I think that in this case, option B is the right choice.
[...]
But, what if someone decides to make just one decision, which is worse in expectation but very improbable to have any negative consequences? Of course, if this person would start to make such decisions repeatedly, then she will predictably end worse off, but if she is able to reliably restrict herself to making just this single decision solely on the basis of its small probability, and following the expected utility otherwise, then for me it seems to be rational.
[...]
It seems to me that the Sam’s strategy achieve the best result at the end.
So, I’m not really seeing any argument in your post. You claim to answer the question “why”, but then just present the cases/stories and go on to say “for me, this is the right choice”. Therefore, it’s difficult to provide any comment on the reasoning.
So, my question would be: in what sense it’s the right choice/rational/achieving best result? The only passage that seems to start addressing that is the one with “it’s worse in expectation, but very improbable”.
(One way to judge a decision rigorously, as you seem to be doing in the “ordinary” case, would be to create a model and a utility function—in your text: a long sequence of decisions, a payoff at each one, aggregated utility measured by a sum or mean).
Thanks for your comment. I’ll try to express myself more clearly. You’ve asked ,, in what sense it’s the right choice/rational/achieving best result?” This is what I had in mind. I regard the decision as rational if from the set of all possible acts it selects first those acts which have probability equal or higher than 0,5 of achieving a net positive result, and then from those acts, the act which has the highest upside. Let’s call this approach the ,,first order” approach (I’m still uncertain about this exact formulation and I may revise it in the future, but let’s stick with this at the moment). For example: I have to choose between options A, B and C
Option A: probability 0,9 of gaining 100 utility points and probability 0,1 of gaining 1000 negative utility points
Option B: probability 0,01 of gaining 100000 utility points and probability 0,99 of gaining 10 negative utility points
Option C: probability 0,75 of gaining 1000 utility points and probability 0,25 of gaining 10000 negative utility points
From this set of options first A and C would be selected and finally option C would be chosen. By this choice I will most probably gain 1000 utility points and loose nothing. However, now is the moment when the expected utility theory (EU) comes into play. Let’s imagine that I know that during my life (let’s say 80 years) I will be confronted with that set of potions many times. If each time I would follow the procedure outlined above, then I would predictably end worse off, since option C has negative EU (indeed, the highest negative EU from all the options). I think that the approach I’ve defined above is not in contradiction with EU if we look at our life holistically. I used to think of it as choosing the best decision framework, which at the end will lead to the best possible outcome. So my rationale for adopting EU is ultimately based on the first order approach that I defined earlier. I’m not sure how exact probabilities and utility points should look like here, but the situation looks roughly like this:
Option A: Adopt EU (with probability above 0,5 will lead to the best possible result overall)
Option B: Use the first order approach in every single decision (with probability above 0,5 will not lead to the best possible result overall)
That shows that the first order approach leads to the acceptance of EU if we look at the situation holistically. Of course, now the question may arise ,,So for what was that whole fancy theorizing about the first order approach? Isn’t it better to just adopt the EU from the start?” Well, at least for mi the EU is not self-evident and it needs some further rationale to be justified. The first order approach tries to capture a fundamental intuition which, I think, stand behind the EU.
So what about Pascal’s wager? In this case the acceptance of wager is the best options according to EU. However, as I’ve tried to show above, EU works only because it pays off to follow it over the long series of choices under uncertainty. If some agent is able to reliably restrict herself to making just one exception to following EU, in the case when it is improbable that it would have any negative consequences, then it seems to me that such exception could be justified. Let’s illustrate it on the same example that I’ve gave above. Suppose that indeed during 80 years of my life I was many times confronted with a choice between the options A, B and C. I followed the EU, so overall I gained a lot of utility points. Now I’m on my deathbed and this is the last hour of my life. Someone approaches me and offers me one more time the choice between options A, B and C. I know that the option B is the best in expectation. However, I have no expectation to life longer than an hour, so there is no more time to make the EU reasoning work. So I decide to choose option C this time. Most probably, I would gain more utility points than if I chose the option B one more time.
Sorry for making this response so long, but I tried to be clear in explaining my reasoning. However, I’m not an expert on probability theory nor on decision theory. If you think that I messed up something in the argument outlined above, fell free to press me on that point. It is really important for me to get things right in this case, so I appreciate the constructive critique.
So, I’m not really seeing any argument in your post. You claim to answer the question “why”, but then just present the cases/stories and go on to say “for me, this is the right choice”. Therefore, it’s difficult to provide any comment on the reasoning.
So, my question would be: in what sense it’s the right choice/rational/achieving best result? The only passage that seems to start addressing that is the one with “it’s worse in expectation, but very improbable”.
(One way to judge a decision rigorously, as you seem to be doing in the “ordinary” case, would be to create a model and a utility function—in your text: a long sequence of decisions, a payoff at each one, aggregated utility measured by a sum or mean).
Thanks for your comment. I’ll try to express myself more clearly.
You’ve asked ,, in what sense it’s the right choice/rational/achieving best result?”
This is what I had in mind.
I regard the decision as rational if from the set of all possible acts it selects first those acts which have probability equal or higher than 0,5 of achieving a net positive result, and then from those acts, the act which has the highest upside. Let’s call this approach the ,,first order” approach (I’m still uncertain about this exact formulation and I may revise it in the future, but let’s stick with this at the moment).
For example: I have to choose between options A, B and C
Option A: probability 0,9 of gaining 100 utility points and probability 0,1 of gaining 1000 negative utility points
Option B: probability 0,01 of gaining 100000 utility points and probability 0,99 of gaining 10 negative utility points
Option C: probability 0,75 of gaining 1000 utility points and probability 0,25 of gaining 10000 negative utility points
From this set of options first A and C would be selected and finally option C would be chosen. By this choice I will most probably gain 1000 utility points and loose nothing.
However, now is the moment when the expected utility theory (EU) comes into play. Let’s imagine that I know that during my life (let’s say 80 years) I will be confronted with that set of potions many times. If each time I would follow the procedure outlined above, then I would predictably end worse off, since option C has negative EU (indeed, the highest negative EU from all the options).
I think that the approach I’ve defined above is not in contradiction with EU if we look at our life holistically. I used to think of it as choosing the best decision framework, which at the end will lead to the best possible outcome. So my rationale for adopting EU is ultimately based on the first order approach that I defined earlier. I’m not sure how exact probabilities and utility points should look like here, but the situation looks roughly like this:
Option A: Adopt EU (with probability above 0,5 will lead to the best possible result overall)
Option B: Use the first order approach in every single decision (with probability above 0,5 will not lead to the best possible result overall)
That shows that the first order approach leads to the acceptance of EU if we look at the situation holistically. Of course, now the question may arise ,,So for what was that whole fancy theorizing about the first order approach? Isn’t it better to just adopt the EU from the start?”
Well, at least for mi the EU is not self-evident and it needs some further rationale to be justified. The first order approach tries to capture a fundamental intuition which, I think, stand behind the EU.
So what about Pascal’s wager? In this case the acceptance of wager is the best options according to EU. However, as I’ve tried to show above, EU works only because it pays off to follow it over the long series of choices under uncertainty. If some agent is able to reliably restrict herself to making just one exception to following EU, in the case when it is improbable that it would have any negative consequences, then it seems to me that such exception could be justified.
Let’s illustrate it on the same example that I’ve gave above. Suppose that indeed during 80 years of my life I was many times confronted with a choice between the options A, B and C. I followed the EU, so overall I gained a lot of utility points. Now I’m on my deathbed and this is the last hour of my life. Someone approaches me and offers me one more time the choice between options A, B and C. I know that the option B is the best in expectation. However, I have no expectation to life longer than an hour, so there is no more time to make the EU reasoning work. So I decide to choose option C this time. Most probably, I would gain more utility points than if I chose the option B one more time.
Sorry for making this response so long, but I tried to be clear in explaining my reasoning. However, I’m not an expert on probability theory nor on decision theory. If you think that I messed up something in the argument outlined above, fell free to press me on that point. It is really important for me to get things right in this case, so I appreciate the constructive critique.