Interestingly, I discovered the Lifespan Dilemma due to this post. While not facing a total breakdown of my ability to do anything else, it did consume an inordinate amount of my thought process.
The question looks like an optimal betting problem- you have a limited resource, and need to get the most return. According to the Kelly Criterion, the optimal percentage of your total bankroll looks like f*=(p(b-1)+1)/b, where p is the probability of success, and b is the return per unit risked. The interesting thing here is that for very large values of b, the percentage of bankroll to be risked almost exactly equals the percentage chance of winning. Assuming a bankroll of 100 units and a 20 percent chance of success, you should bet the same amount if b = 1 million or if b = 1 trillion: 20 units.
Eager to apply this to the problem at hand, I decided to plug in the numbers. I then realized I didn’t know what the bank roll was in this situation. My first thought was that the bankroll was the expected time left- percent chance of success * time if successful. I think this is the mode that leads to the garden path- every time you increase your time of life if successful, it feels like you have more units to bet with, which means you are willing to spend more on longer odds.
Not satisfied, I attempted to re-frame the question into money. Stating it like this, I have 100$, and in 2 hours I will either have 0$, or 1 million, with an 80% chance of winning. I could trade my 80% chance for a 79% chance of winning 1 trillion. So, now that we are in money, where is my bankroll?
I believe that is the trick- in this question, you are already all in. You have already bet 100% of your bankroll, for an 80% chance of winning- in 2 hours, you will know the outcome of your bet. For extremely high values of b, you should have only bet 80% of your bankroll- you are already underwater. Here is the key point- changing the value of b does not change what you should have bet, or even your bet at all- that’s locked in. All you can change is the probability, and you can only make it worse. From this perspective, you should accept no offer that lowers your probability of winning.
Interestingly, I discovered the Lifespan Dilemma due to this post. While not facing a total breakdown of my ability to do anything else, it did consume an inordinate amount of my thought process.
The question looks like an optimal betting problem- you have a limited resource, and need to get the most return. According to the Kelly Criterion, the optimal percentage of your total bankroll looks like f*=(p(b-1)+1)/b, where p is the probability of success, and b is the return per unit risked. The interesting thing here is that for very large values of b, the percentage of bankroll to be risked almost exactly equals the percentage chance of winning. Assuming a bankroll of 100 units and a 20 percent chance of success, you should bet the same amount if b = 1 million or if b = 1 trillion: 20 units.
Eager to apply this to the problem at hand, I decided to plug in the numbers. I then realized I didn’t know what the bank roll was in this situation. My first thought was that the bankroll was the expected time left- percent chance of success * time if successful. I think this is the mode that leads to the garden path- every time you increase your time of life if successful, it feels like you have more units to bet with, which means you are willing to spend more on longer odds.
Not satisfied, I attempted to re-frame the question into money. Stating it like this, I have 100$, and in 2 hours I will either have 0$, or 1 million, with an 80% chance of winning. I could trade my 80% chance for a 79% chance of winning 1 trillion. So, now that we are in money, where is my bankroll?
I believe that is the trick- in this question, you are already all in. You have already bet 100% of your bankroll, for an 80% chance of winning- in 2 hours, you will know the outcome of your bet. For extremely high values of b, you should have only bet 80% of your bankroll- you are already underwater. Here is the key point- changing the value of b does not change what you should have bet, or even your bet at all- that’s locked in. All you can change is the probability, and you can only make it worse. From this perspective, you should accept no offer that lowers your probability of winning.