I don’t think that “the justification for expected utility is that as the number of bets you take approach infinity, it becomes the optimal strategy,” is quite true. Kaj did say something similar to this, but that seems to me a problem with his approach.
Basically, expected utility is supposed to give a mathematical formalization to people’s preferences. But consider this fact: in itself, it does not have any particular sense to say that “I like vanilla ice cream twice as much as chocolate.” It makes sense to say I like it more than chocolate. This means if I am given a choice between vanilla and chocolate, I will choose vanilla. But what on earth does it mean to say that I like it “twice” as much as chocolate? In itself, nothing. We have to define this in order to construct a mathematical analysis of our preferences.
In practice we make this definition by saying that I like vanilla so much that I am indifferent to having chocolate, or to having a 50% chance of having vanilla, and a 50% chance of having nothing.
Perhaps I justify this by saying that it will get me a certain amount of vanilla in my life. But perhaps I don’t—the definition does not justify the preference, it simply says what it means. This means that in order to say I like it twice as much, I have to say that I am indifferent to the 50% bet and to the certain chocolate, no matter what the justification for this might or might not be. If I change my preference when the number of cases goes down, then it will not be mathematically consistent to say that I like it twice as much as chocolate, unless we change the definition of “like it twice as much.”
Basically I think you are mixing up things like “lives”, which can be mathematically quantified in themselves, more or less, and people’s preferences, which only have a quantity if we define one.
It may be possible for Kaj to come up with a new definition for the amount of someone’s preference, but I suspect that it will result in a situation basically the same as keeping our definition, but admitting that people have only a limited amount of preference for things. In other words, they might prefer saving 100,000 lives to saving 10,000 lives, but they certainly do not prefer it 10 times as much, meaning they will not always accept the 100,000 lives saved at a 10% chance, compared to a 100% chance of saving 10,000.
But what on earth does it mean to say that I like it “twice” as much as chocolate?
Obviously it means you would be willing to trade 2 units of chocolate ice cream for 1 unit of vanilla. And over the course of your life, you would prefer to have more vanilla ice cream than chocolate ice cream. Perhaps before you die, you will add up all the ice creams you’ve ever eaten. And you would prefer for that number to be higher rather than lower.
Nowhere in the above description did I talk about probability. And the utility function is already completely defined. I just need to decide on a decision procedure to maximize it.
Expected utility seems like a good choice, because, over the course of my life, different bets I make on ice cream should average themselves out, and I should do better than otherwise. But that might not be true if there are ice cream muggers. Which promise lots of ice cream in exchange for a down payment, but usually lie.
So trying to convince the ice cream maximizer to follow expected utility is a lost cause. They will just end up losing all their ice cream to muggers. They need a system which ignores muggers.
This is definitely not what I mean if I say I like vanilla twice as much as chocolate. I might like it twice as much even though there is no chance that I can ever eat more than one serving of ice cream. If I have the choice of a small serving of vanilla or a triple serving of chocolate, I might still choose the vanilla. That does not mean I like it three times as much.
It is not about “How much ice cream.” It is about “how much wanting”.
I’m saying that the experience of eating chocolate is objectively twice as valuable. Maybe there is a limit on how much ice cream you can eat at a single sitting. But you can still choose to give up eating vanilla today and tomorrow, in exchange for eating chocolate once.
Again, you are assuming there is a quantitative measure over eating chocolate and eating vanilla, and that this determines the measure of my utility. This is not necessarily true, since these are arbitrary examples. I can still value one twice as much as the other, even if they both are experiences that can happen only once in a lifetime, or even only once in the lifetime of the universe.
Sure. But there is still some internal, objective measure of value of experiences. The constraints you add make it harder to determine what they are. But in simple cases, like trading ice cream, it’s easy to determine how much value a person has for a thing.
I don’t think that “the justification for expected utility is that as the number of bets you take approach infinity, it becomes the optimal strategy,” is quite true. Kaj did say something similar to this, but that seems to me a problem with his approach.
Basically, expected utility is supposed to give a mathematical formalization to people’s preferences. But consider this fact: in itself, it does not have any particular sense to say that “I like vanilla ice cream twice as much as chocolate.” It makes sense to say I like it more than chocolate. This means if I am given a choice between vanilla and chocolate, I will choose vanilla. But what on earth does it mean to say that I like it “twice” as much as chocolate? In itself, nothing. We have to define this in order to construct a mathematical analysis of our preferences.
In practice we make this definition by saying that I like vanilla so much that I am indifferent to having chocolate, or to having a 50% chance of having vanilla, and a 50% chance of having nothing.
Perhaps I justify this by saying that it will get me a certain amount of vanilla in my life. But perhaps I don’t—the definition does not justify the preference, it simply says what it means. This means that in order to say I like it twice as much, I have to say that I am indifferent to the 50% bet and to the certain chocolate, no matter what the justification for this might or might not be. If I change my preference when the number of cases goes down, then it will not be mathematically consistent to say that I like it twice as much as chocolate, unless we change the definition of “like it twice as much.”
Basically I think you are mixing up things like “lives”, which can be mathematically quantified in themselves, more or less, and people’s preferences, which only have a quantity if we define one.
It may be possible for Kaj to come up with a new definition for the amount of someone’s preference, but I suspect that it will result in a situation basically the same as keeping our definition, but admitting that people have only a limited amount of preference for things. In other words, they might prefer saving 100,000 lives to saving 10,000 lives, but they certainly do not prefer it 10 times as much, meaning they will not always accept the 100,000 lives saved at a 10% chance, compared to a 100% chance of saving 10,000.
Obviously it means you would be willing to trade 2 units of chocolate ice cream for 1 unit of vanilla. And over the course of your life, you would prefer to have more vanilla ice cream than chocolate ice cream. Perhaps before you die, you will add up all the ice creams you’ve ever eaten. And you would prefer for that number to be higher rather than lower.
Nowhere in the above description did I talk about probability. And the utility function is already completely defined. I just need to decide on a decision procedure to maximize it.
Expected utility seems like a good choice, because, over the course of my life, different bets I make on ice cream should average themselves out, and I should do better than otherwise. But that might not be true if there are ice cream muggers. Which promise lots of ice cream in exchange for a down payment, but usually lie.
So trying to convince the ice cream maximizer to follow expected utility is a lost cause. They will just end up losing all their ice cream to muggers. They need a system which ignores muggers.
This is definitely not what I mean if I say I like vanilla twice as much as chocolate. I might like it twice as much even though there is no chance that I can ever eat more than one serving of ice cream. If I have the choice of a small serving of vanilla or a triple serving of chocolate, I might still choose the vanilla. That does not mean I like it three times as much.
It is not about “How much ice cream.” It is about “how much wanting”.
I’m saying that the experience of eating chocolate is objectively twice as valuable. Maybe there is a limit on how much ice cream you can eat at a single sitting. But you can still choose to give up eating vanilla today and tomorrow, in exchange for eating chocolate once.
Again, you are assuming there is a quantitative measure over eating chocolate and eating vanilla, and that this determines the measure of my utility. This is not necessarily true, since these are arbitrary examples. I can still value one twice as much as the other, even if they both are experiences that can happen only once in a lifetime, or even only once in the lifetime of the universe.
Sure. But there is still some internal, objective measure of value of experiences. The constraints you add make it harder to determine what they are. But in simple cases, like trading ice cream, it’s easy to determine how much value a person has for a thing.