Frankfurt cases and the Newcomb Problem
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Frankfurt cases and the Newcomb Problem Arif Ahmed1
The Author(s) 2019
Abstract A standard argument for one-boxing in Newcomb’s Problem is ‘Why Ain’cha Rich?’, which emphasizes that one-boxers typically make a million dollars compared to the thousand dollars that two-boxers can expect. A standard reply is the ‘opportunity defence’: the two-boxers who made a thousand never had an opportunity to make more. The paper argues that the opportunity defence is unavailable to anyone who grants that in another case—a Frankfurt case—the agent is deprived of opportunities in the way that advocates of Frankfurt cases typically claim. Keywords Decision Theory Newcomb’s Problem Frankfurt cases Free will
Causal Decision Theory (CDT) and Evidential Decision Theory (EDT) are two leading theories of rational choice. Briefly and informally, CDT says that a rational agent does whatever in her view most effectively brings about her ends. EDT says that a rational agent does whatever in her view is the best evidence of what she wants.1 CDT and EDT are both intuitive but not both true. They conflict over cases like: 1 More formally (though still simplifying): let the probability function Cr and the news-value function V respectively represent the agent’s credence (degree of belief) in, and desirability for, an arbitrary proposition, propositions being subsets of the set W of possible worlds. Let O ¼ fo1 . . .on g partition W into propositions describing the agent’s options, and let Z ¼ fz1 . . .zm g partition W into propositions that each specify the outcome in as much detail as concerns the agent. Let some counterfactual-like operator ‘ [ ’ reflect causal dependence: oi [ zj says that if the agent were to realize oi then the outcome would be zj . CDT says that the rational agent maximizes U, and EDT says that the rational agent maximizes V,
& Arif Ahmed [email protected] 1
Faculty of Philosophy, Cambridge University, Cambridge, UK
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A. Ahmed Table 1 Payoffs in Newcomb’s Problem S1 : predicted o1
S2 : predicted o2
o1 : take only opaque box
m
0
o2 : take both boxes
mþk
k
Newcomb’s Problem. You must choose between (o1 ) taking only an opaque box (‘one-boxing’) and (o2 ) taking both it and a transparent one in which $1000 (k) is visible (‘two-boxing’). You get to keep what you take. The opaque box already contains $1 M (m) if and only if (S1 ) a highly reliable predictor foresaw your taking only it. If (S2 ) she foresaw your taking both boxes then the opaque box is empty. So the payoff schedule (in dollars, for which we assume linear increasing utility) is as follows (Table 1). You can’t affect what is in the opaque box, this having been settled at the time of the prediction. So taking both boxes causes you to be richer by k than if you had taken only the opaque box. CDT therefore counts o2 as uniquely rational. But taking only the opaque box is strong evidence that you get m, and taking both boxes is strong evidence that you get k m (since the predictor is very reliable). EDT therefore counts o1 as uniquely rational
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