Proof of Lemma 2(a), p. 28 — the linear strategies with coefficients C^{X_iX_j} solve the best responses (7) and (9)

Let $k>1/3$, $0\le\gamma_C<1$, $t>0$, $\sigma^2>0$, and let $s=t\sigma^2/(1+t\sigma^2)$ be the signal weight, so that $E[\theta\mid Y_i=y]=E[Y_j\mid Y_i=y]=sy$. Let $\bar q=k(a-c)/((\gamma_C+4)k-1)$ and let $C^{X_iX_j}$ be the response coefficients of Lemma 2(a). If chain $j$ plays $q_j=\bar q+C^{X_jX_i}Y_j$, then $E[q_j\mid Y_i=y]=\bar q+C^{X_jX_i}sy$ and $E[q_j]=\bar q$. The claim is that chain $i$'s linear strategy is a best response: for all arrangements $X_i,X_j\in\{S,N\}$ and every signal value $y$,

where $\hat q^S$ is the best response (7) and $\hat q^N$ the best response (9).

This is the existence half of Lemma 2(a) for the quantities: the pair of linear strategies is a Bayesian equilibrium of the quantity stage. The coefficients $C^{X_iX_j}$ are the inputs of every profit formula of the mission.

Formalization Note The conditional expectations are substituted by their values under the linear information structure (p. 8), so the statement is an identity between closed forms. The page cites only (7); for $X_i=N$ the relevant best response is (9), which is what is stated. Uniqueness, delegated by the paper to Ha et al. (2011), is not stated; the wholesale price and cost reduction lines of Lemma 2(a) are not stated.