The Puzzle
In Everettian quantum mechanics, the universe never collapses. Every possible outcome of a quantum measurement actually occurs in some branch of the universal wavefunction. Schrödinger’s equation holds universally and deterministically.
That elegance creates a notorious puzzle: if all outcomes occur, why do we experience them with specific probabilities? Why does the relative frequency of events in our branch follow the Born rule—the squared amplitude of the wavefunction?
Collapse theories dodge the puzzle by fiat: they postulate that outcomes occur randomly with exactly those probabilities. Many-Worlds (Everett’s “relative state” formulation) refuses collapse, so it must explain probability in a deterministic multiverse.
This is the probability problem.
Failed Moves in the Literature
Physicists and philosophers have tried three main strategies:
Declare amplitudes to be probabilities.
Zurek’s envariance program attempts to show that |ψ|² is objectively “the” probability measure built into Hilbert space. The move is elegant but ends up conflating the geometry of the wavefunction with the subjective experience of uncertainty.Decision-theoretic derivations.
Deutsch, Wallace, and Sebens & Carroll argue that rational agents in an Everettian universe must act as if branch weights are probabilities. But critics accuse these derivations of smuggling in the Born rule through rationality axioms that already presuppose it.Instrumental shrugging.
Many working physicists simply “shut up and calculate.” Probability works in practice, so why worry about its foundations? The problem is that this abandons Everett’s promise of a fully coherent, universal quantum theory.
What all of these have in common is a failure to separate two distinct concepts: what the world is like versus how an embedded agent should reason about it.
Measure vs. Credence
Here’s the key move:
Measure is ontological. It’s the squared amplitude of a branch, |ψ|². It quantifies the “weight” of that branch in Hilbert space.
Credence is epistemological. It’s an agent’s subjective degree of belief about which outcome they will experience.
These are not the same thing. Confusing them is what makes so much of the Everettian literature circular.
The Regret/Typicality Lemma
With the distinction in place, the link becomes clear.
Lemma. If an agent assigns credences different from the branch measures, there exists a bet such that almost all of their future selves (weighted by measure) experience regret compared to the strategy that aligned credence with measure.
Proof Sketch:
The agent chooses an action with payoffs contingent on outcomes.
They evaluate it using their credences.
But the actual distribution of payoffs across their future descendants is governed by measure.
If credence ≠ measure, then some bet leads to systematic divergence.
The overwhelming majority of descendants, weighted by measure, will look back and see that the action was suboptimal.
Therefore: to avoid predictable regret in almost all branches, rational agents must align credence with measure.
That is just the Born rule—not as a primitive axiom, nor as an ontological law, but as a normative prescription for agents embedded in a branching universe.
Why This Matters
No circularity. We don’t derive probabilities from determinism or define amplitudes as probabilities. We separate measure and credence, then show why rationality connects them.
Decision-theoretic clarity. The alignment of credence with measure is enforced by the avoidance of regret across descendants, not by smuggled-in axioms.
Philosophical precision. The Born rule is not an unexplained brute fact. It is the rational bridge between the physics of Hilbert space and the epistemology of self-locating agents.
Conclusion
Probability in Everettian quantum mechanics isn’t a metaphysical primitive. It is the rational stance of finite agents navigating an infinite branching structure. The world supplies Measure; we supply Credence; rationality demands we align the two.
That is how the Born rule survives in a world without collapse.