If you bet repeatedly on a gamble in which with probability p you win k times what you bet, and otherwise lose your bet, the fraction of your wealth to bet that maximises your growth rate is p−(1−p)/k. This implies that no matter how enormous the payoff, you should never bet more than p of your wealth. The probability you assign to unsubstantiated promises from dodgy strangers should be very small, so you can safely ignore Pascal’s Wager.
Suppose the mugger says that if you don’t give him $5, he’ll take away 99.999999999999999% of your wealth. I don’t think Kelly bets save you there? The logarithms of Kelly bets help you on the positive side but hurt you on the negative side.
Kelly bets only apply to the situation where you have a choice to gamble or not, and not gambling leaves your wealth unaffected. When the Kelly bet is negative, that means you should decline the bet.
If the mugger is capable of confiscating 99.999999999999999% of your wealth, why is he offering the bet?
TLDR Kelly bets are risk avoidant. I think Kelly bets prevent you from pouring all your money into a pascal-mugging change of winning ungodly sums of money, but Kelly bets will pay a mugger exorbitant blackmail to avoid a pascal-mugging chance losing even a realistic amount money --------- Starting with a pedantic point. None of the Pascal Mugging situations we’ve talked about are true Kelly bets. The mugger is not offering to multiply your cash bet if you win. Your winnings are saved lives, and they cannot be converted into a payroll.
But we can still translate a Pascal’s mugging into the language of a Kelly bet. A translation of the standard Pascal mugging might be: the mugger offers to googolplex-le your money[1], and you think he has a one in a trillion chance of telling the truth. A Kelly bet would say that despite these magnificent EV of the payouts, you should put only ≈a trillionth of your wealth into this bet. So in this case, the one like the original Pascal’s mugging, the one you responded to, the Kelly bet does the “right” thing and doesn’t pay the mugger.
But now suppose the Pascal mugger says “I am a jealous god. If you don’t show your belief in Me by paying Me $90,000 (90% of your wealth), I will send you and a googolplex other people to hell and take all (or all but a googolplexth) of your wealth”. And suppose you think that there’s a 1 in 1 trillion chance he’s telling the truth.
Can we translate this into a Kelly bet? Yes! (I think?) The Kelly criteria tells you how to allocate your portfolio among many assets. Normally we assume there’s a “safe” asset, a “null” asset, one where you are sure to get exactly you money back (the asset into which you put most of your portfolio when you make a small bet). But that asset is optional. We can model this Kelly bet by saying there are two assets into which you can allocate your portfolio. Asset A’s payoff is “return 10% (loses 90%) of the bet with certainty”. Asset B’s payoff is “with probability 999,999,999,999⁄1 trillion (almost 1), return your money even, but with chance 1 in 1 trillion, lose ≈everything”. There is no “safe cash” option—you must split your portfolio between assets A and B.
Here, Kelly criterion really, really hates losing ≈all your bankroll. It says to put almost everything into the safe asset A (pay the mugger), because even a 1 in 1 trillion chance of losing ≈everything isn’t worth it. Log of ≈0 (lost almost everything) is a very negative number.
Perhaps it would be useful to write exact math out.
Good point, this combines the iteratability justification for EV plus the fact that we have finite resources with which to bet. But doesn’t this break down if you are unsure how much wealth you have (particularly if the “wealth” being gambled is non-monetary, for example years of life)? Suppose the devil comes to you and says “if you take my bet you can live out your full lifespan, but there will be a 1 in 1 million chance I will send you to Hell at the end for 100 billion years. If you refuse, you will cease to exist right now.” Well, the wealth you are gambling with is years of life, but it’s unclear how many you have to gamble with. We could use whatever our expected number of years is (conditional on taking the bet) but of course, then we run back into the problem that our expectations can be dominated by tiny probabilities of extreme outcomes. This isn’t just a thought experiment since we all make gambles that may affect our lifespan, and yet we don’t know how long we would have lived by default.
Edit: realized that the devil example has the obvious flaw that as the expected default lifespan increases, so does the amount of years that you’re wagering, so you should always take the bet based on Kelly betting, but this point is more salient with less Pascalian lifespan-affecting gambles. I guess the question that remains is that the gamble is all or nothing, so what do we do if Kelly betting says we should wager 5% of our lifespan? Maybe the answer is: bet your life 5% of the time, or make gambles that will end your life with no more than 5% probability.
The uncertainties that will always be present for a real gamble make the Kelly bet rash, uncertainties about not only the numbers, but about whether the preconditions for the criterion obtain.
Because of this, Zvi recommends that Kelly is the right way to think, and you should evaluate the Kelly recommendation as best you can, but you should then bet no more than 25% to 50% of that amount. Further elaboration here.
Option 10: Kelly betting.
If you bet repeatedly on a gamble in which with probability p you win k times what you bet, and otherwise lose your bet, the fraction of your wealth to bet that maximises your growth rate is p−(1−p)/k. This implies that no matter how enormous the payoff, you should never bet more than p of your wealth. The probability you assign to unsubstantiated promises from dodgy strangers should be very small, so you can safely ignore Pascal’s Wager.
Suppose the mugger says that if you don’t give him $5, he’ll take away 99.999999999999999% of your wealth. I don’t think Kelly bets save you there? The logarithms of Kelly bets help you on the positive side but hurt you on the negative side.
Kelly bets only apply to the situation where you have a choice to gamble or not, and not gambling leaves your wealth unaffected. When the Kelly bet is negative, that means you should decline the bet.
If the mugger is capable of confiscating 99.999999999999999% of your wealth, why is he offering the bet?
TLDR Kelly bets are risk avoidant. I think Kelly bets prevent you from pouring all your money into a pascal-mugging change of winning ungodly sums of money, but Kelly bets will pay a mugger exorbitant blackmail to avoid a pascal-mugging chance losing even a realistic amount money
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Starting with a pedantic point. None of the Pascal Mugging situations we’ve talked about are true Kelly bets. The mugger is not offering to multiply your cash bet if you win. Your winnings are saved lives, and they cannot be converted into a payroll.
But we can still translate a Pascal’s mugging into the language of a Kelly bet. A translation of the standard Pascal mugging might be: the mugger offers to googolplex-le your money[1], and you think he has a one in a trillion chance of telling the truth. A Kelly bet would say that despite these magnificent EV of the payouts, you should put only ≈a trillionth of your wealth into this bet. So in this case, the one like the original Pascal’s mugging, the one you responded to, the Kelly bet does the “right” thing and doesn’t pay the mugger.
But now suppose the Pascal mugger says “I am a jealous god. If you don’t show your belief in Me by paying Me $90,000 (90% of your wealth), I will send you and a googolplex other people to hell and take all (or all but a googolplexth) of your wealth”. And suppose you think that there’s a 1 in 1 trillion chance he’s telling the truth.
Can we translate this into a Kelly bet? Yes! (I think?) The Kelly criteria tells you how to allocate your portfolio among many assets. Normally we assume there’s a “safe” asset, a “null” asset, one where you are sure to get exactly you money back (the asset into which you put most of your portfolio when you make a small bet). But that asset is optional. We can model this Kelly bet by saying there are two assets into which you can allocate your portfolio. Asset A’s payoff is “return 10% (loses 90%) of the bet with certainty”. Asset B’s payoff is “with probability 999,999,999,999⁄1 trillion (almost 1), return your money even, but with chance 1 in 1 trillion, lose ≈everything”. There is no “safe cash” option—you must split your portfolio between assets A and B.
Here, Kelly criterion really, really hates losing ≈all your bankroll. It says to put almost everything into the safe asset A (pay the mugger), because even a 1 in 1 trillion chance of losing ≈everything isn’t worth it. Log of ≈0 (lost almost everything) is a very negative number.
Perhaps it would be useful to write exact math out.
Importantly, I think for the math to work out he has to be offering a payoff proportional to your bet, not a fixed payoff?
Good point, this combines the iteratability justification for EV plus the fact that we have finite resources with which to bet. But doesn’t this break down if you are unsure how much wealth you have (particularly if the “wealth” being gambled is non-monetary, for example years of life)?
Suppose the devil comes to you and says “if you take my bet you can live out your full lifespan, but there will be a 1 in 1 million chance I will send you to Hell at the end for 100 billion years. If you refuse, you will cease to exist right now.” Well, the wealth you are gambling with is years of life, but it’s unclear how many you have to gamble with.We could use whatever our expected number of years is (conditional on taking the bet) but of course, then we run back into the problem that our expectations can be dominated by tiny probabilities of extreme outcomes. This isn’t just a thought experiment since we all make gambles that may affect our lifespan, and yet we don’t know how long we would have lived by default.Edit: realized that the devil example has the obvious flaw that as the expected default lifespan increases, so does the amount of years that you’re wagering, so you should always take the bet based on Kelly betting, but this point is more salient with less Pascalian lifespan-affecting gambles. I guess the question that remains is that the gamble is all or nothing, so what do we do if Kelly betting says we should wager 5% of our lifespan? Maybe the answer is: bet your life 5% of the time, or make gambles that will end your life with no more than 5% probability.
The uncertainties that will always be present for a real gamble make the Kelly bet rash, uncertainties about not only the numbers, but about whether the preconditions for the criterion obtain.
Because of this, Zvi recommends that Kelly is the right way to think, and you should evaluate the Kelly recommendation as best you can, but you should then bet no more than 25% to 50% of that amount. Further elaboration here.