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arxiv: 2503.08833 · v2 · pith:TEQNSQQTnew · submitted 2025-03-11 · 💱 q-fin.TR · q-fin.MF

Randomization in Optimal Execution Games

Steven Campbell , Marcel Nutz This is my paper

Pith reviewed 2026-05-23 00:47 UTC · model grok-4.3

classification 💱 q-fin.TR q-fin.MF
keywords optimal executionNash equilibriumrandomized strategiestransient price impactde-randomizationtransaction costs
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The pith

Randomized strategies in optimal execution games can always be replaced by deterministic ones with strictly lower expected cost via averaging.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper considers games in which N traders each choose trading schedules to minimize their execution costs, where costs arise from a transient price impact that decays over time according to a kernel. It establishes that any randomized trading strategy admits a deterministic counterpart obtained by simple averaging, and that this counterpart has strictly lower expected cost. The argument uses the strict positive definiteness of the impact kernel together with convexity of the trading-cost function. As a direct consequence, no Nash equilibrium can involve randomization, so prior non-existence results for pure equilibria extend immediately to mixed equilibria. The paper separately proves that any equilibrium that does exist is unique.

Core claim

Given a randomized strategy, the deterministic strategy equal to its expectation has strictly lower expected execution cost. Consequently Nash equilibria cannot contain randomized strategies, and non-existence of pure equilibria implies non-existence of randomized equilibria. Equilibria are also unique. Both conclusions hold whenever the impact decay kernel is strictly positive definite and the trading cost is convex.

What carries the argument

The averaging map that replaces any randomized strategy by the deterministic strategy equal to its expectation, which strictly lowers expected cost under the positive-definiteness and convexity assumptions.

Load-bearing premise

The impact decay kernel is strictly positive definite and the trading cost function is convex.

What would settle it

A concrete randomized strategy whose expected cost is not strictly greater than the cost of its averaged deterministic version, or an explicit model with positive-definite kernel and convex cost that admits a mixed equilibrium but no pure equilibrium.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper studies optimal execution games among N traders with transient price impact. It shows that, given a strictly positive definite impact decay kernel and convex trading cost, any randomized strategy admits a deterministic average with strictly lower expected execution cost via a simple averaging procedure. Consequently, Nash equilibria cannot contain randomized strategies (non-existence of pure equilibria implies non-existence of randomized equilibria), and the paper separately establishes uniqueness of equilibria. Both results hold in the general transaction cost model under the stated kernel and convexity assumptions.

Significance. If the central claims hold, the results are significant for the literature on game-theoretic models of optimal execution and market microstructure. The de-randomization argument provides a general reason why mixed strategies cannot restore equilibrium existence, complementing prior negative results on pure equilibria and directing attention to conditions for pure-strategy existence. The uniqueness result adds a positive characterization of equilibria. The modeling assumptions (positive-definite kernel, convex costs) are standard and make the findings applicable beyond specific kernels.

minor comments (1)
  1. [Abstract] The abstract could briefly note that the uniqueness result is established separately from the de-randomization argument, to clarify the logical structure for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept. The provided summary accurately captures the paper's main results on de-randomization of strategies and uniqueness of equilibria.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The central result is a direct mathematical proof that any randomized strategy admits a deterministic average with strictly lower expected cost, using the external assumptions of a strictly positive definite impact decay kernel (to obtain strict inequality via the induced quadratic form) and convex trading costs (to preserve the weak inequality). This is a standard application of Jensen-type arguments on the cost functional and does not reduce to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. Uniqueness is established separately under the same modeling assumptions. The derivation is self-contained against the stated kernel and convexity conditions, with no steps that collapse to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two modeling assumptions that are not derived inside the paper: strict positive definiteness of the impact kernel and convexity of the trading cost. No free parameters or new entities are introduced.

axioms (2)
  • domain assumption The impact decay kernel is strictly positive definite
    Invoked to ensure the averaged strategy yields strictly lower expected cost than any randomized strategy.
  • domain assumption The trading cost function is convex
    Used so that Jensen's inequality or an analogous averaging argument produces a cost improvement.

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Works this paper leans on

26 extracted references · 26 canonical work pages · 2 internal anchors

  1. [1]

    Abi Jaber, E

    E. Abi Jaber, E. Neuman, and M. Voß. Equilibrium in functi onal stochastic games with mean-field interaction. Preprint arXiv:2306.05433v1, 2024. 30

    work page arXiv 2024

  2. [2]

    Alfonsi, A

    A. Alfonsi, A. Schied, and A. Slynko. Order book resilien ce, price manipulation, and the positive portfolio problem. SIAM J. Financial Math. , 3(1):511–533, 2012

    work page 2012

  3. [3]

    Optimal Execution among $N$ Traders with Transient Price Impact

    S. Campbell and M. Nutz. Optimal execution among N traders with transient price impact. Preprint arXiv:2501.09638v1, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  4. [4]

    B. I. Carlin, M. S. Lobo, and S. Viswanathan. Episodic liqu idity crises: Cooperative and predatory trading. J. Finance , 62(5):2235–2274, 2007

    work page 2007

  5. [5]

    Cartea, S

    Á. Cartea, S. Jaimungal, and J. Penalva. Algorithmic and High-Frequency Trading. Cambridge University Press, 2015

    work page 2015

  6. [6]

    Mean Field Games with Partial Information for Algorithmic Trading

    P. Casgrain and S. Jaimungal. Mean field games with partia l information for algorithmic trading. Preprint arXiv:1803.04094v2, 2019

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  7. [7]

    Casgrain and S

    P. Casgrain and S. Jaimungal. Mean-field games with differ ing beliefs for algorithmic trading. Math. Finance , 30(3):995–1034, 2020

    work page 2020

  8. [8]

    Dellacherie and P

    C. Dellacherie and P. A. Meyer. Probabilities and Potential B . North Holland, Amsterdam, 1982

    work page 1982

  9. [9]

    G. Fu, U. Horst, and X. Xia. Portfolio liquidation games w ith self-exciting order flow. Math. Finance, 32(4):1020–1065, 2022

    work page 2022

  10. [10]

    Garleanu and L

    N. Garleanu and L. H. Pedersen. Dynamic portfolio choic e with frictions. J. Econ. Theory , 165:487–516, 2016

    work page 2016

  11. [11]

    Gatheral

    J. Gatheral. No-dynamic-arbitrage and market impact. Quant. Finance, 10(7):749–759, 2010

    work page 2010

  12. [12]

    Gatheral and A

    J. Gatheral and A. Schied. Dynamical models of market im pact and algorithms for order execution. In Handbook on Systemic Risk , pages 579–602. Cambridge University Press, 2013

    work page 2013

  13. [13]

    Gatheral, A

    J. Gatheral, A. Schied, and A. Slynko. Transient linear price impact and Fredholm integral equations. Math. Finance , 22(3):445–474, 2012

    work page 2012

  14. [14]

    Graewe and U

    P. Graewe and U. Horst. Optimal trade execution with ins tantaneous price impact and stochas- tic resilience. SIAM J. Control Optim. , 55(6):3707–3725, 2017

    work page 2017

  15. [15]

    Jacod and A

    J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes . Springer, Berlin, 2nd edition, 2003

    work page 2003

  16. [16]

    C. C. Moallemi, B. Park, and B. Van Roy. Strategic executio n in the presence of an uninformed arbitrageur. J. Financ. Markets , 15(4):361–391, 2012

    work page 2012

  17. [17]

    Neuman and M

    E. Neuman and M. Voß. Optimal signal-adaptive trading w ith temporary and transient price impact. SIAM J. Financial Math. , 13(2):551–575, 2022

    work page 2022

  18. [18]

    Neuman and M

    E. Neuman and M. Voß. Trading with the crowd. Math. Finance , 33(3):548–617, 2023

    work page 2023

  19. [19]

    A. A. Obizhaeva and J. Wang. Optimal trading strategy an d supply/demand dynamics. J. Financial Mark. , 16(1):1–32, 2013

    work page 2013

  20. [20]

    L. H. Pedersen and M. K. Brunnermeier. Predatory trading . J. Finance , 60(4):1825–1863, 2005

    work page 2005

  21. [21]

    Schied, E

    A. Schied, E. Strehle, and T. Zhang. High-frequency lim it of Nash equilibria in a market impact game with transient price impact. SIAM J. Financial Math. , 8(1):589–634, 2017

    work page 2017

  22. [22]

    Schied and T

    A. Schied and T. Zhang. A market impact game under transi ent price impact. Math. Oper. Res., 44(1):102–121, 2019

    work page 2019

  23. [23]

    Schöneborn

    T. Schöneborn. Trade execution in illiquid markets . PhD thesis, TU Berlin, 2008

    work page 2008

  24. [24]

    Schöneborn and A

    T. Schöneborn and A. Schied. Liquidation in the face of a dversity: Stealth vs. sunshine trading. Preprint SSRN:1007014 , 2009

    work page 2009

  25. [25]

    E. Strehle. Optimal execution in a multiplayer model of transient price impact. Market Microstructure and Liquidity , 03(03n04):1850007, 2017

    work page 2017

  26. [26]

    K. Webster. Handbook of Price Impact Modeling . CRC Press, Boca Raton, FL, 2023. 31

    work page 2023