Suppose I am perfectly rational in the decision theoretic sense. A coin is about to be tossed, and I will get five dollars on heads (H) and one dollar on tails (T). I have a choice whether to leave the coin fair (F) or load it (L) in favor of tails so that the probability of tails is 3/4.
It is obvious what I do. I calculate the expected utilities of my options F and L as follows.
- EU(F) = P(H|F) ⋅ $5 + P(T|F) ⋅ $1 = (1/2) ⋅ $5 + (1/2) ⋅ $1 = $3
and
- EU(L) = P(H|L) ⋅ $5 + P(T|L) ⋅ $1 = (1/4) ⋅ $5 + (3/4) ⋅ $1 = $2.
And then I choose F.
Except it’s not so simple. For I am perfectly rational. But since, as we just saw, the perfectly rational agent has to choose F, it follows that P(L) = 0, and so P(H|L) and P(T|L) are undefined. So I can’t decide! So now there is no guarantee how I will act, and P(H|L) and P(T|L) once again make sense. And then again they don’t. Oops!
What can be done? Causal decision theorists will note that I reasoned like an evidential decision theorist above. But this makes no difference in this case. The causalist’s story will be a bit more complicated but will end up with the same problem.
We might want to introduce primitive conditional probabilities like Popper functions that let you conditionalize on events with zero probability, and then have P(H|L) = 1/4 and P(T|L) = 3/4, even though P(L) = 0. But that is introducing a lot of complications. Primitive conditional probabilities are not unproblematic.
What should we do? Maybe we should suppose something like primitive suppositional decision theory, where what we are primitively given are the suppositional probabilities PF and PL, without them being defined in terms of conditional and unconditional credences as in evidential and causal decision theories. But this seems problematic. Do we have to suppose that in addition to conditional and unconditional credences, we have suppositional credences? Maybe.
Or perhaps decision theory only applies to agents that have non-zero credences of going for all the options.
The notation above is just passive probabilities, and you are allowing dynamic changes which it does not capture. If you instead use Pearl's do notation, P(H | L) can be rewritten as P (H | do(L)) which is still defined even if you do not do L.
ReplyDeleteThanks. I think that's another version of the suppositional formalism?
ReplyDeleteAnother option, which I think is close to the idea of suppositional probabilities, would be to weight utilities by the probabilities of counterfactuals. In this case, we would for example use P(if L would occur then H would occur) instead of P(H|L).
ReplyDeleteYeah, but doesn't that require something like Molinism?
DeleteAll that said, I don't think formalisms--whether suppositional or do()--actually solve the problem. For if a perfectly rational agent *cannot* do anything irrational, then we are conditioning on an impossibility ("classic" CDT/EDT), supposing an impossibility (suppositional formalism), or forcing a variable to a value it can't have (Pearl's do).
ReplyDeleteWhen we do something like that, we are doing something quite fishy. We can bring out some of the fishiness by tweaking the reward system by adding something like a bonus of one cent if you're essentially perfectly rational. Now what are your payoffs? Since you're essentially perfectly rational, whether you get that bonus is guaranteed by your nature, so you can count on it no matter what, and your payoffs are now $3.01 and $2.01. On the other hand, you are not perfectly rational, and hence not essentially so, if you go for the lower payoff, so you only get $2.00. Which is it?
And if you want to make heads spin, you can suppose a bonus of $10 for not having essential perfect rationality.
The problem arises when you use the fact that P(L) = 0. That value makes P(H|L) and P(T|L) undefined; but P(H|L) and P(T|L) are not undefined, they are 1/4 and 3/4. They are defined by the scenario's description. The problem is vicious circularity; could the solution be something like a hierarchy? You can say that P(L) = 0, when you have answered the question of what to do, but why not rule out putting that value into the equations that determine what you do? Would that be problematic?
ReplyDeleteAs far as I can tell, it requires something weaker than Molinism, namely that the probabilities of counterfactuals in the EU equation always sum to one. In this case, this would mean that we require P(if L would H) + P(if L would ~H) = 1. (This principle is not true in Lewis' similarity-based theory of counterfactuals. So if someone likes that theory they won't like this proposal.)
ReplyDeleteSo, basically "Molinism modulo null sets".
DeleteIn any case, it requires there to almost surely be a fact of the matter about how a non-actual coin would have gone.
The Pearl type notation can be supposed to correspond to paths on an acyclic graph model of decisions and outcomes. The probabilities should correspond to a movement in the graph whether that path is actually taken or not. In fact, since the acyclic graph is just a model of what could happen, all such paths could be considered to be counterfactual (including the actual one determined to be the one our rational agent is going to take or has taken), since they are all hypothetical when the graph is considered as a model prior to any particular decision path through it.
ReplyDelete"In any case, it requires there to almost surely be a fact of the matter about how a non-actual coin would have gone."
ReplyDeleteI don't see why. For example, it could be that P(if F would H) = P(if F would ~H) = 1/2, which seems reasonable. But obviously the approach I am suggesting would be more appealing if we had a general and formal way of assigning probabilities to counterfactuals, rather than just going by what seems reasonable.
If you can not or are rather incapable of making a decision on your own, then just let a coin/pseudo random number generator such as "random()" make that decision for you:
ReplyDeleteI choose and "decide" for F, if and only if X=random() ≤ Cr, and
I choose and "decide" for L, if and only if the same X = random() > Cr
with some critical number 0≤Cr≤1.
The general and total expected utility E[U] is now dependent on this free variable/critical number Cr, which you can choose/"decide" to be, what ever you like.
But if you want to be "rational", then better choose/"decide" Cr to be such, that the expected utility E[U] being maximal/optimal.
So it follows, that here
E[U] = 5$•P(H|F)•P(F)+1$•P(T|F)•P(F)+5$•P(H|L)•P(L)+1$•P(T|L)•P(L)
= 5$•0.5•Cr+1$•0.5•Cr+5$•0.25•(1-Cr)+1$•0.75•(1-Cr)
= 1$•Cr+2$.
So the general and total expected utility E[U] is maximal/optimal for Cr=1.
Sure, this then means and implies, that "P(L)=1-Cr=1-1=0", but then so what?!?
This doesn't invalidate the choice for the critical free number Cr or the decision for the expected utility E[U] being optimal/maximal for that choice or the general and total decision being "rational".
If you can not or are rather incapable of making a decision on your own, then just let a coin/pseudo random number generator such as "random()" make that decision for you:
ReplyDeleteI choose and "decide" for F, if and only if X=random() ≤ Cr, and
I choose and "decide" for L, if and only if the same X = random() > Cr
with some critical number 0≤Cr≤1.
The general and total expected utility E[U] is now dependent on this free variable/critical number Cr, which you can choose/"decide" to be, what ever you like.
But if you want to be "rational", then better choose/"decide" Cr to be such, that the expected utility E[U] being maximal/optimal.
So it follows, that here
E[U] = 5$•P(H|F)•P(F)+1$•P(T|F)•P(F)+5$•P(H|L)•P(L)+1$•P(T|L)•P(L)
= 5$•0.5•Cr+1$•0.5•Cr+5$•0.25•(1-Cr)+1$•0.75•(1-Cr)
= 1$•Cr+2$.
So the general and total expected utility E[U] is maximal/optimal for Cr=1.
Sure, this then means and implies, that "P(L)=1-Cr=1-1=0", but then so what?!?
This doesn't invalidate the choice for the critical free number Cr or the decision for the expected utility E[U] being optimal/maximal for that choice or the general and total decision being "rational".