Chapter 4 — Local vs Global Optimisation
Richard Bellman's Principle of Optimality, published in 1957, is one of the most powerful ideas in applied mathematics. It states that an optimal policy has the property that, regardless of the initial state and decision, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.
In plain language: if you are taking the best possible path, then every sub-path must also be the best possible path from wherever you currently stand.
This is beautiful. It is also conditionally true — and the conditions matter enormously.
Bellman's principle holds when you can observe the full state of the system. When you know where you are, where you can go, and what each transition costs. Dynamic programming — the computational method built on this principle — works precisely because it requires the state space to be fully enumerable.
Now consider the real world.
Locally optimal steps can produce globally catastrophic states when:
- Feedback loops accelerate. A locally optimal fishing harvest triggers a population decline that makes next year's locally optimal harvest even more extractive. Each step is rational. The trajectory is collapse.
- State space expands beyond observation. A bank optimises its own risk exposure by packaging mortgages into securities. Locally optimal. But the systemic risk — the state that matters — is invisible to any individual bank.
- Path dependence locks trajectories. Infrastructure built around fossil fuels makes each incremental investment in fossil fuels locally optimal, even as the cumulative trajectory becomes globally catastrophic. The sunk cost is not a fallacy — it is a real constraint on the state space.
- Boundary conditions are ignored. Bellman's principle says nothing about what happens when you run out of states. When the petri dish is full. When the fish are gone.
Fishing until no fish is not madness. It is optimisation without systemic awareness. Every individual fisher, optimising their own catch, is acting rationally within their observable state space. The tragedy — Hardin's tragedy — is not that people are greedy. It is that the objective function each agent optimises does not include the boundary.