Any randomized strategy in N-trader optimal execution games can be strictly improved by its non-randomized average, so equilibria must be pure and are unique when the impact kernel is strictly positive definite and trading costs are convex.
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Randomization in Optimal Execution Games
Any randomized strategy in N-trader optimal execution games can be strictly improved by its non-randomized average, so equilibria must be pure and are unique when the impact kernel is strictly positive definite and trading costs are convex.