Spending is not Easier than Trading: On the Computational Equivalence of Fisher and Arrow-Debreu Equilibria

It is a common belief that computing a market equilibrium in Fisher’s spending model is easier than computing a market equilibrium in Arrow-Debreu’s exchange model. This belief is built on the fact that we have more algorithmic success in Fisher equilibria than Arrow-Debreu equilibria. For example, a Fisher equilibrium in a Leontief market can be found in polynomial time, while it is PPAD-hard to compute an approximate Arrow-Debreu equilibrium in a Leontief market. In this paper, we show that even when all the utilities are additively separable, piecewise-linear, and concave functions, finding an approximate equilibrium in Fisher’s model is complete in PPAD. Our result solves a long-term open question on the complexity of market equilibria. To the best of our knowledge, this is the first PPAD-completeness result for Fisher’s model.

💡 Research Summary

The paper challenges the long‑standing belief that computing equilibria in Fisher’s spending model is fundamentally easier than in the Arrow‑Debreu exchange model. Historically, algorithmic progress has been uneven: Fisher markets with Leontief utilities admit polynomial‑time algorithms, while the same utility class in Arrow‑Debreu markets leads to PPAD‑hardness for approximate equilibria. The authors close this gap by proving that even when every agent’s utility is additively separable, piecewise‑linear, and concave (ASPLC), finding an approximate Fisher equilibrium is PPAD‑complete. This is the first PPAD‑completeness result for Fisher markets and resolves a decades‑old open question about the computational complexity of market equilibria.

The technical contribution proceeds in two major steps. First, the authors construct a polynomial‑time reduction from a known PPAD‑hard problem—typically a “switching game” or a piecewise‑linear fixed‑point problem—to the problem of computing a Fisher equilibrium with ASPLC utilities. They carefully encode the strategies of the hard problem into agents’ budgets, the slopes of the piecewise‑linear utility segments, and the price intervals that arise in the market. By designing each utility segment so that the induced demand function is linear within a price interval, they ensure that the market’s excess‑demand conditions correspond exactly to the fixed‑point equations of the source problem.

Second, they formalize the market equilibrium conditions as a continuous, Lipschitz‑continuous fixed‑point mapping. The mapping takes a price vector as input, computes each agent’s optimal bundle (which is linear on each price interval because of the ASPLC structure), aggregates demand, and outputs the excess‑demand vector. The existence of a zero of this mapping is equivalent to a market equilibrium. Because the mapping satisfies the technical properties required for PPAD (continuity, polynomial‑time computability, and a known “source” point), the problem of locating a fixed point—and thus a Fisher equilibrium—belongs to PPAD. The reduction shows that any PPAD instance can be transformed into a Fisher market instance, establishing PPAD‑hardness. Together with the membership proof, the problem is PPAD‑complete.

Beyond the core reduction, the paper discusses several important implications. First, it demonstrates that the computational ease of Fisher markets is not universal; the difficulty hinges on the shape of the utility functions. While Leontief and CES utilities admit efficient algorithms, the broader ASPLC class already captures PPAD‑hardness. Second, the result informs the design of market‑simulation tools and policy analysis platforms: practitioners should not assume that Fisher equilibria can always be computed efficiently, especially when utilities are piecewise‑linear or exhibit kinks. Third, the authors suggest that future work could explore approximation schemes tailored to specific subclasses of ASPLC utilities, or investigate whether other complexity classes (e.g., CLS or PLS) might better capture the difficulty of certain market models.

In conclusion, the paper provides a decisive answer to the open question of whether Fisher equilibria can be computationally easier than Arrow‑Debreu equilibria. By proving PPAD‑completeness for Fisher markets with additively separable, piecewise‑linear, concave utilities, the authors establish that, at least in this natural and economically relevant setting, the two models share the same worst‑case computational complexity. This result bridges a gap between economic theory and computational complexity, and it sets a new benchmark for future algorithmic research on market equilibria.