Paper: arXiv 2503.08833

Abstract

We study optimal execution in markets with transient price impact in a competitive setting with $N$ traders. Motivated by prior negative results on the existence of pure Nash equilibria, we consider randomized strategies for the traders and whether allowing such strategies can restore the existence of equilibria. We show that given a randomized strategy, there is a non-randomized strategy with strictly lower expected execution cost, and moreover this de-randomization can be achieved by a simple averaging procedure. As a consequence, Nash equilibria cannot contain randomized strategies, and non-existence of pure equilibria implies non-existence of randomized equilibria. Separately, we also establish uniqueness of equilibria. Both results hold in a general transaction cost model given by a strictly positive definite impact decay kernel and a convex trading cost.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is highly mathematical, featuring advanced stochastic calculus, filtrations, and strict positive definiteness proofs, placing it in the ‘Lab Rats’ quadrant due to its theoretical depth without empirical validation. It lacks any backtesting, datasets, or implementation details, focusing purely on theoretical game-theoretic analysis of optimal execution.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Existence of Equilibria in<br>Optimal Execution Games"] --> B{"Can Randomization Restore<br>Nash Equilibria?"}
  B --> C["Model: Competitive Traders<br>Transient Price Impact"]
  C --> D["Key Analysis:<br>Randomized vs. Non-randomized Strategies"]
  D --> E{"Comparison of<br>Expected Execution Costs"}
  E -- Result --> F["Randomized strategy is strictly dominated by<br>a non-randomized strategy via averaging"]
  F --> G["Outcome: Nash Equilibria<br>cannot contain randomized strategies"]
  G --> H["Conclusion:<br>Non-existence of pure equilibria implies<br>non-existence of randomized equilibria"]