The value of adaptive commitment: option value = information value

Parent: derivations

0. Claim and contribution

A risk-neutral decision-maker faces an uncertain continuation value $X$ and may choose between continue (get $X$ ) and abandon (get a constant, normalized to $0$ ). We prove rigorously:

Information monotonicity — finer information before committing weakly raises the expected payoff (Thm 1); "decide-after ≥ decide-before" is the special case.
Risk monotonicity — under full information, a mean-preserving spread of $X$ raises the value of the gate (Thm 2).
Equivalence — this gate value equals the Expected Value of Perfect Information (EVPI) of the two-action problem (Thm 3). Every non-trivial step is proved; the one cited deep result is Strassen's theorem, isolated explicitly.

1. Setup (measure-theoretic)

Fix $(\Omega,\mathcal F,P)$ and $X\in L^1(\Omega,\mathcal F,P)$ . Information available before committing is a sub-σ-algebra $\mathcal G\subseteq\mathcal F$ . The optimal expected payoff under $\mathcal G$ :

V(\mathcal G):=E\big[\max(E[X\mid\mathcal G],\,0)\big].

Extremes: $\mathcal G=\{\varnothing,\Omega\}$ gives $\max(E[X],0)$ (blind); $\mathcal G=\mathcal F$ gives $E[\max(X,0)]$ (gate). Write $\varphi(t):=\max(t,0)$ : convex, $0\le\varphi(t)\le|t|$ , so $\varphi(X)\in L^1$ .

2. Lemma 1 (supporting line of a convex function) — proved

Lemma 1

Claim. For convex $\varphi:\mathbb R\to\mathbb R$ and any $a$ , there is $m$ with $\varphi(x)\ge\varphi(a)+m(x-a)\ \forall x$ .

Proof

For $h>0$ let $D(h)=\frac{\varphi(a+h)-\varphi(a)}{h}$ . For $0<h_1<h_2$ , $t=h_1/h_2$ , $a+h_1=(1-t)a+t(a+h_2)$ , so $\varphi(a+h_1)\le(1-t)\varphi(a)+t\varphi(a+h_2)$ , i.e. $\varphi(a+h_1)-\varphi(a)\le t(\varphi(a+h_2)-\varphi(a))$ ; dividing by $h_1=th_2$ gives $D(h_1)\le D(h_2)$ . So $D$ is nondecreasing and bounded below; $\varphi'_+(a)=\inf_{h>0}D(h)$ exists, and similarly $\varphi'_-(a)\le\varphi'_+(a)$ . Pick $m\in[\varphi'_-(a),\varphi'_+(a)]$ . For $x>a$ : $\frac{\varphi(x)-\varphi(a)}{x-a}\ge\varphi'_+(a)\ge m$ ; for $x<a$ : $\le\varphi'_-(a)\le m$ , and $x-a<0$ flips it.

Corollary

A finite convex $\varphi$ on $\mathbb R$ is continuous with locally bounded subgradients; picking a supporting slope $m_q$ at each rational $q$ and $\ell_q(x)=\varphi(q)+m_q(x-q)$ gives $\varphi(x)=\sup_{q\in\mathbb Q}\ell_q(x)$ (a countable sup of affine functions).

3. Lemma 2 (conditional Jensen) — proved

Lemma 2

Claim. $Y\in L^1$ , $\varphi$ convex, $\varphi(Y)\in L^1$ $\Rightarrow$ $E[\varphi(Y)\mid\mathcal G]\ge\varphi(E[Y\mid\mathcal G])$ a.s.

Proof

With $\varphi=\sup_q\ell_q$ : each $\ell_q$ affine, so $E[\ell_q(Y)\mid\mathcal G]=\ell_q(E[Y\mid\mathcal G])$ a.s., and $\ell_q(Y)\le\varphi(Y)$ gives $E[\varphi(Y)\mid\mathcal G]\ge\ell_q(E[Y\mid\mathcal G])$ a.s. The exceptional null set depends on $q$ , but $\mathbb Q$ is countable, so off a single null set $E[\varphi(Y)\mid\mathcal G]\ge\sup_q\ell_q(E[Y\mid\mathcal G])=\varphi(E[Y\mid\mathcal G])$ .

4. Theorem 1 (information monotonicity)

Theorem 1

$\mathcal G_1\subseteq\mathcal G_2\subseteq\mathcal F\Rightarrow V(\mathcal G_1)\le V(\mathcal G_2)$ . In particular $\max(E[X],0)\le E[\max(X,0)]$ (gate ≥ blind).

Proof

$Y=E[X\mid\mathcal G_2]$ ; tower: $E[X\mid\mathcal G_1]=E[Y\mid\mathcal G_1]$ . Then
$V(\mathcal G_1)=E[\varphi(E[Y\mid\mathcal G_1])]\overset{\text{Lem 2}}{\le}E[E[\varphi(Y)\mid\mathcal G_1]]=E[\varphi(Y)]=V(\mathcal G_2).$

5. Theorem 2 (risk monotonicity)

Theorem 2

$V_\ast(X):=E[\max(X,0)]$ . Call $X_2$ a mean-preserving spread of $X_1$ if there is a coupling with $X_2\overset{d}{=}X_1+Z$ , $E[Z\mid X_1]=0$ .

Claim. Then $V_\ast(X_2)\ge V_\ast(X_1)$ .

Proof

$\mathcal G=\sigma(X_1)$ , $E[X_2\mid\mathcal G]=X_1$ . By Lemma 2, $V_\ast(X_2)=E[E[\varphi(X_2)\mid\mathcal G]]\ge E[\varphi(E[X_2\mid\mathcal G])]=E[\varphi(X_1)]=V_\ast(X_1)$ .

Remark

That this construction is equivalent to the convex order $X_1\preceq_{cx}X_2$ is Strassen's theorem / Rothschild–Stiglitz (1970) — the one external deep result; I prove only the direction I use.

6. Theorem 3 (option value = information value)

Theorem 3

Two-action problem: $U(\text{continue},\omega)=X(\omega)$ , $U(\text{abandon},\omega)=0$ . Then
$\mathrm{EVPI}:=E[\max_a U(a,\cdot)]-\max_a E[U(a,\cdot)]=E[\max(X,0)]-\max(E[X],0)=V(\mathcal F)-V(\{\varnothing,\Omega\}).$

Proof

Pointwise $\max_a U(a,\omega)=\max(X(\omega),0)$ and $\max_a E[U(a,\cdot)]=\max(E[X],0)$ .

So option value = information value = value of adaptivity: one quantity, three names.

7. Corollary (two-point closed form)

Corollary

$X=\mu\pm s$ each w.p. $\tfrac12$ , $\mu,s\ge0$ :
$V(\mathcal F)-V(\{\varnothing,\Omega\})=\max\!\Big(\tfrac{s-\mu}{2},\,0\Big).$
Zero until uncertainty $s$ exceeds the mean $\mu$ (until a bad outcome is possible), then linear in $s$ . Verifies all three theorems.

8. Scope

Risk-neutral (else replace $\max$ by $E[u(\cdot)]$ ; Thms 1,3 unchanged, Thm 2 needs $u\circ\max(\cdot,0)$ convex); static single gate (dynamic optimal stopping is separate); exogenous information (no acquisition cost); abandon value normalized to 0.

9. Cited vs proved

Proved: supporting line, conditional Jensen, information & risk monotonicity, equivalence, closed form. Cited: Strassen / Rothschild–Stiglitz (construction ⟺ convex order); continuity & local boundedness of finite convex functions.