Suppose it is true that, each time we decide to act, every moment of our lives, the expected value to each is the highest possible. No matter what you decide to do, let's suppose, you should expect an infinite positive payoff. It's tempting to think that such a world is the best possible. Al Hajek remarks (offhandedly) that, if ours was such a world, Leibniz would have been right: we live in the best of all possible worlds.
But I think this is false and it's interesting that it is false, since a world whose expected value is infinitely great is no less valuable than one whose actual value is infinitely great. Suppose my entire life is a gamble in which I always have some small chance of coming to believe that God exists. No matter what I decide to do, there is always a small chance that I wind up a theist. And suppose that Pascal was right that belief in God has infinite expected value. So, no matter what I decide to do--I decide to sing or yell or fight or . . etc., there is always some chance that I wind up believing that God exists. It follows that no matter what I decide to do and everytime I decide (and everytime everyone else does) it has infinite expected value. Would that make this the best possible world?
Change the example. Suppose there is always some small chance, no matter what I decide, that I wind up taking the following bet. I pay $100 for a chance at an infinite payoff. But the infinite payoff is the following,
SPP. (.5 x $2) + (.25 x $4) + (.125 x $8) .....infinitely.
Now suppose no matter what anyone decided, they too always had some small chance of taking that bet. It would be true of everyone that no matter what they decided, their expected value is infinitely high. You can reinterpret the dollar sign any way you'd like. Make it a hedon sign if you want, or make it a happiness sign, or whatever you think is valuable. Though everyone always rightly expects an infinitely valuable payoff, it is pretty clear that this is not the best of all worlds. I don't think that bet is worth the $100.00 to play, despite the fact that it has an infinite expected value.
I conclude that if everyone always rightly expects an infinitely valuable payoff, no matter what they decide to do, for their entire lives, it does not follow that this is the (or, even, a) best possible world. But why is that result strange? It is strange because a world in which every action has an infinite expected value is equal in value to one that has infinite actual value. As I said above, a world that is infinitely valuable is not more valuable than one that has infinite expected value: V(w) =oo+ = V(w') = p(oo+) for any positive probability p. More interestingly, the fact that a world w is infinitely valuable, and contains no evil, does not entail that w is the (or a) best possible world.
Hey Mike,
I'm having a hard time following. Help me clear things up. You interpret Hajek as saying that:
If the expected value of any action performed by an individual S in world w is infinite, then w is the best possible world.
You think this is wrong. For there is a non-infinitesimal probability that anything I (or anybody) do could lead me to believe that God exists where the value of such belief is infinite. Thus, the expected value is infinite. So the expected value of anything I (or anyone) do is infinite. But it is still reasonable not to believe in God (or to take the bet you lay out) even if it has infinite expected value.
That seems right. And it also seems that anything I do has infinite expected disvalue too.
You then say It follows further that the fact that a world is infinitely valuable, and contains no evil, does not entail that it is the (or a) best possible world.
How does this relate to your previous points? I'm wondering, why can't we object to Hajek as follows: a world w could be worse than another w' if w contains less good than w' even if the expected value of w and w' is the same, for everyone in either w or w'?
Take your own kind of example. Take a St. Petersburg case in a world with only one individual in it. Suppose the expected value of the bet on tails is infinite, but as a matter of fact, the coin lands heads on the second toss. The actual value of the world is then not infinite even though its expected value is infinite for everyone in it. Thus, Hajek's principle is false (as you're suggesting).
If the expected value of any action performed by an individual S in world w is infinite, then w is the best possible world.
He actually says that, so there isn't much interpretation. But he says it in a joking way, so I have no dispute with him.
And it also seems that anything I do has infinite expected disvalue too.
Hajek's version of Pascal's wager does not include infinite disvalues for failing to believe. In any case, this is not something I want to dispute either.
You then say It follows further that the fact that a world is infinitely valuable, and contains no evil, does not entail that it is the (or a) best possible world. How does this relate to your previous points?
Ok, great. Yes this is what I want to say. Take any such world (i.e. one with infinite value and no instances of evil) w. So, we put the value of w at infinity.
1. V(w) = oo+
But any world that w' has an infinitely positive expected value is equal in value to w. Take any positive probability p, it is true that,
2. p(oo+) = oo+
So,
3. V(w) = V(w') = p(oo+) = oo+
But now I argue by modus tollens, giving a counterexample to the view that w' is a best possible world. w' might be a world in which I always have the opportunity to take the gamble in SPP at the cost of $100.00. I say that the gamble in SPP is not one I would take for $100.00. Would you? You would almost certainly wind up with much less than $100.00 in return despite the fact that it has infinitely positive expected utility. But if that is right, then w+ is not a best possible world. But since w' = w in value, it cannot be true that w is a best world either. I'm assuming that if two worlds have exactly the same value, then if one of them is a best world, then so is the other.
. . . a world w could be worse than another w' if w contains less good than w' even if the expected value of w and w' is the same, for everyone in either w or w'?
It could be difficult to make this fly. For instance, I would never take 10 cents in exchange for a gamble that has a 95% chance of a billion dollar payoff, and a 5% chance at nothing, despite the fact that I would have the positively valued 10 cents in the exchange and you would just have the gamble. Give me the gamble any day; it is clearly more valuable. No?
I'm trying to think of motivation for (2). The way I understand expected value, it applies to actions and not worlds. It's just an epistemic probability multiplied by the value of an outcome of an action. Value and expected value will coincide only when the probability = 1, unless the value is infinite or zero, then they will coincide when the probability is positive and real-valued. So I see that, in your case, the value of action = the expected value, thus supporting (2) and (3). But (2) and (3) are talking about the value of worlds.
Maybe you could say more about how a world could have expected value. It seems to me that expected value and actual value could come apart. I also think that "expected value" is a misnomer. That something has expected value does not imply that it has value. It implies that it would have value were it to obtain....
No I wouldn't pay much for the SPP gamble either. And I agree with your last point.
This is an interesting question,
Maybe you could say more about how a world could have expected value. It seems to me that expected value and actual value could come apart.
I take the value of worlds to be derivative. Among the sources of world value is the experience of inhabitants. Worlds where sentient beings suffer are, to that extent, worsened. Worlds where agents are happier or better satisfied, and so on, are to that extent improved. Worlds where agents experience an eternity of complete happiness are extremely valuable worlds. That's what I had in mind.
There is not much question (at least not much that I know about) about the value equivalence of the utility or value of gambles and the utility or value of unweighted sums. A rational agent for whom utility is linear with cash should prefer .5 chance at ($100) to a certain $1 gain, if given a choice between the two. The value V(.5($100)) pretty clearly exceeds V($1). So worlds in which such agents get the former gamble are worth more to worlds in which each such agent gets $1. There is more value in the former worlds.
About the world's having derivative value in the way you suggest--agreed.
I'm not sure how to put my point. I'm uncertain about this: The value V(.5($100)) pretty clearly exceeds V($1). So worlds in which such agents get the former gamble are worth more to worlds in which each such agent gets $1. There is more value in the former worlds.
I would agree that the expected value of "V(.5($100)) pretty clearly exceeds V($1)." The first has an expected value of $50 and the second of only $1. And $50 is greater than $1. But I would also say that whether the objective value of V(.5($100)) exceeds V($1) depends upon whether, as a matter of objective fact, one wins the bet. If one doesn't win, the value of the bet = $0 and the other = $1 and so the judgment reverses.
So whether there is more actual value in the former world will depend upon how the world turns out. If one wins, it has more value, if one loses, it doesn't. And, whether one loses or wins, the former has greater expected value. Perhaps this is just another way of saying that I think the value of these worlds can come apart, one world can have more expected value while having less actual value. I would then extend this reasoning to the infinite case.
The first has an expected value of $50 and the second of only $1. And $50 is greater than $1. But I would also say that whether the objective value of V(.5($100)) exceeds V($1) depends upon whether, as a matter of objective fact, one wins the bet. If one doesn't win, the value of the bet = $0 and the other = $1 and so the judgment reverses.
Hard for me to follow this. You agree that it is rational to exchange $1 for a gamble which includes a .5 chance at $100. So, in fact, you get to choose the world that gets actualized. Which world will it be? Which is better? You clearly think that the gamble world is better, and would opt to actualize such a world. Do you know that you might lose the bet? Of course you do, and yet you think it is a better world, you take the bet. Are you telling me that the bet was not worth taking if you happen to lose? I don't understand that. The fact that you didn't get the payoff does not entail that you actualized a worse world. That's part of what makes this interesting. You did actualize a better world, despite the expected payoff not having occurred. It's more or less the point I'm after. The fact that the payoffs do occur does not make it best.
You agree that it is rational to exchange $1 for a gamble which includes a .5 chance at $100.
Certainly.
So, in fact, you get to choose the world that gets actualized.
Only in the sense that I actualize a world in which I take the bet, but not in the sense that I actualize a world in which that bet is won.
Which world will it be? Which is better?
The better world is the one in which I take the bet and win.
You clearly think that the gamble world is better, and would opt to actualize such a world.
I do not. I think the better world is the one in which I take the bet and win. If I take the bet and lose, the better world is the one in which I do not take the bet at all.
Do you know that you might lose the bet? Of course you do, and yet you think it is a better world, you take the bet.
No. I don't think it is a better world. I think it is expectedly better, but whether it is actually better depends upon whether I take the bet and win.
Are you telling me that the bet was not worth taking if you happen to lose? I don't understand that.
No I'm not suggesting that. Expected betterness makes the bet worth taking. The idea is that whether it is actually better to take the bet, not in the sense that it is rational to take the bet depends upon whether I win.
The fact that you didn't get the payoff does not entail that you actualized a worse world.
I agree. I'm saying the fact that I didn't get the payoff when betting entails that a worse world is actualized, but not by me, the coin flip or card dealing does the actualizing.
You did actualize a better world, despite the expected payoff not having occurred. It's more or less the point I'm after. The fact that the payoffs do occur does not make it best.
I guess I'm still struggling with this. First, I don't see how I'm actualizing the better world. The outcome is independent of what I do. I agree that the fact that the payoff does not occur does not make the outcome best, or the choosing to bet best. Second, I agree it is better to bet. It is better in the sense that given what I have to go on, it is rational. But that's because I think it is the expected value that governs rationality, not the objective value.
Is your view idiosyncratic? Not that it matters to your argument, but are there any defenses of the view you're putting forward in the literature?
Hi Christian,
No, the view I'm defending about the relation of sums and gambles for sums is pretty orthodox stuff. What is new (I think) is it's application to the value of worlds. If there are just two possible worlds to choose from, w1 and w2, and you had to choose the better world, then I'm saying that you should choose w1 under the following assumption.
1. V(w1) = .5($100)
2. V(w2) = $1
If you had a third choice in w3 (this is the world you allude to), then you should choose w3.
3. V(w3) = $100
I'm supposing, for the moment, that w3 is not among the choices for actualization. So, my claim is that w1 is better than w2 (that is, God would choose to actualize w1 rather than w2, given their relative value) and w3 is better than w1 (that is, God would choose w3 over w1, given their relative value). If w3 is not among the options available, then God would actualize w1. Of course, for every valuable sum, there is some equivalent gamble. So God should not prefer w3 to w4,
4. V(w4) = .5($200) = $100
And when it comes to infinitely valuable worlds, then, for just the reasons noted above, there is no difference in value between w' and w'',
5. V(w') = p(oo+)
6. V(w'') = oo+
Hi Mike,
That's clear, thanks. It is also clear now where I disagree.
1. V(w1) = .5($100)
I suggest (1) is equivocal. Either 'V' denotes value (utility in this case) or expected value. If expected value, then I agree that (1) is true when '$100' denotes value. If 'V' denotes value, then I deny that (1) is true when '$100' denotes value. A simple case to make the point.
Take Smith and his duplicate. They face two decisions. Smith gets two gambles and loses them both. His duplicate gets those same two gambles and wins them both.
Smith's first gamble: .5(100 utiles)
Smith's second gamble: .5(100 utiles)
Smith's expected utility = 100 utiles
Smith's actual utility = 0 utiles since he loses both gambles.
Duplicate's first gamble: .5(100 utiles)
Duplicate's second gamble: .5(100 utiles)
Duplicate's expected utility = 100 utiles
Duplicate's actual utility = 200 utiles since he wins both gambles.
The argument: Duplicates world is better than Smith's world. In Smith's world there is 0 utiles and in Duplicate's world there is 200 utiles. All else is equal. Nonetheless, in both worlds the expected value is the same. Thus, EV does not equal V, or, as you're putting it, V(w) need not equal V(p)(w'). The same point applies to outcomes with infinite value.
I would a think a simpler example would make the point wrt God. He creates people each with a positive probability of getting infinite value, say, by going to heaven by performing some act. However, nobody goes to heaven. Call such a world in which that happens w. Contrast w with a world w' in which everyone goes to heaven. Clearly, w' is better that w. However, the expected value of w = w'. Thus, EV need not equal V.
I don't see where anything you say above is related to my claim that, if V(w1) = .5($100) and V(w2) = $1, and you could choose one of the two worlds to actualize on the basis of the value of those worlds, you should choose w1. This is painfully obvious, isn't it? Look, forget about worlds and just think of what it is rational to bet on. If I offer you the gamble [.5($100);.5(0)] at the cost of $1, then you would be terribly irrational not to take the bet.
But in taking the bet you have chosen to actualize w1 over w2. You have just shown that it is better to be in w2 than in w1. If you did not think that was true, even with the chance of losing, you would never take the bet. And that's exactly what God should do, given just those alternatives. QED.
All the talk about the duplicate's actual value is beside the point. You are again smuggling into the discussion the world w3 where bets are won. I'm not asking about w3. That is not one of the options.
I don't see where anything you say above is related to my claim that, if V(w1) = .5($100) and V(w2) = $1, and you could choose one of the two worlds to actualize on the basis of the value of those worlds, you should choose w1. This is painfully obvious, isn't it? Look, forget about worlds and just think of what it is rational to bet on. If I offer you the gamble [.5($100);.5(0)] at the cost of $1, then you would be terribly irrational not to take the bet.
Please don't take me to be denying this. What you say above is clearly true.
But in taking the bet you have chosen to actualize w1 over w2. You have just shown that it is better to be in w2 than in w1. If you did not think that was true, even with the chance of losing, you would never take the bet. And that's exactly what God should do, given just those alternatives. QED
And I am not denying this either. Though I am unclear what it means to claim I'm actualizing w1 over w2. I'm actualizing a world in which I bet, so I'm actualizing a part of w1 over a part of w2. But I'm not actualizing a world in which I win, since that is not in my control. So there is a part of w1 that I am not actualizing.
I'm not (or not trying to) smuggle into the discussion w3. I agree, w3 is besides the point. Let me set aside the point I was trying to make above which was simply that the value of an outcome or world is not a weighted sum or probabilities of values of actions, while the value of an action may be the weighted sum of such probabilities.
It would be rational to prefer (1) .5 chance at getting infinite value and .5 chance at getting nothing to (2) a .05 chance at getting infinite value and a .95 chance at getting nothing. However, the expected value of both options seems to be the same, i.e. infinite.
Question: How do you think one should defend preferring (1) to (2)? For I think we could criticize God for a setting up a world in which the we are faced with option (2) rather than option (1).
Christian,
I don't think it is too difficult to defend the choice of world w1. There are risks that we are are happy to take, and prefer much more than certain small payoffs. Indeed, the expected value of w1 = $50, which means that a rational person should be indifferent between choosing w1 and choosing w1' which we can assume includes a sure thing payoff of $50. One important lesson about rational choice from Newcomb--among worlds or among actions--is that the best choice is not in general the one that pays. The best choice in this case (between w1 and w2) is the risky one, and I'd prefer a world in which I get to take that risk to one in which I have a sure thing of $1. The criticism of God should come from forcing w2 on us, which is worth less than w1 You might be disagreeing with this (I'm not sure), but in rational choice, if EV(w1) = $50, then w1 has a value greater than w1'' = $48. Expected values are values, and rational agents should be willing to exchange $48 for the chance at $100 in world w1.
Mike
Sorry to join the discussion mid-stream and one that is between Christian and you. But, given what you have argued, does a rational agent run the risk of never being satisfied? If it is possible to construct a choice situation where the risk is better regarding expected value then what one has (the bird in the hand so to speak) should not the rational agent choose the risk? If this is so, are there not some undesirable consequences regarding morality? Example; I make a promise to you to do something for 1.00 but then someone else offers me the opportunity to get 50.00 with some marginal risk of getting nothing. Should I not then break my promise to you and take the risk? If we were to answer that promises made take priority over potential risks/gains then does this not pose a problem for your argument regarding what a rational agent should do?
John,
I'm not sure what to say about the moral case, but my guess is that morality overrides consideration of rational self-interest in this case. But to your other question, it is true that continuing buy gambles might lead to less satisfaction. But (unless you are very risk averse) the possiblity of less satisfaction should not be a deterent to buying a gamble when the expected value is sufficiently high (setting aside cases like the St. Petersburg paradox).
Mike
My point is not that I might be less satisfied, I grant that, but rather, in principle, I will never be satisfied. I am assuming that if it is rational for me to choose X then I should choose X. It would seem, re your argument, that there could always be a risky choice that it would be rational for me to choose. Once I made that choice another set of choices would arise, one that would be more rational for me to choose then the other. There would never be a point where I would be satisfied with what I have, but would always be in a situation where I ought to be rationally willing to put what I have at risk re another choice available to me. This does seem to me to have ethical implications regarding the primacy of rational self-interest that cannot be over-ridden by appealing to something that is not in my self-interest.
On a theological point, does this mean that God can never be satisfied? It seems to me that the problem of the 'best of all possible worlds' is missing an important point. The question is not, 'is this the best of all possible worlds?', but rather, "could this world have been better then it is without sacrificing something of greater value?' It seems that your argument has implications for this question as well as the best of all possible worlds one.
Mike,
When I said: Question: How do you think one should defend preferring (1) to (2)? I think we could criticize God for setting up a world in which the we are faced with option (2) rather than option (1).
(1) referred to a .5 chance at getting infinite value and .5 chance at getting nothing, and
(2) referred to a .05 chance at getting infinite value and a .95 chance at getting nothing.
Both options have infinite expected value. Clearly (1) is more rational. And it would be wrong, all else equal, for God to create a world where all of us face (2) rather than (1).
Responding to your recent post, no, I still disagree. But what I disagree with is not the claim about what bets people should take, or how to calculate their value--that's obvious. What I disagree with is your use of 'best' and 'value'. I think you are equivocating two senses of each each word. You are equivocating between expected value and actual value, and between containing the most value and containing the most expected value. And I deny that "expected value" is any kind of value at all, it's not, it's hypothetical and value is a property of obtaining states of affairs. But since I can't convince you of any of that, no matter, I thought I would try raising the question I did above.
I think you are equivocating two senses of each each word. You are equivocating between expected value and actual value, and between containing the most value and containing the most expected value.
If this is what you think, then your quarrel is with rational choice theory generally. On every version of rational chocie theory I know of, a rational person should be willing to place value n on a gamble whose expected value is n. And I should, as a rational agent, prefer getting a .5 chance at $10 over a sure thing at $4.
1. $4
2. [.5($10);.5(0)]
I simply should not exchange (1) for (2) unless the value of (2) is at least as high as (1). It is otherwise senseless to make the exchange. Why would I trade $4 for something whose value I think is less or incomparable? It's incoherent.
In any case, your quarrel is not with me, but with everyone else working in that field.
There would never be a point where I would be satisfied with what I have, but would always be in a situation where I ought to be rationally willing to put what I have at risk re another choice available to me.
If you in fact reach a point where you are satisfied, rational choice does not force you to keep taking gambles. If you're satisfied, you might not find a gamble appealing anymore.
It seems to me that the problem of the 'best of all possible worlds' is missing an important point. The question is not, 'is this the best of all possible worlds?', but rather, "could this world have been better then it is without sacrificing something of greater value?
If a world is the best possible, then it could not be improved without sacrificing something greater. Showing that a world can be improved without comparable loss is just showing that it's not the best. So it's hard to see the different between these questions.
I simply should not exchange (1) for (2) unless the value of (2) is at least as high as (1). It is otherwise senseless to make the exchange. Why would I trade $4 for something whose value I think is less or incomparable? It's incoherent.
This is another instance of the equivocation I pointed out. It's true that one should not exchange your (1) for (2). But not "unless the value of (2) is at least as high as (1)" Rather, unless the expected value of (2) is at least as high as (1). Since the expected value of (2) is greater than (1) that is why it's rational. Whether it's value simpliciter is greater is independent of whether it is rational to choose one rather than the other.
In any case, your quarrel is not with me, but with everyone else working in that field.
As I understand the field people in it are talking about expected value (which involve probability assignments to outcomes), not value simpliciter (which do not). You are equating value simpliciter with expected value, but people in the field (as far as I am aware) don't do that. That's not to say there's no connection between the two, there might be a connection between the two kinds of "value", but I don't see what it is yet other than the trivial connection that the outcomes given an act, i.e. those states conditional on the performace of an act have some value simpliciter.
But what about my question? Do you think God could be criticized for bringing about a world where everyone has my option (2) rather than (1) when they both have the same expected value? I think the answer is yes, if you agree, what could ground such a criticism on your view?
Rather, unless the expected value of (2) is at least as high as (1).
Look, what is the current value of the gamble [.5($100); .5(0)]? The equivocation talk is a red herring. What are you willing to pay for it right now (not in the future)? You will say (I hope) more than $10. That's because the expected value (= the value) of the gamble is $50. That is not a future value, that is a current, actual value. It is precisely that value which determines what rational agents should pay for it NOW, not in the future, in cold cash. You seem to be thinking of expected value as a value it will have or could have. It's not. It is the value it does have.
You're moving back and forward between the claim that a world or outcome has value and the claim that an action has value. Do you mean to claim that the EV of a world equals it's actual value and the EV of action equals it's actual value, or only one of these, and if so, which?
Even if you were to establish that the value of an action equals it's expected value--which I deny, for another reason: possibly, two acts can have the same EV though one is worse than the other since it could be performed from bad motives while the other would not be--that doesn't show the value of a world equals it's expected value, right?
Here's another objection: the probabilities in calculating EV are epistemic. So EV is epistemic. So, if you were right, then all value would depend upon the beliefs of agents. So, subjectivism (in some variety) is true. But subjectivism is false. QED.
Here's another objection: there could be a world with happy birds flying around that has positive value. In this world there are no epistemic probabilities since there are no believers. Thus if you were right, this world would be impossible. But it's possible. QED.
Then there was my infinity case: there is a world w where everybody is in some valuable qualitative state with finite value, but members in w also have a .5 chance of infinite bliss. w is better than another world w' that contains duplicates of those in w, but where members in w' have a .00005 chance at getting that infinity bliss. If EV were to equal value, then these worlds would have the same value. But they don't. QED.
There is also the point that if worlds that failed to obtained had EV and if EV = those world's actual value, then non-obtaining worlds would have actual value. But that's false. For it would entail that every world has the same value since it is true at every world that infinitely many good and bad worlds fail to obtain there. QED.
Thus, if you want to equate the EV of an action with the value of that action, that doesn't seem crazy to me, but very debateable. If, however, you want to equate the value of the world with it's EV, I've done what I can to convince you otherwise...
Do you mean to claim that the EV of a world equals it's actual value and the EV of action equals it's actual value, or only one of these, and if so, which?
Both, of course.
Even if you were to establish that the value of an action equals it's expected value--which I deny, for another reason: possibly, two acts can have the same EV though one is worse than the other since it could be performed from bad motives while the other would not be--that doesn't show the value of a world equals it's expected value, right?
If bad motives are actually bad things, then their disvalue would be discounted by their improbability, just as all value is. It would be a part of the EU calculation.
Here's another objection: the probabilities in calculating EV are epistemic. So EV is epistemic. So, if you were right, then all value would depend upon the beliefs of agents. So, subjectivism (in some variety) is true. But subjectivism is false
Huh? There is both, objective and subjective probability. There is room and need for both.
Here's another objection: there could be a world with happy birds flying around that has positive value. In this world there are no epistemic probabilities since there are no believers. Thus if you were right, this world would be impossible. But it's possible.
You've lost me. When did I ever say that a world without epistemic probability estimates is impossible?
Then there was my infinity case: there is a world w where everybody is in some valuable qualitative state with finite value, but members in w also have a .5 chance of infinite bliss. w is better than another world w' that contains duplicates of those in w, but where members in w' have a .00005 chance at getting that infinity bliss. If EV were to equal value, then these worlds would have the same value. But they don't.
If I follow this, you are claiming that if G and G' are two gambles with the same positive expected utility, but the probabilities for positive payoff in G are higher, then G is preferable to G'. But that's false, consider,
1. G = [.6($100), .4(0)] = 60
2. G' = [.1($600), .9(0)] = 60
These have exactly the same expected utility, as in the case you describe, and the probability of positive payoff is higher in G. Your view commits you to the bizarre position that G is rationally preferable to G'. But G is obviously not rationally preferable to G'.
There is also the point that if worlds that failed to obtained had EV and if EV = those world's actual value, then non-obtaining worlds would have actual value.
Once again, hard to follow. How could a non-obtaining world have actual value? It would have value at itself, maybe. It is no more than a possible, non-actual world.
Your view commits you to the bizarre position that G is rationally preferable to G'. But G is obviously not rationally preferable to G'.
No. My claim is restricted to outcomes with the same value. Yours involves outcomes with different value, i.e. $100 in one case and $600 in the other.
The comments above are edited to what might be worth pursuing.
No. My claim is restricted to outcomes with the same value. Yours involves outcomes with different value, i.e. $100 in one case and $600 in the other
My question is why I should prefer the gamble in (1) to the gamble in (2) when I know that the expected utility of each is identical? You seem to have one salient reason available and that's that (1) makes it more probable that I get some positive value.
1. [.9(oo+); 0]
2. [.1(oo+); 0]
If that's the reason to prefer (1) to (2), then in any case in which expected utility is identical--as it clearly is in (1) and (2)--I should prefer the gamble which gives me the greater probability of getting positive value. But then I should prefer (3) to (4), which is nonsense.
3. [.6($100), 0] = 60
4. [.1($600), 0] = 60
So maybe you'd want to say that (1) gives me a greater chance at an infinite amount and (2) gives me a smaller chance at the same infinite amount. But this is not quite right, either. (2) entails (6) (and vice versa), since (oo+) = (9 x oo+). So the same reasons you give for prefering (1) to (2) would commit you to preferring (5) to (6). But it is easy to show by associative laws that there is also no difference between (5) and (6). (5) entails (6) and (6) entails (5). So it cannot be true that (5) is preferable to (6), by your standards. But, if that is so, then it also cannot be true that (1) is preferable to (2), by your standards.
5. [.9(oo+); 0]
6. [.1(9 x oo+);0]
So, what then is the reason for preferring (1) to (2)?
Mike,
I'm really not sure what happened??? I responded to each of your points when attempting to publish my response. Did you delete part of my response before posting it? I apologize for even suggesting that, I only do so because you said, "The comments above are edited to what might be worth pursuing." If no, my recent comments do give the impression that I was ignoring those other points your were making. But somehow they didn't get posted, I'm not sure why...maybe I accidentally deleted them.
As to this:
5. [.9(oo+); 0]
6. [.1(9 x oo+);0]
So, what then is the reason for preferring (1) to (2)?
I really don't know. I suspect the following principle is reasonable:
Is S is indifferent between outcomes A and B, then if the probability that A obtains given that S does C is greater than the probability that B obtains given that S does D, then it is rational for S to do C rather than D. That's all I got.
Christian,
I have the rest of the comments, no worries, but I wanted to focus on one that I think progress can be made on. I can't talk about all of them at once. I'm happy to post them, if you want. I took this to be the most interesting comment.
5. [.9(oo+); 0]
6. [.1(9 x oo+);0]
So, what then is the reason for preferring (1) to (2)?I suspect the following principle is reasonable:
Is S is indifferent between outcomes A and B, then if the probability that A obtains given that S does C is greater than the probability that B obtains given that S does D, then it is rational for S to do C rather than D. That's all I got.
But that cannot be right, since by associative laws, (6) is equivalent to (7).
7. [(.1 x 9)(oo+); 0]
But (7) is just equivalent to (8),
8. [.9(oo+); 0]
But (8) is just (5). And (5) cannot be preferable to itself.
About the comments: feels like a slight to be quite honest. Though, I certainly agree that it pays neither of us to spin wheels. So let's focus on the point you have just raised.
First, I was too quick in my last response. Here's my general worry. We can propose a choice situation where the following two propositions appear true. (1) the expected utility of two exclusive and exhaustive acts are identical, and (2) one of these acts is irrational to perform.
I'm inclined to infer that, if it makes sense to talk about infinite utility, then the rule which says it is not rational to fail to maximize expected utility is false. Maximizing expected utility is a necessary, though insufficient condition on rational action. But you offer a choice situation:
5. [.9(oo+); 0]
6. [.1(9 x oo+);0]
8. [.9(oo+); 0]
You claim that by associative laws (6) = (8). And since (6) = (8), it follows from transitivity that (6) = (5). Thus, I cannot say that (5) is preferable (6) even though the probability of getting the outcome in (5) is greater than that in (6). It is irrational to prefer an action to itself.
I find this problem to be very difficult. But I don't see how it raises a problem for my earlier contention which was:
If S is indifferent between outcomes A and B, then if the probability that A obtains given that S does C is greater than the probability that B obtains given that S does D, then it is rational for S to do C rather than D.
In your example this principle does not apply. I am not indifferent between (9 times oo+) and (oo+). I prefer the former. If math tells me the utility of both is the same, then OK, but then I don't understand what (9 times oo+) is supposed to mean.
I do know that I would prefer an .9 chance at infinite utility rather than a .00005 chance at it. I want a theory of infinite quantities and rationality that vindicates that such a preference is rational. I suspect this means we need to appeal to non-standard mathematics where it makes sense to say that (9 times oo+) > (oo+). But I'm not qualified to evaluate the merits of this move.
Christian,
No slight intended. The idea is to moderate comments a bit to keep dicussion from diffusing. I usually just ask commentors to do this in a longish thread, if it does not happen naturally. I know you well enough, I think, to just select the best.
In your example this principle does not apply. I am not indifferent between (9 times oo+) and (oo+). I prefer the former.
I don't understand how you are not indifferent between 9 x oo+ and oo+. They are exactly the same size (not just sort of the same size, exactly the same size, and provably so.). Given these equivalences in size, we can run the argument in several ways. You are presumably indifferent between (5) and (5').
5. [.9(oo+); 0]
5'. [.9(oo+); 0]
By substitution of equivalents (oo+ = (.1 x oo+)), (5') equals (6'),
6'. [.9(.1 x oo+); 0]
By association and commutation, (6') is equivalent to (7'),
7'. [.1(.9 x oo+); 0]
But again by substitution of equivalents, (7') is just (8'),
8'. [.1(oo+); 0]
The upshot is that your principle would have you indifferent between (5) and (5'). But by some simple arithmetical operations on (5'), we can show that it is equivalent to (8'). And your principle says that you are not indifferent between (5) and (8'). The consequence is that you both are and are not indifferent between (5) and (5').
I don't understand how you are not indifferent between 9 x oo+ and oo+. They are exactly the same size (not just sort of the same size, exactly the same size, and provably so.). Given these equivalences in size, we can run the argument in several ways.
Don't know why everything is getting italicized. Anyway, agreed there are numbers of ways to make your point, all very persuasive arguments. That is, if you assume a Cantorian notion of measuring cardinality. I find the notion suspicious and the proofs depend upon it. Of course, if you're point above is correct, then you could have argued in the beginning:
Mike*: I don't understand how you are not indifferent between .9 x oo+ and oo+. They are exactly the same size (not just sort of the same size, exactly the same size, and provably so.)
Well, my response would be I'm just not indifferent. Call me crazy. I'd rather have infinite utility for certain than having a .9 chance at it. I modus tollens your argument. Thus, I'm also not indifferent between 9 x oo+ and oo+.
These issues are too deep for me. Interesting though.