(A.1): $\bar\omega_i^{II}(\beta^*/\beta_i) + A_i = \bar\omega_0(\beta^*/\beta_i)$

Setting. $\iota$ is a finite set of subunits. $S_0$ and each $S_k$ are finite. The weights $P_0$ and $P_k$ are assumed to be probability weights (nonnegative, each summing to $1$), with $P(s)=P_0(s_0)\prod_kP_k(s_k)$ and $\mathbb E[X]=\sum_sP(s)X(s)$. The payoffs are $v_0:D_0\times S_0\to\mathbb R$ and $v_k:D_k\times D_0\times S_k\to\mathbb R$.

A joint strategy $\beta$ consists of:

• head functions $\zeta_0:S_0\to Z_0$, $\gamma_0^k:Z_0\to M_{0k}$ and $\delta_0:Z_0\times\prod_kM_k\to D_0$;
• subunit functions $\zeta_k:S_k\to Z_k$, $\gamma_k:Z_k\times M_{0k}\to M_k$ and $\delta_k:Z_k\times M_{0k}\to D_k$.

Its information maps are $y_k^\beta(s)=(\zeta_k(s_k),\gamma_0^k(\zeta_0(s_0)))$ and $y_0^\beta(s)=(\zeta_0(s_0),(\gamma_k(y_k^\beta(s)))_k)$. The payoff components are

$$u_k^\beta(s)=v_k(\delta_k(y_k^\beta(s)),\delta_0(y_0^\beta(s)),s_k),\qquad u_0^\beta(s)=v_0(\delta_0(y_0^\beta(s)),s_0),$$

and the expected organization payoff is

$$\bar\omega_0(\beta)=\mathbb E\Big[\sum_ku_k^\beta+u_0^\beta\Big].$$

The functions built from $\beta^*$. Fix $\beta^*$ and real constants $(A_k)_{k\in\iota}$. For $\bar y=(z_0,m)$:

• the head factor is

$$H(\bar y)=\operatorname{cavg}_{P_0}\big(\{t:\zeta^*_0(t)=z_0\},\ t\mapsto v_0(\delta^*_0(\bar y),t)\big);$$

• for each $j$, the subunit factor is

$$F_j(\bar y)=\operatorname{cavg}_{P_j}\Big(\{t:\gamma^*_j(\zeta^*_j(t),\gamma_0^{j*}(z_0))=m_j\},\ t\mapsto v_j\big(\delta^*_j(\zeta^*_j(t),\gamma_0^{j*}(z_0)),\delta^*_0(\bar y),t\big)\Big),$$

where $\operatorname{cavg}_w(E,X)=\sum_EwX/\sum_Ew$, with value $0$ when the denominator is $0$;

• the incentive payoff is

$$W^{II}_i(\beta,s)=u_i^\beta(s)+H(y_0^\beta(s))+\sum_{j\ne i}F_j(y_0^\beta(s))-A_i.$$

Theorem. Take a joint strategy $\beta^*$, constants $A$, a subunit $i$, and a subunit-$i$ strategy $b\in B_i$. Let $\beta^*/b$ be $\beta^*$ with subunit $i$'s strategy replaced by $b$. Then

$$\mathbb E\big[W^{II}_i(\beta^*/b,\cdot)\big]+A_i=\bar\omega_0(\beta^*/b).$$

$\beta^*$ is not required to lie in any strategy set, and no positivity of any conditioning event is assumed.

Degenerate cases.

• The weight hypothesis forces every state set to be nonempty.
• Wherever a conditioning event has weight $0$, the corresponding $H$ or $F_j$ takes the value $0$.
• If $\iota=\{i\}$, the sum over $j\ne i$ is empty. The claim then reads $\mathbb E[u_i^{\beta^*/b}+H(y_0^{\beta^*/b})-A_i]+A_i=\bar\omega_0(\beta^*/b)$.