conversion). We show that state-independent strategies incur linear regret even without uncertainty of conversions. We complement this by showing that there are state-independent strategies that, while still having linear regret, achieve a $(1-\frac 1 e)$ approximation to the optimal reward.

In repeated games, such as auctions, players typically use learning algorithms to choose their actions. The use of such autonomous learning agents has become widespread on online platforms. In this paper, we explore the impact of players incorporating monetary transfers into their agents' algorithms, aiming to incentivize behavior in their favor. Our focus is on understanding when players have incentives to make use of monetary transfers, how these payments affect learning dynamics, and what the implications are for welfare and its distribution among the players. We propose a simple game-theoretic model to capture such scenarios. Our results on general games show that in a broad class of games, players benefit from letting their learning agents make payments to other learners during the game dynamics, and that in many cases, this kind of behavior improves welfare for all players. Our results on first- and second-price auctions show that in equilibria of the ``payment policy game,'' the agents' dynamics can reach strong collusive outcomes with low revenue for the auctioneer. These results highlight a challenge for mechanism design in systems where automated learning agents can benefit from interacting with their peers outside the boundaries of the mechanism.

Many real-life contractual relations differ completely from the clean, static model at the heart of principal-agent theory. Typically, they involve repeated strategic interactions of the principal and agent, taking place under uncertainty and over time. While appealing in theory, players seldom use complex dynamic strategies in practice, often preferring to circumvent complexity and approach uncertainty through learning. We initiate the study of repeated contracts with a learning agent, focusing on agents who achieve no-regret outcomes. Optimizing against a no-regret agent is a known open problem in general games; we achieve an optimal solution to this problem for a canonical contract setting, in which the agent's choice among multiple actions leads to success/failure. The solution has a surprisingly simple structure: for some $\alpha > 0$, initially offer the agent a linear contract with scalar $\alpha$, then switch to offering a linear contract with scalar $0$. This switch causes the agent to ``free-fall'' through their action space and during this time provides the principal with non-zero reward at zero cost. Despite apparent exploitation of the agent, this dynamic contract can leave \emph{both} players better off compared to the best static contract. Our results generalize beyond success/failure, to arbitrary non-linear contracts which the principal rescales dynamically. Finally, we quantify the dependence of our results on knowledge of the time horizon, and are the first to address this consideration in the study of strategizing against learning agents.

We study Proportional Response Dynamics (PRD) in linear Fisher markets where participants act asynchronously. We model this scenario as a sequential process in which in every step, an adversary selects a subset of the players that will update their bids, subject to liveness constraints. We show that if every bidder individually uses the PRD update rule whenever they are included in the group of bidders selected by the adversary, then (in the generic case) the entire dynamic converges to a competitive equilibrium of the market. Our proof technique uncovers further properties of linear Fisher markets, such as the uniqueness of the equilibrium for generic parameters and the convergence of associated best-response dynamics and no-swap regret dynamics under certain conditions.

Understanding emerging behaviors of reinforcement learning (RL) agents may be difficult since such agents are often trained in complex environments using highly complex decision making procedures. This has given rise to a variety of approaches to explainability in RL that aim to reconcile discrepancies that may arise between the behavior of an agent and the behavior that is anticipated by an observer. Most recent approaches have relied either on domain knowledge that may not always be available, on an analysis of the agent's policy, or on an analysis of specific elements of the underlying environment, typically modeled as a Markov Decision Process (MDP). Our key claim is that even if the underlying model is not fully known (e.g., the transition probabilities have not been accurately learned) or is not maintained by the agent (i.e., when using model-free methods), the model can nevertheless be exploited to automatically generate explanations. For this purpose, we suggest using formal MDP abstractions and transforms, previously used in the literature for expediting the search for optimal policies, to automatically produce explanations. Since such transforms are typically based on a symbolic representation of the environment, they can provide meaningful explanations for gaps between the anticipated and actual agent behavior. We formally define the explainability problem, suggest a class of transforms that can be used for explaining emergent behaviors, and suggest methods that enable efficient search for an explanation. We demonstrate the approach on a set of standard benchmarks.

This paper deals with the following common type of scenario: several users engage in some strategic online interaction, where each of them is assisted by a learning agent. A typical example is advertisers that compete for advertising slots over some platform. Typically, each of these advertisers enters his key parameters into some advertiser-facing website, and then this website's "agent" participates on the advertiser's behalf in a sequence of auctions for ad slots. Often, the platform designer provides this agent as its advertiser-facing user interface. In cases where the platform's agent does not optimize sufficiently well for the advertiser (but rather, say, for the auctioneer), one would expect some other company to provide a better (for the advertiser) agent.

This paper deals with the following common type of scenario: several users engage in a repeated online auction, where each of them is assisted by a learning agent. A typical example is advertisers that compete for ad-slots: typically, each of these advertisers fills in his key parameters into some advertiser-facing website, and then this website's "agent" participates on the advertiser's behalf in a sequence of auctions for ad slots. Often, the auction's platform designer provides this agent as its advertiser-facing user interface. In cases where the platform's agent does not optimize sufficiently well for the advertiser (but rather, say, for the auctioneer) one would expect some other company to provide a better (for the advertiser) agent. A typical learning algorithm of this type will have at its core some regret-minimization algorithm [33, 9], such as multiplicative weights (see [2] and references therein) or some variant of fictitious play [12, 54], such as follow the perturbed leader [31, 40]. In particular, for ad auctions, there is empirical data showing that bids are largely consistent with no-regret learning [46, 52].