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.
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?
1Option 10: Kelly betting.
If you bet repeatedly on a gamble in which with probability you win times what you bet, and otherwise lose your bet, the fraction of your wealth to bet that maximises your growth rate is . This implies that no matter how enormous the payoff, you should never bet more than 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.
McGee's argument is akin to the following piece of mathematics, separate from utility theory. What is ?
Clearly, it is equal to , which equals , which is .
Clearly, it is equal to , which equals , which is .[1]
We do not respond to this paradox by supposing there must be a maximum and a minimum integer. We cannot, because mathematics, even more than physics, is a coherent whole in which we cannot change one thing without changing everything. We must instead accept the fact that not all sequences converge.
It may seem like one can answer McGee's paradox by saying "oh well, I guess my utility function's bounded", but coherence problems will still arise, which I was alluding to in asking how you might find the bounds and what sort of magnitude you imagine for them. What happens in someone's quest for maximising utility when they begin to succeed? To approach the maximum possible utility they could ever have? One would have to find oneself caring less and less about every new thing, until one's future is the torpor of "utility death", jaded beyond all caring. This is indeed a standard trope in fiction, but if death is a solvable problem, I would expect utility death to be so also.
By choosing different sequences, one can produce examples where either or both of the limits of the regrouped sequences is finite. ↩︎
If your utillity is finite, how would you determine its bounds?
If the bound is a practical figure, and not up in the ↑↑↑↑-sphere, that looks rather like scope insensitivity. If it is up there, it is indistinguishable from unbounded in the practical realm.
Necroposting, but for the knowledge of posterity, this was originally posted by Patrick Nielsen Hayden to Usenet on 9 August 2000, in rec.arts.sf.fandom.
What difference does it make whether I die in 60 years or in 10,000? In the end, I’ll still be dead.
Does it make a difference whether you die in 60 years or in 60 days? If it does, at what point do you despair of the value of further years, and why?
My own preferred lifespan is "more". (Although despite that, I also have no plans to sign up for cryonics.)
You have half a dozen job offers. Which one do you take? The one with the biggest potential upside? The lowest potential downside? Expected value? Expected log value? Depends on your attitude to risk.
You have half a dozen schemes for how society should be organised. From behind the veil of ignorance, are you willing to risk the chance of a poor position for the chance of a better one? Depends on your attitude to risk. Even behind the veil, different people may rate the same proposal differently.
I must admit to not having read Rawls, only read about his veil of ignorance, but I am sceptical of the very idea of designing a society, other than for writing fiction in. Any designed civilisation will drift immediately on being set in motion, as people act as they see fit in the circumstances they are in. Planned societies have a poor track record. The static societies of the past, where once a cobbler, always a cobbler, are generally not regarded as something we would want to bring back.
As a pure thought experiment, one can imagine whatever one likes, but would this bridge, or that distant rocky tower, stand up?
Artist credit: Rodney Matthews. I love his art, and that of the very similar Roger Dean, but I appreciate it as fantasy, and am a little sad that these things could never be built. Notice that the suspension bridge is only suspended on the nearer half.
I know where it comes from, and it has always seemed arbitrary to me. Why Rawls's maximin rather than average, or some other weighting? In everyday life, people make decisions for themselves and for others all the time without being certain that they will turn out well. Why not the same for ideas for how society as a whole should be organised?
You can just not do things.