Combinatorial Auctions with Submodular Valuations: No Ascending Auction Can Find Equilibrium

In this article, we explore the limitations of a particular type of algorithmic approach in economics, known as ascending auctions. These are used to find an equilibrium in situations where agents have submodular valuations, meaning that their preferences for certain items can be represented by a non-linear function. However, the authors prove that no ascending auction can guarantee finding an equilibrium in this context, even with unlimited time and information.
To understand why this is the case, let’s consider an analogy. Imagine you are at a buffet with many delicious food options, but you have a limited budget to spend. You want to allocate your budget in a way that maximizes your overall satisfaction. Now imagine that each dish has a non-linear preference function, meaning that the more of one dish you eat, the less satisfied you become with it. This is similar to how submodular valuations work in economics.
The authors show that finding an equilibrium in this situation using ascending auctions is like trying to find the perfect point on a mountain where all the slopes leading up to it are equal. In other words, it’s a difficult problem that can’t be solved by simply bidding higher and higher over time. The reason is that the submodular valuations create a non-linear relationship between the items, making it hard to find the optimal allocation of resources.
The authors also discuss related results in the field of economics, including the concept of NP-completeness, which shows that many economic problems are inherently difficult and cannot be solved quickly even with the most advanced computers. They also reference other works on submodular valuations and their implications for economic modeling.
In summary, this article highlights the challenges of using ascending auctions to find equilibrium in situations where agents have submodular valuations. It demonstrates that these problems are NP-complete and cannot be solved quickly, even with unlimited time and information. This has important implications for economists seeking to model and analyze complex systems involving non-linear preferences.