# A Martian makes a bet

I’m a Martian.

As a Martian, we’ve long ago licked this “singularity” thing, so I’m logically omniscient, but I’m also bored. I decide to invade Earth, mostly for the lulz. I land my flying saucer, pull out my ray gun, and prepare to get invading.

I come across two earthlings in heated debate. They turn to me and look excited.

“Ah, excellent. A Martian! You can serve as an impartial adjudicator to our debate.”

I am, naturally perplexed.

“Uh, you do know I’m here to invade you right? I have this ray gun?”

“Yes yes. It’s very impressive, but this is much more important.”

“More important than the fact that I’m planning to enslave you all and force you to obey my every whim?”

“Sure, sure. That’s just politics. We don’t care about that. We’re too concerned with matters of real import. We want to know what you think about the sportsball tournament“.

“The what?”

“Ah ha! Excellent. A truly impartial adviser! Tell me, what do you think the chances of the Las Venturas Bandits winning the sportsball tournament are?”

Obviously I have no idea about sportsball. I’m a Martian! We don’t play such things. Still, from a position of complete ignorance I must presume that each team is equally likely to win, so I need only find out how many teams there are to determine my probabilities. So I ask them.

“Oh no. We couldn’t tell you that. If we give you information about sportsball you’d no longer be an impartial judge.”

Their reasoning is odd, but I shall for now adhere to their quaint local customs and apply the obvious prior of $$\frac{1}{n(n+1)}$$ on the number of sportsball teams there could be (I leave it as an exercise to the reader why this would be obvious. If you’re logically omniscient like me you already know, if not it should be a simple calculation). From there, the answer is a straightforward calculation.

“Well, naturally the probability is $$\frac{\pi^2}{6} – 1$$, or about $$0.645$$ if you want to be all numerical about it.”

“Gosh, so confident! But what, then, is the chance of the Beaneaters winning?”

The answer is equally immediate: About $$0.290$$.

“Interesting, interesting. But what, pray tell, is the chance of the Jersey Boomers winning?”

“0.185, naturally”

At this point the quieter of the two is starting to look slightly confused, but the loud one keeps talking.

“Truly Martians are very insightful. Can you tell me,  how likely do you think it is that the Swedish Meatballs will win?”

“Oh, about 0.135 I would say”

At this point the one who has been quiet can no longer contain himself.

“I say! These probabilities you’re giving us don’t make any sense. I’ve been counting, and they add up to much more than one! You’re at 1.25 already!”

“Well, naturally. I had to update my probabilities as you gave me new information. At this point the probability of any individual one of the teams you’ve mentioned winning is about 0.135, with about a 0.459 chance that some other team you haven’t mentioned yet wins”.

“But we didn’t give you any information! The whole point was you were supposed to be impartial!”

“Of course you did. You gave me a lower bound on how many teams there were.”

At this point they both turn bright red and start shouting again at each other about how they’ve ruined everything. Thankfully, this gives me the insight I needed to calculate a truly accurate probability for who will win this game.

“Excuse me?”, I interrupt. They both turn to me.

“I’ve now understood the problem more thoroughly and can tell you the chances of each of your teams winning”

“Really? Do tell”.

They both look very eager to have the debate resolved.

So I shoot them.

And then I conquer the planet and ban the whole silly game of sportsball.

The answer is of course, zero. When you make me play silly guessing games, everyone loses.

What’s going on here?

Well, the Martian’s prior doesn’t matter really. If you must know, it’s based on a hierarchical Bayesian model where the number of teams is geometric, but the geometric parameter is also unknown and is distributed uniformly on $$[0, 1]$$. Really this example would work with almost any reasonable prior though (say, any proper prior which assigns non-zero probability to every positive integer). I just wanted something concrete so I could put numbers to it.

So we have a prior on the number of teams. Call this number of teams $$T$$. The probability of a given team winning given $$T = n$$ is $$\frac{1}{n}$$, so the probability of a single named team winning is $$E(\frac{1}{T})$$.

But we don’t have a single named team. We’ve been asked a series of questions, each of which tells us the existence of a named team. Therefore our answer to the nth question is not $$E(\frac{1}{T})$$ but instead $$E(\frac{1}{T} | T \geq n)$$, because we know at that point that there must be at least $$n$$ named teams.

For our particular example this is always decreasing, but even in general it must eventually head towards zero because $$E(\frac{1}{T} | T \geq n) \leq \frac{1}{n}$$. So as time progresses and we become aware of the existence of more and more teams your probability that you assign to any given one of them must go down.

This is a neat example because it demonstrates how if you insist on Bayesian rather than Knightian uncertainty here, the mere asking of questions can itself be a source of information which causes you to update your probabilities. As we are asked questions we are given an idea of the scope of possibilities available, and this forces us to adjust our updates. This happens even though we are logically omniscient – the questions are not forcing us to consider new possibilities, they are genuinely providing us with new information about the world.

(It is of course possible that the questioners are making names up – say by some ludicrous procedure like just picking names at random from Wikipedia’s list of fictional sports teams – but you could include a term in your probability distribution for that if you wished. I didn’t because that would have made the example harder to follow)

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