The Bellman equation
A problem stretching over all of time, rewritten as the same small problem faced once, again and again.
The claim
A decision taken today changes what you can do tomorrow. Choosing a whole path at once — consumption in every year of a life, say — looks impossibly large.
Bellman’s insight is that you do not have to. Define $V(s)$ as the best you can do from state $s$ onward. Then
\[V(s) = \max_{a} \; \Big\{ u(s, a) + \beta \, \mathbb{E}\big[V(s')\big] \Big\}\]Today’s payoff, plus the discounted value of wherever today’s choice leaves you. One infinite problem becomes one finite problem, repeated.
The principle underneath
Whatever you did to arrive at $s$, the remainder of an optimal plan must itself be optimal from $s$ onward. The past is a sunk fact; only the state matters. This is why sunk costs are irrelevant to the next decision, stated as mathematics rather than as advice.
What it quietly assumes
That the state really does capture everything relevant about the past. Write down too small a state and the equation is simply false: the problem is no longer Markov, and today’s best choice depends on history you have thrown away.
It also assumes the future is discounted ($\beta < 1$) and payoffs are bounded, which is what makes the operator a contraction and guarantees a unique $V$.
Where it breaks
Habits, learning and reputation all carry the past forward. Each must be added to the state, and the state grows.
The curse of dimensionality. Each extra state variable multiplies the space to be searched. Problems that are elegant on paper become numerically brutal with four or five dimensions.
Time-inconsistent preferences. Under hyperbolic discounting, the “you” of next year disagrees with the plan the “you” of today would choose. There is no single $V$ for everyone to agree on, and the standard machinery quietly stops applying.