Proposition 7(a), p. 21 — Cournot: (S,S) unique below k_N, (S,S) and (N,N) with (S,S) Pareto-better between, (N,N) unique above k_S

Fix the competition intensity $0<\gamma_C<1$, the signal accuracy $t>0$ and the prior variance $\sigma^2>0$. There exist thresholds $k^N_C<k^S_C$ with

such that, for all $a,c,k$ satisfying the standing assumptions $0<c<a$, $k>a/(4c)$, $k>1/3$, the set $\mathcal E$ of pure-strategy equilibria of the stage-one information-sharing game between the two manufacturers (Table 1) is:

1. if $k<k^N_C$: $\mathcal E=\{(S,S)\}$;
2. if $k^N_C<k<k^S_C$: $\mathcal E=\{(S,S),(N,N)\}$, and each manufacturer's payoff at $(S,S)$ is strictly larger than at $(N,N)$;
3. if $k>k^S_C$: $\mathcal E=\{(N,N)\}$.

Here manufacturer $i$ chooses whether chain $i$ shares ($S$) or not ($N$), pays the retailer the side payment $\hat m$ needed to induce sharing, and earns $\Pi^{SX_j}_{M_i,C}-\hat m^{X_j}$ if it shares and $\Pi^{NX_j}_{M_i,C}$ otherwise.

This is the main result of the Cournot half of the paper: information sharing is the unique outcome when cost reduction is efficient, a coordination problem with a Pareto-better sharing equilibrium at intermediate efficiency, and absent when cost reduction is inefficient.

Formalization Note The thresholds depend only on $(\gamma_C,t,\sigma^2)$ and are chosen before $a,c,k$; their bounds are those of Proposition 6(a)(2), and they make all three cases nonempty. The page's "otherwise" is stated as $k>k^S_C$: at $k=k^N_C$ and $k=k^S_C$ both $(S,S)$ and $(N,N)$ are equilibria, so the boundary points are left out. Only pure-strategy equilibria are counted, as in the paper (the middle region also has a mixed equilibrium). "Pareto optimal" is the strict inequality of the payoffs at $(S,S)$ over $(N,N)$, stated for manufacturer 1; by symmetry it is the same for manufacturer 2. The hypothesis $\gamma_C>0$ is added (at $\gamma_C=0$ the middle region is empty).