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.
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?