A plan for Pascal's mugging?

yttrium

The idea is to compare not the results of actions, but the results of decision algorithms. The question that the agent should ask itself is thus:

"Suppose everyone¹ who runs the same thinking procedure like me uses decision algorithm X. What utility would I get at the 50th percentile (not: what expected utility should I get), after my life is finished?"
Then, he should of course look for the X that maximizes this value.

Now, if you formulate a turing-complete "decision algorithm", this heads into an infinite loop. But suppose that "decision algorithm" is defined as a huge table for lots of different possible situations, and the appropriate outputs.

Let's see what results such a thing should give:

If the agent has the possibility to play a gamble, and the probabilities involved are not small, and he expects to be allowed to play many gambles like this in the future, he should decide exactly as if he was maximizing expected utility: If he has made many decisions like this, he will get a positive utility difference in the 50th percentile if and only if his expected utility from playing the gamble is positive.
However, if Pascal's mugger comes along, he will decline: The complete probability of living in a universe where people like this mugger ought to be taken seriously is small. In the probability distribution over expected utility at the end of the agent's lifetime, the possibility of getting tortured will manifest itself only very slightly at the 50th percentile - much less than the possibility of losing 5 Dollars.

The reason why humans will intuitively decline to give money to the mugger might be similar: They imagine not the expected utility with both decisions, but the typical outcome of giving the mugger some money, versus declining to.

¹I say this to make agents of the same type cooperate in prisoner-like dilemmas.

The idea is to compare not the results of actions, but the results of decision algorithms. The question that the agent should ask itself is thus:

Let's see what results such a thing should give:

If the agent has the possibility to play a gamble, and the probabilities involved are not small, and he expects to be allowed to play many gambles like this in the future, he should decide exactly as if he was maximizing expected utility: If he has made many decisions like this, he will get a positive utility difference in the 50th percentile if and only if his expected utility from playing the gamble is positive.
However, if Pascal's mugger comes along, he will decline: The complete probability of living in a universe where people like this mugger ought to be taken seriously is small. In the probability distribution over expected utility at the end of the agent's lifetime, the possibility of getting tortured will manifest itself only very slightly at the 50th percentile - much less than the possibility of losing 5 Dollars.

¹I say this to make agents of the same type cooperate in prisoner-like dilemmas.

Yes that is a good insight. I'll rephrase it to perhaps make it clear to a somewhat different set of people. If your strategy is to have a good median outcome of your life, you will still get to bet on longshots with high payoffs, as long as you expect to be offered a lot of those bets. The fewer bets you expect to be offered of a certain type, the more likely winning must be for you to take it, even if the "expected" pay out on these is very high.

A quantification of this concept in somewhat simple cases was done by Jim Kelly and is called the Kelly Criterion. Kelly asked a question: given you have finite wealth, how do you decide how much to bet on a given offered bet in order to maximize the rate at whcih your expected wealth grows? Kelly's criterion, if followed, also has the side-effect of insuring you never go completely broke, but in a world of minimum bet sizes, you might go broke enough to not be allowed to play anymore.

Of course, all betting strategies, where you are betting against other presumed rational actors, require you to be smarter, or at least more correct thant the people you are betting against, in order to allow you to win. In Kelly's calculation, the size of your bet depends on both the offered odds and the "true" odds. So how do you determine the true odds? Well that is left as an exercise for the reader!

And so it goes with Pascal's muggings. As far as my study has taken me, I know of no way to reliably estimate whether the outcomes in offered in Pascal's muggings are one in a million, one in a google, one in a googleplex, or one in 3^^^3. And yet the "correct" amount to bet using the Kelly criterion will vary by as big a factor as those probability estimates vary one from the other.

There is also the result that well-known cognitive biases will cause you to get infinitesimal probabilities wrong by many orders of magnitude, without properly estimating your probable error on them. For any given problem, there is some probability estimate below which all further attempts to refine the estimate are in the noise: the probability is "essentially zero." But all the bang in constantly revisiting these scenarios comes from the human biases that allow us to think that because we can state a number like 1 in a million or 1 in a google or 1 in 3^^^3 that we must be able to use it meaningfully in some probabilistic calculation.

If you are of the bent that hypotheses such as the utility of small probabilities should be empirically checked before you start believing the results of these calculations, it may take a few lifetimes of the universe (or perhaps a google lifetimes of the universe) before you have enough evidence to determine whether a calculation involving a number like 1 in a google means anything at all.

google

Googol. Likewise, googolplex.

4CarlShulman14y

The Kelly criterion doesn't maximize expected wealth, it maximizes expected log wealth, as the article you linked mentions: [...] Suppose that I can make n bets, each time wagering any proportion of my bankroll that I choose and then getting three times the wagered amount if a fair coin comes out Heads, and losing the wager on Tails. Expected wealth is maximized if I always bet the entire bankroll, with an expected wealth of (initial bankroll)(3^n)(the probability of all Heads=2^-n). The Kelly criterion trades off from that maximum expected wealth in favor of log wealth. A utility function that goes with log wealth values gains less, but it also values losses much more, with insane implications at the extremes. With log utility, multiplying wealth by a 1,000,000 has the same marginal utility whatever your wealth, and dividing wealth by 1,000,000 has the negative of that utility. Consider these two gambles: Gamble 1) Wealth of $1 with certainty. Gamble 2) Wealth of $0.00000001 with 50% probability, wealth of $1,000,000 with 50% probability. Log utility would favor $1, but for humans Gamble 2 is clearly better; there is very little difference for us between total wealth levels of $1 and a millionth of a cent. Worse, consider these gambles: Gamble 3) Wealth of $0.000000000000000000000000001 with certainty. Gamble 4) Wealth of $1,000,000,000 with probability (1-1/3^^^3) and wealth of $0 with probability 1/3^^^3 Log utility favors Gamble 3, since it assigns $0 wealth infinite negative utility, and will sacrifice any finite gain to avoid it. But for humans Gamble 4 is vastly better, and a 1/3^^^3 chance of bankruptcty is negligibly worse than wealth of $1. Every day humans drive to engage in leisure activities, eat pleasant but not maximally healthful foods, and otherwise accept small, go white-water rafting, and otherwise accept small (1 in 1,000,000, not 1 in 3^^^3) probabilities of death for local pleasure and consumption. This is not my utility function. I

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