Strategic Multi-Armed Bandit Problems Under Debt-Free Reporting

Dear Reviewer VQPm,

Thank you very much for your detailed and insightful review. We greatly appreciate the time and effort you dedicated to it.

We are computing regret with reference to $\mu_{2}$, which involves incentivizing the best arm with $O(T \Delta_{12})$. This approach is not harmful, as it offsets the difference between $\mu_{1}$ and $\mu_{2}$ while preserving $\mu_{2}$ for the player. Essentially, we allocate all values greater than $\mu_{2}$ to the best arm throughout the game. Additionally, the incentive should address the second phase. If not properly incentivized during this phase, the best arm might withhold a portion of the reward (especially since the second-best mean is not communicated), leading to reported values that are less than $\mu_{2}$ on average. This could result in linear regret, given that the second phase lasts much longer than the first. Thus, incentivizing the best arm with $O(T \Delta_{12})$ appears necessary (as also done by Braverman et al.), which is acceptable given the definition of regret. However, improvements have been achieved through the new bonus structure assigned to suboptimal arms, which has been optimized due to the adaptiveness of the new algorithm.

We appreciate your feedback on the weaknesses in our representations, language, and typos. We will address these issues and make the necessary corrections in the final version.

We hope that we have addressed your concerns and answered your questions clearly, contributing positively to your evaluation.

Dear Reviewer 3dvn,

Thank you for your insightful feedback and for the time you dedicated to reviewing our paper.

In the current version of our paper, we did not include experimental results, which is indeed quite common in game theory, due to the fact that the dynamics of the game are intrinsically dependent on arm strategies that are generally intractable in real-world scenarios. Consequently, we provided a theoretical analysis of the dominant Subgame Perfect Equilibrium (SPE) and discussed the tightness of its regret. However, inspired by your review, we have conducted a simulated experiment where we fix the arm strategies from the beginning and track the cumulative regret. Specifically, we consider six strategic arms with $\mu_{1} > \mu_{2} > \mu_{3} > \mu_{4} > \mu_{5} > \mu_{6}$. The experiment is run over a horizon of 10,000 and averaged over 100 epochs. We examine three scenarios:

1. Untruthful Arbitrary Reporting: At each round, the selected arm randomly reports 100%, 60%, 40%, 10%, or 0% of its observed reward.
2. Truthful Reporting: This corresponds to the Dominant SPE.
3. "Optimal" Reporting: Here we use the term "optimal" somewhat loosely. In this scenario, only the two best arms report truthfully, while the remaining suboptimal arms report 0.

The results are detailed in the attached PDF. As expected, the worst regret corresponds to the first scenario. However, for the two remaining scenarios, both exhibit logarithmic cumulative regret, with the "optimal" reporting scenario demonstrating a better factor since the algorithm eliminates suboptimal arms more quickly and transitions faster to the second phase where it achieves better performance. While "optimal" reporting is challenging to achieve, S-SE effectively incentivizes arms to report truthfully, creating a dominant SPE and guaranteeing the player logarithmic regret.

Our model is inspired by several real-world applications and can be seen as a multi-agent extension of the principal-agent problem in contract theory. It deals with dynamic agency issues where a principal must select one of K agents to carry out tasks on their behalf, with the cost of these tasks remaining unknown to the principal (e.g., choosing among K contractors for a job). A crucial aspect is that the principal does not know in advance the exact cost or benefit of the actions performed. The concept of debt-free reporting of strategic arms is inspired by e-commerce transactions, where a platform may choose to cancel a sale but cannot create one. This generalizes to binary bandits, where it is easy to hide a success and report a fake failure but difficult to create a success. Additionally, it is motivated by repeated trades with budget constraints, where an arm cannot report more than it possesses.

We also agree with your suggestion regarding the relabeling of the limitations section and will make the necessary corrections.

We hope that we have addressed your concerns and answered your questions satisfactorily, contributing positively to your evaluation.

Dear Reviewer JBtJ,

We appreciate your review and the time you've dedicated to it.

In this paper, we concentrate on strategic arms under debt-free reporting, driven by various real-world applications such as interactions on e-commerce platforms and repeated trades with budget constraints [9, 10], where reported values cannot surpass observed ones. Under the unconstrained payment setting, where arms can report any value, [8] presented an optimal mechanism with constant regret. However, when this assumption is relaxed to restricted payments (i.e., debt-free reporting), [8] proposes a mechanism that suffers from a regret of $T^{2/3}$. Hence, moving from unrestricted payment to debt-free reporting increases the regret from a constant value to $T^{2/3}$. This observation motivated us to focus on this setting.

While [8] employs additional rounds at the end of the game as incentives, we propose using payments at the end. Both types of bonuses are allocated at the end of the game, and bonuses are subtracted from player revenue and considered in the regret calculation, regardless of their nature. By using payments as incentives, defining truthful reporting as a dominant subgame perfect equilibrium (SPE) is straightforward. However, with additional rounds, we refer to a pseudo-truthful strategy since truthfulness applies only to rounds excluding the additional bonus rounds, specifically during the initial phase when the algorithm is learning the statistics of each arm. During the additional bonus rounds, which occur at the end of the game, the player ignores the values reported by the arms, as clearly stated in [8]. Given that actual payment is a common practice in typical mechanism designs for procurement auctions, we chose actual payments to maintain a more direct intuition and facilitate the presentation. In fact, a modified version of our algorithm that includes playing each arm $\frac{\Psi_k}{\mu_k}$ at the end effectively regains the payment through rounds. As already mentioned in Section 6, the improvement in regret does not depend on the type of incentives, whether they are additional bonus rounds or bonus payments. This is because bonuses are subtracted from player revenue and considered in the regret calculation, irrespective of their nature. Therefore, replacing the additional rounds in [8] with payments will not lead to an improvement in regret. The improvement is mainly due to the adaptivity of the algorithm, which allows for more tailored and less harmful bonuses. However, this adaptivity also necessitates a more technical solution concept due to the extensive nature of the game, as opposed to the simultaneous one used in [8].

If the player knows the minimum gap $\Delta$, an explore-then-commit-based algorithm similar to [8] can achieve $\log T$ regret. In fact, it suffices to explore each arm for $\log(T)/\Delta^2$ rounds instead of $T^{2/3}$. However, the purpose of bandit algorithms is to overcome the lack of this knowledge and effectively address cases where $\Delta$ is unknown.

We thank you for pointing out the representation issues, and we assure you that we will adjust the final version of our work accordingly.

We hope this addresses your concerns and contributes positively to your evaluation.

Dear Reviewer byTp,

We express our sincere gratitude for your invaluable review and the time you have dedicated to it.

This paper, motivated by various real-world applications such as e-commerce and repeated trades with balanced budgets, delves into strategic arms under debt-free reporting. Our primary objective is to address an open problem in [8], which involves improving the regret under debt-free reporting, aiming to progress from $T^{2/3}$ to $\sqrt{T \log(T)}$. Therefore, we adopt the same setting as in [8], including a fixed number of strategic arms. On one hand, this setting inherently includes the scenario of arms exiting the market, equivalent to arms deciding to report 0 for all subsequent steps. On the other hand, accommodating arms entering the market would necessitate significant changes to the setting (compared to [8]) and would require introducing concepts of adversarial rewarding or even non-stationary systems. While we agree that this is a fascinating proposal, we demonstrate that even within the same setting as [8], there are substantial improvements (such as a more suitable incentive-compatible algorithm, better/tighter regret bounds, robustness to small deviations, etc.), which form the core of this paper. We assure you that the notion of a time-variant market will be included in the future work section, as we recognize that it will motivate further significant research.

In the strategic setting, under the most unfavorable circumstances, it is impossible to guarantee gains surpassing $\mu_2$. This limitation arises because the optimal arm only needs to report a marginally higher value than the second-best arm to be chosen as the superior option by the player, as formally presented in Lemma 14 of [8]. Therefore, providing a bonus of $O(T \Delta_{12})$ to the best arm to ensure truthful reporting remains optimal for the player, as it guarantees a return of $\mu_{2}$ each time the best arm is played. Generally speaking, the harmful bonus in the strategic case is the one subtracted from $\mu_{2}$, i.e., the ones paid to the suboptimal arms, which is sublinear in our case (on the order of $\log(T) / \Delta$). As a side note, even [8] rewards the best arms with $O(T \Delta_{12})$ (refer to Mechanism 2, Line 3), which is entirely acceptable since it does not contribute to the regret.

In the paragraph "Tightness of Regret" (line 271 of our paper), we discussed that the upper bound on the regret is tight and that the lower bound is of the same order as the result given in (12).

Additionally, we want to highlight the contributions of our paper. There is a notable technicality worth mentioning regarding the definition and proof of our solution concept. Our setting involves an extensive game, as opposed to a simultaneous one (as in Braverman et al., 2019, where learning occurs once at the beginning). In our extensive game, the equilibrium strategy of each arm at each node must consider the updated history of the entire game up to that point. This task is generally intractable (Ben-Porath, 1990; Conitzer and Sandholm, 2002). Interestingly, we demonstrate that our mechanism ensures truthfulness as the best response to adversarial strategies independent of history. Our approach entails consistent truthful reporting, regardless of the equilibrium path. Notably, this remains an equilibrium irrespective of past events, resembling sub-game perfect equilibrium. While the adaptivity of our algorithm introduces a more complicated solution concept for the analysis, it is crucial for the improvement in regret that we present. This adaptivity necessitates well-tailored and less harmful incentivizing bonuses, which are key to better regret. The paper also includes an additional bound which nicely characterizes the regret under deviations giving an idea about the robustness of the algorithm.

We appreciate your feedback and assure you that this discussion will be incorporated into the final version of our work. We hope this addresses your concerns and contributes positively to your evaluation, leading to a better score.