Deutsch Wallace Decision Theory

The Deutsch–Wallace program asks how an agent should evaluate actions whose quantum outcomes lead to different future records and rewards. Within an Everettian account, all nonzero branches persist. The target is therefore a rule for rational preferences, rather than a stochastic law selecting one surviving branch.

Required background. Born rule: Translate a specified state and measurement into outcome probabilities.

Quantum games and rewards

A game specifies an initial state, a physically available unitary action, and rewards attached to orthogonal macroscopic outcome subspaces. For

$$\lvert\psi\rangle=\sum_i c_i\lvert i\rangle,$$

let outcome $i$ deliver reward $r_i$. A Born-weighted representation takes the form

$$V(G)=\sum_i |c_i|^2u(r_i),$$

where $u$ is a utility function. Utility need not equal money: diminishing utility can make two games with equal expected cash value unequally attractive.

The rigorous program makes assumptions about the richness of available acts and the structure of preferences. Ordering, continuity, state supervenience, branching indifference, and diachronic consistency play distinct roles. The exact theorem must be read with its definitions; this page gives the mechanism, not a substitute formal proof.

Equal weights and refinement

Suppose a game has two equal-amplitude branches with rewards $r$ and $s$. A symmetry interchanging branches while preserving the reward structure motivates equal decision weight. For rational squared amplitudes, an ancillary system refines a branch into equal-amplitude pieces. If splitting into same-reward subbranches is irrelevant to preferences, regrouping yields Born weights.

For example, a game with rewards 10 and 0 and squared amplitudes $3/4,1/4$ can be refined into four equal-amplitude branches, three receiving 10. With linear utility its value is $7.5$. A naive count of the original two branches would give 5 and would change after an irrelevant refinement.

This illustrates why branch counting is problematic. It also displays the substantive premise: an agent must be indifferent to extra branching that changes no reward-relevant feature.

Consistency across time

Suppose a future agent in each branch prefers act $A_i$ to $B_i$. Diachronic consistency constrains the present agent’s preferences between compound actions assembled from those future choices. Without a suitable principle, an agent could knowingly plan actions that every future continuation rejects.

Whether this is the right rationality requirement in a branching universe is part of the philosophical debate. The theorem does not derive its axioms from unitary dynamics alone. Nor does accepting its mathematics require treating each axiom as compulsory for every possible theory of personal identity or concern.

What the result explains

A representation theorem can identify which weights a coherent preference system must use. Further interpretation connects those weights to uncertainty, caring about descendants, or empirical confirmation. These roles should be discussed separately.

The decision program does not predict new laboratory probabilities distinct from Born quantum mechanics. Its proposed achievement is explanatory: showing why agents in an Everettian world should reason with those probabilities. It also relies on sufficiently definite macroscopic reward structures, typically justified through decoherence.

Exercises

A game pays 100 with squared amplitude $1/4$ and 0 otherwise. Compare its value with a certain payment of 16 when $u(r)=\sqrt r$.

Solution

The game’s expected utility is $(1/4)\sqrt{100}=2.5$, whereas the certain reward gives $\sqrt{16}=4$. The agent prefers certainty. The conclusion concerns Born-weighted utility, not expected money, which would be 25 for the gamble.

References

• D. Deutsch, “Quantum theory of probability and decisions,” Proceedings of the Royal Society A 455, 3129–3137 (1999), doi:10.1098/rspa.1999.0443.
• D. Wallace, “A formal proof of the Born rule from decision-theoretic assumptions” (2009), arXiv:0906.2718.