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8) One reasonable utility function here is U = log(W): approximating your happiness as logarithmic in your wealth. That would mean going from $10k to $100k is worth about as much as going from $100k to $1m, which feels…. reasonable? (this is what the Kelly Criteria assumes)

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9) So, if you have $100k, Kelly would suggest you risk half of it ($50k). This is a lot! But also 75% odds are good.

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10) What about a wackier bet? How about you only win 10% of the time, but if you do you get paid out 10,000x your bet size? (For now, let’s assume you only get to do this bet once.)

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11) Kelly suggests you only bet $10k: you’ll almost certainly lose. And if you kept doing this much more than $10k at a time, you’d probably blow out. That this bet is great expected value; you win 1,000x your bet size, way better than the first one! It’s just very risky.

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12) In many cases I think $10k is a reasonable bet. But I, personally, would do more. I’d probably do more like $50k. Why? Because ultimately my utility function isn’t really logarithmic. It’s closer to linear.

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Matt Hollerbach

@breakingthemark

Replying to

and

Are you saying you would bet more if you only ha $100,000? or you would bet more because you do have more than $100,000 to start?

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@breakingthemark

more if I only had $100k

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Matt Hollerbach

@breakingthemark

Replying to

and

You do realize the reason you shouldn't bet more than $10,000 is the same reason you gave for not betting it all in #22? You need funding to provide value with future opportunities.

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@breakingthemark

eh they're to different degrees; losing 10% of your wealth is a lot less likely to cripple your opportunity to do future things than losing > 50%. life isn't exactly a game of iterated "exactly the same bet". And in practice it matters a lot roughly how much $ you'll need!

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Matt Hollerbach

@breakingthemark

Replying to

and

The Kelly criteria isn't about iterating the exact same bet either. It's about maximizing your long term growth rate in games of random chance. The bet can change each time and you still shouldn't bet above the Kelly value.

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@breakingthemark

long term *geometric* growth rate, not long term growth rate all-in maximizes long term growth rate

4:37 PM · Dec 14, 2020Twitter Web App

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@breakingthemark

and it only works if you face nothing in your life but a series of bets that are both (a) uncorrelated and (b) each let you bet up to your entire wealth without any decreasing returns

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@breakingthemark

which isn't to say Kelly isn't cool! It is cool! but cool is different from "absolutely the right thing to do in all circumstances, the math proves it". there's no such thing as ^ when it comes to betting, at least not in general, without knowing more context.

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Matt Hollerbach

@breakingthemark

Replying to

and

Absolutely not. All long term growth rates are geometric. There is no other kind.

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@breakingthemark

uhhhh how are you defining growth rate? "Growth rates refer to the percentage change of a specific variable within a specific time period" from the first link: investopedia.com/terms/g/growth I'd interpret that as lim t --> inf EV[ W_{t+1}/W_t ]

investopedia.com

Growth Rates: Formula, How to Calculate, and Definition

Growth rates are the percent change of a variable over time. It can be applied to GDP, corporate revenue, or an investment portfolio. Here's how to calculate growth rates.

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@breakingthemark

All-in has an absorbing barrier. Kelly doesn't, assuming no min bet, and guarantees geometric growth rate at the limit of sufficient samples (assuming you get there in time) provided the process is stationary in hit rate and win/loss. <- does not apply to financial markets.

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@breakingthemark

What would you apply Kelly?