Researchers prove bundling goods is optimal even with heavy-tailed, infinite variance valuations
New game theory framework shows bundling remains the best sales strategy when you only know the mean and MAD, not variance.
A team of researchers from Tilburg University and Eindhoven University of Technology has published a significant paper on arXiv titled 'Robust Optimality of Bundling Goods Beyond Finite Variance.' The work tackles a classic problem in mechanism design: how should a seller price many independent goods when they have only limited knowledge about how buyers value them? The authors move beyond the traditional assumption that the seller knows the mean and *finite* variance of valuations. Instead, they develop a novel 'distributionally robust' game-theoretic model where the seller, armed only with knowledge of the mean and Mean Absolute Deviation (MAD), chooses a revenue-maximizing mechanism. A hostile 'nature' then selects the worst-case, revenue-minimizing distribution that fits those limited statistics.
The key finding is that bundling all goods together at a single price remains the asymptotically optimal strategy for the seller, even when dealing with heavy-tailed value distributions that have *infinite variance*. This extends a foundational result in economics. However, the guaranteed revenue is strictly less than the mean valuation, reflecting the increased uncertainty. Another critical insight is the 'indifference to the order of play'—the seller's guaranteed revenue is the same whether they move first or nature does, a property unique to bundling. Finally, the authors demonstrate the universality of their derived optimal bundling price, showing it simultaneously optimizes for absolute revenue, absolute regret, and a ratio objective, making it a powerfully robust tool for practical pricing strategy.
- Proves bundling is optimal with only mean & MAD knowledge, extending beyond finite variance assumptions to handle heavy-tailed, infinite-variance distributions.
- Establishes 'indifference to order of play': max-min and min-max game outcomes are identical, a property unique to bundling versus separate sales.
- The derived optimal bundling price is universal, effectively optimizing for revenue, regret, and ratio objectives simultaneously.
Why It Matters
Provides a more robust, practical pricing framework for digital goods, SaaS, and content bundles where customer value distributions are uncertain and potentially erratic.