DeepSeek made its name offering far cheaper AI services than its pricier western competitors, but the cheap ride appears to be at an end. The company has started telling customers to prepare for significant price rises with the advent of its latest model. With the announcement of DeepSeek V4 Pro, the company is raising its API pricing fourfold. The company said it's adopting a new peak and off-peak pricing to "allocate resources more reasonably." Starting on August 16, the DeepSeek V4 Pro model will cost 3.96 for 1 million output tokens at peak hours, more than four times the current rate of 0.87.

Firms' algorithm development practices are often homogeneous. Whether firms train algorithms on similar data or rely on similar pre-trained models, the result is correlated predictions. In the context of personalized pricing, correlated algorithms can be viewed as a means to collude among competing firms, but whether or not this conduct is legal depends on the mechanisms of achieving collusion. We investigate the precise mechanisms through a formal game-theoretic model. Indeed, we find that (1) higher correlation diminishes consumer welfare and (2) as consumers become more price sensitive, firms are increasingly incentivized to compromise on the accuracy of their predictions in exchange for coordination. We demonstrate our theoretical results in a stylized empirical study where two firms compete using personalized pricing algorithms. Our results demonstrate a new mechanism for achieving collusion through correlation, which allows us to analyze its legal implications. Correlation through algorithms is a new frontier of anti-competitive behavior that is largely unconsidered by US antitrust law.

We study contextual online pricing with biased offline data. For the scalar price elasticity case, we identify the instance-dependent quantity δ2 that measures how far the offline data lies from the (unknown) online optimum. We show that the time length T, bias bound V, size N and dispersion λmin(ˆΣ) of the offline data, and δ2 jointly determine the statistical complexity.

We initiate the study of contextual dynamic pricing with a heterogeneous population of buyers, where a seller repeatedly posts prices (over T rounds) that depend on the observable d-dimensional context and receives binary purchase feedback. Unlike prior work assuming homogeneous buyer types, in our setting the buyer's valuation type is drawn from an unknown distribution with finite support size K . We develop a contextual pricing algorithm based on optimistic posterior sampling with regret eO(K dT), which we prove to be tight in dand T up to logarithmic terms. Finally, we refine our analysis for the non-contextual pricing case, proposing a variance-aware zooming algorithm that achieves the optimal dependence on K .

We provide an algorithm with regret O(CdloglogT) for contextual pricing with C corrupted rounds, improving over the previous bound of O(d3Clog2(T)) of Krishnamurthy et al. (2020). The result is based on a reduction that calls the uncorrupted algorithm as a black-box, unlike the previous approach that modifies the inner workings of the uncorrupted algorithm. As a result, it leads to a conceptually simpler algorithm. Finally, we provide a lower bound ruling out a O(C +dloglogT)algorithm. This shows that robustifying contextual pricing is harder than robustifying contextual search with ϵ-ball losses, for which it is possible to design algorithms where corruptions add only an extra additive term C to the regret.

Dynamic pricing algorithms typically assume continuous price variables, which may not reflect real-world scenarios where prices are often discrete. This paper demonstrates that leveraging discrete price information within a semi-parametric model can substantially improve performance, depending on the size of the support set of the price variable relative to the time horizon. Specifically, we propose a novel semi-parametric contextual dynamic pricing algorithm, namely BayesCoxCP, based on a Bayesian approach to the Cox proportional hazards model. Our theoretical analysis establishes high-probability regret bounds that adapt to the sparsity level γ, proving that our algorithm achieves a regret upper bound of eO(T(1+γ)/2 + dT) for γ < 1/3 and eO(T2/3 + dT) for γ 1/3, where γ represents the sparsity of the price grid relative to the time horizon T. Through numerical experiments, we demonstrate that our proposed algorithm significantly outperforms an existing method, particularly in scenarios with sparse discrete price points.

We study contextual dynamic pricing when a target market can leverage Kauxiliary markets--offline logs or concurrent streams--whose mean utilities differ by a structured preference shift. We propose Cross-Market Transfer Dynamic Pricing (CM-TDP), the first algorithm that provably handles such model-shift transfer and delivers minimax-optimal regret for both linear and nonparametric utility models. For linear utilities of dimension d, where the difference between source-and targettask coefficients is s0-sparse, CM-TDP attains regret eO (dK 1 + s0) log T .

Training data is the backbone of large language models (LLMs), yet today's data markets often operate under exploitative pricing - sourcing data from marginalized groups with little pay or recognition. This paper introduces a theoretical framework for LLM data markets, modeling the strategic interactions between buyers (LLM builders) and sellers (human annotators). We begin with theoretical and empirical analysis showing how exploitative pricing drives high-quality sellers out of the market, degrading data quality and long-term model performance. Then we introduce fairshare, a pricing mechanism grounded in data valuation that quantifies each data's contribution.

Dynamic pricing with resource constraints is a critical challenge in online learning, requiring a delicate balance between exploring unknown demand patterns and exploiting known information to maximize revenue. We propose three tailored algorithms to address this problem across varying levels of prior knowledge: (1) a Boundary Attracted Re-solve Method for the full information setting, achieving logarithmic regret without the restrictive non-degeneracy condition; (2) an online learning algorithm for the no information setting, delivering an optimal $O(\sqrt{T})$ regret; and (3) an estimate-then-select re-solve algorithm for the informed price setting, leveraging machine-learned prices with known error bounds to bridge the gap between full and no information scenarios. Moreover, through numerical experiments, we demonstrate the robustness and practical applicability of our approaches. This work advances dynamic pricing by offering scalable solutions that adapt to diverse informational contexts while relaxing classical assumptions.

We study contextual online pricing with biased offline data. For the scalar price elasticity case, we identify the instance-dependent quantity $\delta^2$ that measures how far the offline data lies from the (unknown) online optimum. We show that the time length $T$, bias bound $V$, size $N$ and dispersion $\lambda_{\min}(\hat{\Sigma})$ of the offline data, and $\delta^2$ jointly determine the statistical complexity.