3 Evidential and causal decision theory

Let us begin with Evidential Decision Theory. The easiest way to set this view out is to impose the requirement of probabilistic independence on the set of states of nature. Then each state has one credence no matter which act we are evaluating, and an evidential expected utility theorist will say that it is rational to choose an act which has the greatest (or has among the greatest) expected utility out of all of the available acts, with expected utility given by the formula below:

Note that both theories accept this equation, if it is applied over the “right” set of states of nature. In that sense, it is common ground. It turns out, however, that evidential decision theorists can accept another characterisation of evidential expected utility which works over any set of states of nature, without imposing an independence requirement, provided that the other three conditions are satisfied.

With this formula, we don’t actually need the states to be probabilistically independent of the acts.

All we now require is that each state, with each act, settles everything you care about. And where the states are probabilistically independent of the acts, \Cr(S_i \mid A) = \Cr(S_i) — so the two agree. Notice that dominance reasoning does not apply for partitions that are not probabilistically independent of the acts.

It turns out that whichever way we slice up the possibilities into states of nature, we always get the same result from the formula in Principle 4. This is known as partition invariance; for a proof, see Joyce (1999: 121–2, Theorem 4.1), and for discussion, Ahmed (2014: 40).¹ We will not prove the result in this course, but a fact is still a fact.

Let’s apply this to the Newcomb case. Suppose, to have some numbers, that the demon is right 95\% of the time. The prediction states are not probabilistically independent of the acts, so Principle 3 does not apply. But Principle 4 does, and it tells us to use the credences conditional on the act: given that you One Box, the demon predicted One Box with probability 0.95; given that you Two Box, it predicted Two Box with probability 0.95. So:

One Box comes out far ahead. So evidential decision theory says One Box.

Now for the other camp. Causal decision theorists read condition (ii) as requiring causal independence: the states of nature must be things your choice makes no difference to. The hard part is saying, in general, which states those are. The standard answer is due to David Lewis: the states should be dependency hypotheses.

A dependency hypothesis says everything there is to say about what your actions would bring about, and nothing about which action you will perform. Whichever one is true is true regardless of what you do. So your choice cannot make any difference to it, which is exactly what condition (ii), on the causal reading, requires. The source is Lewis (1981).²

Take one counterfactual for each option: if I were to do A, the outcome would be O, written A \;\square\!\rightarrow\; O. A dependency hypothesis is a whole pattern of these — one per option, all at once.

Here is Lewis’s example. You have three options: brush Bruce the cat, stroke him, or leave him alone. Bruce might purr loudly, purr softly, or not purr at all. One dependency hypothesis is this pattern:

Here is a quite different one:

There are many such patterns, and exactly one of them is true. Which one is true is a fact about Bruce, not something you can bring about: you have no influence over what he would do if you brushed him. So each pattern is causally independent of what you do, by construction.

Every possible pattern of links from acts to consequences is a dependency hypothesis. Take any way of assigning an outcome to each of your options: the proposition that this is what each option would bring about is one of the dependency hypotheses. So the dependency hypotheses are all of the ways in which the consequences could depend on your acts.

In the Newcomb case there are two options, and the demon has already acted, so there are two hypotheses to consider:

K_1 says the million is in the blue box, so that One Box would get you \$1{,}000{,}000 and Two Box would get you \$1{,}001{,}000. K_2 says it isn’t. Neither of these says what you will choose — only what your choosing would bring about.

These are not the only dependency hypotheses there are, but they are the only ones an agent should entertain in this scenario. Any other pattern would have the contents of the blue box depend on which box you take, and the story rules that out: the demon put the money in, or didn’t, last night.

Suppose we have either the set of dependency hypotheses, or alternatively some other causally independent set of states of nature, to hand. Let’s say this set is indexed, so that its elements are K_i. We can then write down the formula for causal expected utility — the thing that causal decision theorists think rational agents should maximise:

Note that the credences here are unconditional, and that is the whole point. Your act might well be evidence about which K_i is true, just as it was evidence about the prediction. But conditioning the credence in K_i on the act would let that evidence back in, and the causal decision theorist’s view is that evidence about what you cannot affect should play no part in deciding what to do.

Applied to Newcomb’s problem, write k for \Cr(K_1), so that \Cr(K_2) = 1 - k:

Whatever k is, Two Box comes out \$1{,}000 ahead. So causal decision theory says Two Box. This is just dominance reasoning again, now applied to states which really are causally independent of your choice.