Consider an example of a 10-people world ( not counting the madman). What happens there? What happens in a 10^n-people world? what happens in the limit of n->infinity?

An earlier analysis (previous blog post) looks at the case of a finite world population and shows that, using either the "proximate risk" or "proportion murdered" approach, you still get P(you die | kidnapped) = 1/36, because the fact that you are kidnapped is strong evidence that the murderer never murders at all due to running out of people to kidnapped.

Go to a casino. Bet $1 on something with a 50% chance of winning. If you win, you have won $1; try again. If you lose, double your bet size (which means that winning will leave you having won $1 total over the sequence of doubled bets) and repeat.

One argument says that in the long run, you will come out a winner, because every bet you make is part of a sequence and at the end of that sequence, you are $1 richer. Another argument says that in the long run, you will only break even, because each bet has a 50% chance of winning and a 50% chance of losing the same amount of money.

Of course, the answer is that you can't increase your bet infinitely, and when you stop increasing your bet, the statistical loss at the point where you stop increasing your bet exactly makes up for the statistical win all the other times you finished the sequence and won $1.

Furthermore, if you *could* increase your bet infinitely, this problem wouldn't happen, *but* if you could increase your bet infinitely, the expectation *isn't well defined*, because you are trying to compute it for a non-converging infinite series.

All this problem is is the same idea applied to probability of death instead of expectation of win. If the madman ever runs out of people, the overall probability depends exactly on what the madman does when he runs out of people (since it's not as well defined as it is for bets). If the madman never runs out of people, the probability involves a non-converging infinite series and so is not well defined.

If this is a metaphor for extinction, then when the madman runs out of people, he keeps rolling the dice on the remaining people until it eventually comes up snake eyes, in which case the chance of extinction is 100%. On the other hand, they can last arbitrarily long given an arbitrarily small probability of extinction.

An earlier version of my analysis (the previous blog post) looked at the case of finite n and found, as you suggest, that the possibility of running out of people to kidnap is an important consideration. You can choose the number of batches n to be so large that it is virtually certain a priori that the madman will eventually murder:

P(eventually murders) = 1 - epsilon for some small epsilon

However, it turns out that conditioning on the fact that you are kidnapped changes the probability dramatically:

P(eventually murders | you are kidnapped) = about 10/9 * 1/36

The reason for this is that there are about 9 times as many people in the final batch as in all other batches combined, so the fact that you are kidnapped is strong evidence that the madman is on his last batch of potential victims.

## The Dice Room, Human Extinction, and Consistency of Bayesian Probability Theory

I'm sure that many of you here have read *Quantum Computing Since Democritus*. In the chapter on the anthropic principle the author presents the Dice Room scenario as a metaphor for human extinction. The Dice Room scenario is this:

1. You are in a world with a very, very large population (potentially unbounded.)

2. There is a madman who kidnaps 10 people and puts them in a room.

3. The madman rolls two dice. If they come up snake eyes (both ones) then he murders everyone.

4. Otherwise he releases everyone, then goes out and kidnaps 10 times as many people as before, and returns to step 3.

The question is this: if you are one of the people kidnapped at some point, what is your probability of dying? Assume you don't know how many rounds of kidnappings have preceded yours.

As a metaphor for human extinction, think of the population of this world as being all humans who ever have or ever may live, each batch of kidnap victims as a generation of humanity, and rolling snake eyes as an extinction event.

The book gives two arguments, which are both purported to be examples of Bayesian reasoning:

1. The "proximate risk" argument says that your probability of dying is just the prior probability that the madman rolls snake eyes for your batch of kidnap victims -- 1/36.

2. The "proportion murdered" argument says that about 9/10 of all people who ever go into the Dice Room die, so your probability of dying is about 9/10.

Obviously this is a problem. Different decompositions of a problem should give the same answer, as long as they're based on the same information.

I claim that the "proportion murdered" argument is wrong. Here's why. Let pi(t) be the prior probability that you are in batch t of kidnap victims. The proportion murdered argument relies on the property that pi(t) increases exponentially with t: pi(t+1) = 10 * pi(t). If the madman murders at step t, then your probability of being in batch t is

pi(t) / SUM(u: 1 <= u <= t: pi(u))

and, if pi(u+1) = 10 * pi(u) for all u < t, then this does indeed work out to about 9/10. But the values pi(t) must sum to 1; thus they cannot increase indefinitely, and in fact it must be that pi(t) -> 0 as t -> infinity. This is where the "proportion murdered" argument falls apart.

For a more detailed analysis, take a look at

http://bayesium.com/doomsday-and-the-dice-room-murders/

This forum has a lot of very smart people who would be well-qualified to comment on that analysis, and I would appreciate hearing your opinions.

I'm not sure that "jack of all trades" is a helpful identity, given the known benefits of economic specialization. Remember the origin of that term: "Jack of all trades, and master of none." It's often more useful to be really, really good at one thing and trade for what you need in other areas.

It can often be useful to have a "T-shaped" expertise, though: some level of familiarity with a wide variety of topics, and deep expertise in one area. The cross bar of the T helps you when your existing expertise and skills are not enough -- you know enough to find someone who can help you, or to know what new skills / knowledge you need to pick up. (Or, perhaps more importantly, you know what you don't know.)

## Predicting Organizational Behavior

Can someone recommend a good introduction to the topic of organizational behavior? My interest is in *descriptive* rather than *prescriptive* models -- I'm interested in what is known about predicting the behavior of organizations, rather than guidance on what they should do to achieve their goals. This kind of prediction strikes me as something of substantial practical use, especially to business; being able to work out the plausible range of future actions of city hall, the state legislature, Congress, regulatory agencies, competitors in the marketplace, large customers, and important suppliers would be a valuable capability in making one's own plans.

Do you know why this book is on the MIRI course list? What is the connection to Friendly AI?

I've certainly found this to be a useful strategy when dealing with complicated problems in software development. Sometimes a problem is just too big, and I can't quite see how all the pieces need to fit together. If I allow myself to leave some important design problems unresolved while I work on the parts that I do understand well enough to write, I often find that the other pieces then fall into place straightforwardly.

"Absence of evidence isn't evidence of absence" is such a ubiquitous cached thought in rationalist communities (that I've been involved with) that its antithesis was probably the most important thing I learned from Bayesianism.

I find it interesting that Sir Arthur Conan Doyle, the author of the Sherlock Holmes stories, seems to have understood this concept. In his story "Silver Blaze" he has the following conversation between Holmes and a Scotland Yard detective:

Gregory (Scotland Yard detective): "Is there any other point to which you would wish to draw my attention?"

Holmes: "To the curious incident of the dog in the night-time."

Gregory: "The dog did nothing in the night-time."

Holmes: "That was the curious incident."

There are many publicly available data sets and plenty of opportunities to mine data online, yet we see little if any original analysis based on them here. We either don't have norms encouraging this or we don't have enough people comfortable with statistics doing so.

In my case, I'm comfortable with statistics but don't know where to find the data for questions that interest me. The fact that much research is nearly inaccessible if you're not affiliated with a university or other large institution is also a problem.

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The madman murders only almost always. It is possible but vanishingly unlikely that he just never rolls snake eyes (or he runs outside of the total population with the growth so he can't get a full patch). Option 1 doesn't care whether the doom ultimately happens while option 2 assumes that the doom will happen.

The proper enlish version of option two would be "Given that the dice came up snake eyes and that you were kidnapped at some point what is the probabilty that it did so while you were kidnapped?". Notice also that this is independent off what dice readings result in doom. That is if the world is only saved on snake eyes the chance is still "only" 9/10.

Note that

P(you are in batch t | murders batch t & you are kidnapped)

cannotbe 9/10 for all t; in fact, this probabilitymustgo to 0 in the limit as t -> infinity, regardless of what prior you use.