Proposition 3.2, p. 17 — the market equilibrium is unique: ω* = max{0, (ϕ − d − β⁻¹(ϕ − 1))/(1 − d)}, N*_r = (ρ/ν)(1−ω*)/(1+ω*), N*_s = (2ρ/c)(1−ω*)/(1+ω*)

Consider the limit-order market with protected midpoint pegs, with flow cost of speed $c>0$, midpoint execution fee $0\le d<1$, investor gross surplus $\varphi>1$, discount rate $\delta>0$, and arrival rates $\rho,\nu>0$; the discount factor is $\beta=\exp(-\delta/\rho)\in(0,1)$. Then there is a unique market equilibrium $(\omega^*,N_r^*,N_s^*)$ in the sense of Definition 3.1. The equilibrium fraction of investors choosing midpoint pegs rather than market orders is

and the equilibrium masses of proprietary traders acting as market makers and as snipers are

Uniqueness means: this triple is a market equilibrium, and every market equilibrium $(\omega,N_r,N_s)$ equals it.

The result shows that, when pegs are protected from sniping, the peg share does not depend on the sniping intensity $\nu$, and gives closed forms for the populations of market makers and snipers.

Formalization Note The paper prints the restriction $\varphi\ge1$, but Assumption A4a (one of the hypotheses A1–A5 of the proposition) requires $\varphi>1$, and at $\varphi=1$ formula (3.6) gives $\omega^*=1$, which Definition 3.1 does not admit, while $\pi_p>\pi_m$ on $[0,1)$; so the statement uses $\varphi>1$. The fee restriction is the proposition's $d\ge0$. The condition $\beta\in(0,1)$ is not a separate hypothesis: it follows from $\delta,\rho>0$. The payoffs in Definition 3.1 are the defining sums under the steady state $q$, not their closed forms.