$$\max_{\pi = (f \in \mathcal{F},\, \theta \in \Theta)}\; F^\pi(S_0) \;=\; \mathbb{E}\!\left[\,\sum_{t=0}^{T} \pi(t) \cdot J\!\left(S_t,\, X_t^\pi(S_t \mid \theta)\right)\right]$$
You are not choosing decisions $x_t$ directly — those are random variables that fall out of whatever policy you are running.
What you are actually searching over is the policy itself: the type of decision function $f \in \mathcal{F}$
and its tunable parameters $\theta \in \Theta$. That search is deterministic. That is what you control.
The decisions $X_t^\pi(S_t \mid \theta)$ are the output, not the input.
Component 1 — The Joy Function
$$J(S_t, x_t) \;=\; \lambda_h h_t \;+\; \lambda_w w_t \;+\; \lambda_r r_t \;+\; \lambda_p p_t$$
Subject to $\,\lambda_h + \lambda_w + \lambda_r + \lambda_p = 1\;$ and $\;\lambda_i > 0$.
Only you know your $\lambda$s -- and they change as you age.
Component 2 — The Mortality Discount
$\pi(t)$ is your survival probability at age $t$. It starts near 1 and decays.
$\mu(t)$ is the Gompertz mortality rate -- it compounds exponentially with age.
This is the decay term. It is not negotiable, but it is foreseeable.
Component 3 — State Transition
$$S_{t+1} \;=\; S^M\!\bigl(S_t,\; X_t^\pi(S_t \mid \theta),\; W_{t+1}\bigr)$$
$S_t = (h_t,\, w_t,\, r_t,\, p_t)$ is your state vector today — already realized.
$X_t^\pi(S_t \mid \theta)$ is the decision your policy produces given that state.
$W_{t+1}$ is exogenous information: the world's move, not yours.
Component 4 — The Policy Search
$$\pi^* \;=\; \underset{f \in \mathcal{F},\; \theta \in \Theta}{\arg\max}\; \mathbb{E}\!\left[\sum_{t=0}^{T} \pi(t) \cdot J\!\left(S_t,\, X_t^{f,\theta}(S_t)\right)\right]$$
The search has two levels: (1) choose the class of policy $f \in \mathcal{F}$ — the structure of how you make decisions —
then (2) tune its parameters $\theta$ within that class. Most people do neither deliberately.
What This Actually Tells You
01
Timing is not neutral. Because $\pi(t)$ decays, a unit of joy at 35 is worth more to the sum than a unit of joy at 75. This is not pessimism. It is the math of a finite horizon -- and it argues for front-loading the things money can't buy, like
relationships and
purpose, not deferring them.
02
Wealth cannot substitute for health past a threshold. $w_t$ tends to rise while
$h_t$ tends to fall. But $\lambda_h$ and $\lambda_w$ are your weights -- if you underinvest in health today, no future $w_t$ accumulation fully compensates inside $J$.
03
The policy is the lever, not the decision. $x_t$ is a random variable — it is what your policy produces given the current state. You cannot optimize over random variables directly. What you can search over deterministically is $(f \in \mathcal{F},\, \theta \in \Theta)$: the class of decision function and its tunable parameters. This is the Stoic point rendered precisely — you do not control outcomes, you design the policy that produces them.
04
Most people never choose their policy. They inherit one — from upbringing, habit, culture — and run it without inspection. The search over $\mathcal{F}$ is what humans do ad-hoc. The framework makes it explicit. How are you going to make decisions? That choice is upstream of every $x_t$ you will ever make.
05
Your $\lambda$s are the metaproblem. Most people never explicitly define what they are actually maximizing. Without knowing your weights, the objective function is just decoration. The Six Steps start with the metaproblem — and this is the metaproblem of your life.