Newcomb's Paradox: Decision Theories

About Newcomb's Problem

You face two boxes: one transparent containing $1,000, one opaque. A superintelligent predictor has already decided whether to put $1,000,000 in the opaque box based on its prediction of your choice. If it predicted you'll take only the opaque box, it put the million in. If it predicted you'll take both, it left the opaque box empty. The predictor is almost always right.

The Dilemma

Two compelling but contradictory arguments:

• One-box reasoning: The predictor is almost always right. People who take both boxes almost always get $1,000. People who take one box almost always get $1,000,000. So take one box.
• Two-box reasoning: The prediction already happened—the money is either there or not. Either way, taking both boxes gets you $1,000 more than taking one. So take both boxes.

Decision Theories

Evidential Decision Theory (EDT)

Choose the action with the highest evidential expected value—what the action is evidence for:

EU_EDT(A) = Σ P(O | A) × U(O)

EDT one-boxes: taking one box is strong evidence you'll get the million, since the predictor saw what kind of person you are.

Causal Decision Theory (CDT)

Choose the action with the highest causal expected value—what the action actually causes:

EU_CDT(A) = Σ P(O || A) × U(O)

CDT two-boxes: your choice doesn't causally affect what the predictor already did. The || symbol denotes a causal intervention, not mere conditioning.

Functional Decision Theory (FDT)

Choose as if you're determining the output of the decision algorithm that both you and the predictor are computing:

EU_FDT(A) = Σ P(O | Source(You) = Source(Predictor) = A) × U(O)

FDT one-boxes: the predictor simulated your decision procedure. By choosing one-box, you're determining what that procedure outputs, which determines what the predictor predicted.

Philosophical Issues

• Dominance: Two-boxing strictly dominates one-boxing in every state of the world. Isn't violating dominance irrational?
• Retrocausality: Does EDT require your choice to causally affect the past? Or is correlation enough for decision-making?
• Counterfactuals: What does "if I took both boxes" mean when there's a predictor who foresaw your choice? Are counterfactuals about your algorithm or your action?
• Free will: If the predictor is always right, do you have genuine choice? Can you "prove it wrong"?