[Note: If you have a degree in maths or statistics or economics, some of this might be ill informed. Forgive me, I’m a political science graduate just muddling through! Similarly, I’m not too well-versed in decision theory so much of this is likely well-trodden ground. Future posts also won’t be so maths heavy.]

Math/econ here. On Pascal's mugger, our probability that the mugger will keep their promise is allowed to decrease with how much money they promise. So, there is no reason the expected return should increase with their promise.

For the lottery example, we need to distinguish between expected value and expected utility. Even if the expected return on the lottery were positive, almost everyone is risk-averse. Hence, why you said you would still probably not buy such a lottery ticket.

An effective altruist is already working directly with utilities in their calculations and might not need to make this distinction. However, perhaps the appropriate societal utility function (as a function of everyone's each individual utility) is somewhat risk averse.

Pascal's wager makes a lot of assumptions over uncertainty. As an atheist conditioning on my being wrong, I have no idea what God would want. If I had to guess, God would probably prefer a humanist over a selfish and disingenuous monkey dart at the wall. Similarly, I'm doubtful God would want me to subscribe on the selfish microscope chance you put in a good word.

> I’m used to using EV and Expected Utility when thinking about what the rational thing to do is [...]

I think the problem lies in *merely* using an EV calculation to determine what the rational decision is. Rather, a better procedure uses both an EV calculation *as well as* a bet-sizing calculation. For an example of a bet-sizing method, see the Kelly criterion <https://en.wikipedia.org/wiki/Kelly_criterion>. [1]

In essence, a decision to act involves a cost, i.e., an expenditure of some resources (often called the "bet size" in discussions about the Kelly strategy, which tend to involve examples about gambling). For example, in the mugger case, the cost is whatever dollar amount you are handing over to the mugger. In the Pascal's wager case, the cost is the utility loss one takes from adopting a Christian lifestyle (if there is such a utility loss). For voting, the cost is the time you take to read up on candidates and go vote.

The important insight is that if you want to maximize the growth of your wealth (or your cumulative net "utils"), there is an optimal expenditure size for each betting opportunity. Pay either more or less than the optimal amount and the expected value of the logarithm of your capital balance [2] goes *down*, even if the individual transaction has a positive expected value! The expected value of the logarithm of your capital balance can even become negative (your capital can go to zero) if you are too far off the optimal bet size.

A simple case demonstrating this would be a positive EV lottery ticket. In the United States, there are a few national lotteries which work on a system whereby if there is no winner during one time period, the prize money is rolled forward to the next time period, increasing the prize amount for the next time. In these types of lotteries, it is not too uncommon for there to be a string of no-winner periods, which eventually results in the prize amount being so large that lottery tickets have a positive EV. Suppose for example that the Mega Millions lottery has accumulated a prize of $1 billion USD, the cost of a ticket is $2, and the chance of winning is 1 in 250 million. The expected value of a ticket is thus $4, twice the cost of the ticket. Most asset classes have ROIs nowhere near 100%, so does this mean that if you have $2 to invest, it should go into buying a lottery ticket instead of into a stock index fund? If you merely compare the one-year-forward EV of any given stock index versus the 100% ROI from a lottery ticket, then the answer appears to be, yes, buy the lottery ticket.

But when you look at the situation with your Kelly strategy glasses on, you see that the optimal fraction of your wealth to bet, given the parameters above, is [1/250,000,000 - (1 - 1/250,000,000)/500,000,000] =

1/500,000,000. Let's say your net worth is $1 million USD. The optimal bet size is then 1 / 5th of a penny. (And if, like most people, you have less than $1 million USD, the optimal bet size is even less.) Since the minimum "investment" is a $2 ticket, and tickets cannot be bought in fractional quantities, after rounding to the nearest whole number, we see the rational number of tickets to buy is 0. That is, despite the huge expected ROI for "investing" in a lottery ticket, you should not buy one. A strategy of repeatedly buying lottery tickets under these circumstances produces a negative growth rate for your capital, which eventually leads to bankruptcy with 100% probability, in the limit of an infinite sequence of such transactions. (Though under other circumstances, such as a vastly greater initial wealth OR the ability to buy up a large fraction of the tickets at once, it could be rational to buy lottery tickets.)

Applying this reasoning to the mugger scenario: since the win probability is extremely low, the optimal bet size may well be so small that it isn't reasonable to give them as much as penny.

And of note to the Pascal's wager scenario: even the promise of an infinite reward does not necessarily increase the optimal bet size past a certain maximum. If there is a non-zero probability of losing the bet, there is no reward large enough to make the optimal bet size your entire capital. (However, I did once hear a well-known preacher say that the proper interpretation of Pascal's argument was that the rewards for Christian living *in this life* were such that it was worth doing even if we could not calculate what would happen to us in the next life. Under this interpretation, the "cost" of the bet is negative. That is, being a Christian is a free lunch, and it is rational to take the bet for that reason. The reader may judge the salience of this view for themselves.)

[1] For technical reasons, the Kelly criterion is a bit simplistic should only be applied in real life with caution. It is nevertheless a great way to introduce oneself to the underlying concept of bet sizing. See the wikipedia article for more discussion about some nuances.

[2] Or the logarithm of your net cumulative "utils".

Similarly, the utilitarians sole purpose in life should be maximizing utility by making people convert to Christianity in any way possible.

One thing worth mentioning is risk aversion. You could have an expected value that is positive and not take the choice because you are risk averse. I think effective altruists should be risk neutral in their charitable investments. For this reason, they should make extremely risky investments. See here: https://parrhesia.substack.com/p/should-effective-altruists-make-risky

edited Dec 31, 2021Math/econ here. On Pascal's mugger, our probability that the mugger will keep their promise is allowed to decrease with how much money they promise. So, there is no reason the expected return should increase with their promise.

For the lottery example, we need to distinguish between expected value and expected utility. Even if the expected return on the lottery were positive, almost everyone is risk-averse. Hence, why you said you would still probably not buy such a lottery ticket.

An effective altruist is already working directly with utilities in their calculations and might not need to make this distinction. However, perhaps the appropriate societal utility function (as a function of everyone's each individual utility) is somewhat risk averse.

Pascal's wager makes a lot of assumptions over uncertainty. As an atheist conditioning on my being wrong, I have no idea what God would want. If I had to guess, God would probably prefer a humanist over a selfish and disingenuous monkey dart at the wall. Similarly, I'm doubtful God would want me to subscribe on the selfish microscope chance you put in a good word.

> I’m used to using EV and Expected Utility when thinking about what the rational thing to do is [...]

I think the problem lies in *merely* using an EV calculation to determine what the rational decision is. Rather, a better procedure uses both an EV calculation *as well as* a bet-sizing calculation. For an example of a bet-sizing method, see the Kelly criterion <https://en.wikipedia.org/wiki/Kelly_criterion>. [1]

In essence, a decision to act involves a cost, i.e., an expenditure of some resources (often called the "bet size" in discussions about the Kelly strategy, which tend to involve examples about gambling). For example, in the mugger case, the cost is whatever dollar amount you are handing over to the mugger. In the Pascal's wager case, the cost is the utility loss one takes from adopting a Christian lifestyle (if there is such a utility loss). For voting, the cost is the time you take to read up on candidates and go vote.

The important insight is that if you want to maximize the growth of your wealth (or your cumulative net "utils"), there is an optimal expenditure size for each betting opportunity. Pay either more or less than the optimal amount and the expected value of the logarithm of your capital balance [2] goes *down*, even if the individual transaction has a positive expected value! The expected value of the logarithm of your capital balance can even become negative (your capital can go to zero) if you are too far off the optimal bet size.

A simple case demonstrating this would be a positive EV lottery ticket. In the United States, there are a few national lotteries which work on a system whereby if there is no winner during one time period, the prize money is rolled forward to the next time period, increasing the prize amount for the next time. In these types of lotteries, it is not too uncommon for there to be a string of no-winner periods, which eventually results in the prize amount being so large that lottery tickets have a positive EV. Suppose for example that the Mega Millions lottery has accumulated a prize of $1 billion USD, the cost of a ticket is $2, and the chance of winning is 1 in 250 million. The expected value of a ticket is thus $4, twice the cost of the ticket. Most asset classes have ROIs nowhere near 100%, so does this mean that if you have $2 to invest, it should go into buying a lottery ticket instead of into a stock index fund? If you merely compare the one-year-forward EV of any given stock index versus the 100% ROI from a lottery ticket, then the answer appears to be, yes, buy the lottery ticket.

But when you look at the situation with your Kelly strategy glasses on, you see that the optimal fraction of your wealth to bet, given the parameters above, is [1/250,000,000 - (1 - 1/250,000,000)/500,000,000] =

1/500,000,000. Let's say your net worth is $1 million USD. The optimal bet size is then 1 / 5th of a penny. (And if, like most people, you have less than $1 million USD, the optimal bet size is even less.) Since the minimum "investment" is a $2 ticket, and tickets cannot be bought in fractional quantities, after rounding to the nearest whole number, we see the rational number of tickets to buy is 0. That is, despite the huge expected ROI for "investing" in a lottery ticket, you should not buy one. A strategy of repeatedly buying lottery tickets under these circumstances produces a negative growth rate for your capital, which eventually leads to bankruptcy with 100% probability, in the limit of an infinite sequence of such transactions. (Though under other circumstances, such as a vastly greater initial wealth OR the ability to buy up a large fraction of the tickets at once, it could be rational to buy lottery tickets.)

Applying this reasoning to the mugger scenario: since the win probability is extremely low, the optimal bet size may well be so small that it isn't reasonable to give them as much as penny.

And of note to the Pascal's wager scenario: even the promise of an infinite reward does not necessarily increase the optimal bet size past a certain maximum. If there is a non-zero probability of losing the bet, there is no reward large enough to make the optimal bet size your entire capital. (However, I did once hear a well-known preacher say that the proper interpretation of Pascal's argument was that the rewards for Christian living *in this life* were such that it was worth doing even if we could not calculate what would happen to us in the next life. Under this interpretation, the "cost" of the bet is negative. That is, being a Christian is a free lunch, and it is rational to take the bet for that reason. The reader may judge the salience of this view for themselves.)

[1] For technical reasons, the Kelly criterion is a bit simplistic should only be applied in real life with caution. It is nevertheless a great way to introduce oneself to the underlying concept of bet sizing. See the wikipedia article for more discussion about some nuances.

[2] Or the logarithm of your net cumulative "utils".

Similarly, the utilitarians sole purpose in life should be maximizing utility by making people convert to Christianity in any way possible.

One thing worth mentioning is risk aversion. You could have an expected value that is positive and not take the choice because you are risk averse. I think effective altruists should be risk neutral in their charitable investments. For this reason, they should make extremely risky investments. See here: https://parrhesia.substack.com/p/should-effective-altruists-make-risky