I’m having difficulty understanding exactly what an answer of “such a probability does not exist” means in this context. Assuming we both were subjected to the same experiment, but I then assigned a 50% probability to being the Original, how would our future behaviour differ? In what concrete scenario (other than answering questions about the probability we were the Original) would you predict us to act differently as a result of this specific difference in belief?
Our behavior should be different in many cases. However, base on my past experience, people who accept self-locating probabilities would often find various explanations so our decisions would still be the same.
For example, in “Repeating the Experiment” the relative frequency of Me being the Original won’t converge on any particular value. If we bet on that, I will say there is no strategy to maximize My personal gain. (There is a strategy to max the combined gain of all copies if everyone abides by it. As reflected by the probability of a randomly sampled copy being Original is 1⁄2)
On the other hand, you would say if I repeat the experiment long enough the relative frequency of me being the Original would converge on 50%, and the best strategy to max my personal gain is to bet accordingly.
The problem of this example is that personal gain can only be verified by the first-person perspective of the subject. A verifiable example would be this: change the original experiment slightly. The Mad scientist would only perform the cloning if a fair coin toss landed on Tails. Then after waking up how should you guess the probability of Heads? What’s the probability of Heads if you learn you are the Original? (Essentially the sleeping beauty problem).
If you endorse self-locating probability, then there are two options. First, the thirder. After waking up the probability of I am the Original is 2⁄3. The probability of Heads is 1⁄3. After learning I am the Original the probability of Heads updates to 1⁄2.
The other option is to say after waking the probability of Heads is 1⁄2, the probability of I am the Original is 3⁄4. After learning I am the Orignal the probability of Heads needs to be updated. (How to do this update is very problematic, but let’s skip it for now. The main point is the probability for Heads would have to be smaller than 1⁄2. And this is a very weak camp compare to the thirders)
Because I reject self-locating probability, I would say the probability of Heads is 1⁄2. And it is still 1⁄2 after learning I am the Original. No update because there is no probability in the first place.
This should result in different betting strategies. Say you have just experienced 100 iterations of this toss and cloning and haven’t learned whether you were the Orignal or the Clone in any of those iterations. Now you are offered to enter a bet for 2 dollars that will pay 5 dollars if the coin landed on Heads for each of those 100 iterations. Assuming you are a thirder, then you should not enter these bets, since you believe the probability of Heads is only 1⁄3. Whereas I would enter all these bets. But again, base on past experience thirders would come up with some explanation as to why they would also enter these bets. So our decisions would still be the same.
This should result in different betting strategies.
Probabilities are not strategies. Strategy development may make use of probabilities, but only according to models that link various probabilities to outcomes, risk, and heaps of other factors. You can often formulate exactly the same strategies using different models employing different probabilities. Sometimes there is a single simplest model that employs an obvious probability space to yield a clear winning strategy, sometimes there is not.
Depending upon how the payoffs are structured in your bets and how results affect future states, I might or might not enter into any of those bets. You also use the term “thirder” and “halfer” as if this is a fixed personality trait, and not a choice of which probability space to employ in each particular scenario.
I’m having difficulty understanding exactly what an answer of “such a probability does not exist” means in this context. Assuming we both were subjected to the same experiment, but I then assigned a 50% probability to being the Original, how would our future behaviour differ? In what concrete scenario (other than answering questions about the probability we were the Original) would you predict us to act differently as a result of this specific difference in belief?
Our behavior should be different in many cases. However, base on my past experience, people who accept self-locating probabilities would often find various explanations so our decisions would still be the same.
For example, in “Repeating the Experiment” the relative frequency of Me being the Original won’t converge on any particular value. If we bet on that, I will say there is no strategy to maximize My personal gain. (There is a strategy to max the combined gain of all copies if everyone abides by it. As reflected by the probability of a randomly sampled copy being Original is 1⁄2)
On the other hand, you would say if I repeat the experiment long enough the relative frequency of me being the Original would converge on 50%, and the best strategy to max my personal gain is to bet accordingly.
The problem of this example is that personal gain can only be verified by the first-person perspective of the subject. A verifiable example would be this: change the original experiment slightly. The Mad scientist would only perform the cloning if a fair coin toss landed on Tails. Then after waking up how should you guess the probability of Heads? What’s the probability of Heads if you learn you are the Original? (Essentially the sleeping beauty problem).
If you endorse self-locating probability, then there are two options. First, the thirder. After waking up the probability of I am the Original is 2⁄3. The probability of Heads is 1⁄3. After learning I am the Original the probability of Heads updates to 1⁄2.
The other option is to say after waking the probability of Heads is 1⁄2, the probability of I am the Original is 3⁄4. After learning I am the Orignal the probability of Heads needs to be updated. (How to do this update is very problematic, but let’s skip it for now. The main point is the probability for Heads would have to be smaller than 1⁄2. And this is a very weak camp compare to the thirders)
Because I reject self-locating probability, I would say the probability of Heads is 1⁄2. And it is still 1⁄2 after learning I am the Original. No update because there is no probability in the first place.
This should result in different betting strategies. Say you have just experienced 100 iterations of this toss and cloning and haven’t learned whether you were the Orignal or the Clone in any of those iterations. Now you are offered to enter a bet for 2 dollars that will pay 5 dollars if the coin landed on Heads for each of those 100 iterations. Assuming you are a thirder, then you should not enter these bets, since you believe the probability of Heads is only 1⁄3. Whereas I would enter all these bets. But again, base on past experience thirders would come up with some explanation as to why they would also enter these bets. So our decisions would still be the same.
Probabilities are not strategies. Strategy development may make use of probabilities, but only according to models that link various probabilities to outcomes, risk, and heaps of other factors. You can often formulate exactly the same strategies using different models employing different probabilities. Sometimes there is a single simplest model that employs an obvious probability space to yield a clear winning strategy, sometimes there is not.
Depending upon how the payoffs are structured in your bets and how results affect future states, I might or might not enter into any of those bets. You also use the term “thirder” and “halfer” as if this is a fixed personality trait, and not a choice of which probability space to employ in each particular scenario.