Tuesday, November 24, 2015

Dutch Books and infinite sequences of coin tosses

Suppose we have an infinite sequence of independent and fair coins. A betting portfolio is a finite list of subsets of the space of outcomes (heads-tails sequences) together with a payoff for each subset. Assume:

  1. Permutation: If a rational agent would be happy to pay x for a betting portfolio, and A is one of the subsets in the betting portfolio, then she would also be happy to pay x for a betting portfolio that is exactly the same but with A replaced by A*, where A* is isomorphic to A under a permutation of the coins.
  2. Equivalence: A rational agent who is happy to pay x for one betting scenario, will be willing to accept an equivalent betting scenario---one that is certain to give the same payoff for each outcome---for the same price.
  3. Great Deal: A rational agent will be happy to pay $1.00 for a betting scenario where she wins $1.25 as long as the outcome is not all-heads or all-tails.
Leveraging the Axiom of Choice and using the methods of the Banach-Tarski Paradox, one can then find two betting portfolios that the agent would be happy to accept for $1.00 each such that it is certain that if she accepts them both, she will win at most $1.25; hence she will accept a sure loss of at least $0.75. For details of a closely related result, see Chapter 6 in Infinity, Causation and Paradox (draft temporarily here).

So what to do? I think one should accept Causal Finitism, the doctrine that causal antecedents are always finite. Given Causal Finitism, one can't have a real betting scenario based an infinite number of coin tosses. Moreover, the only known way to operationalize the use of the Axiom of Choice in the proof in a betting scenario also involves infinite causal dependencies.


Angra Mainyu said...


I have a few questions, but I'm afraid I can't read the whole paper at the moment, so maybe the paper answers it.

Q1. If I'm getting this right, a betting portfolio is like a finite list of sets of sequences of ones and zeros, so it's like a finite list of subsets of the interval [0,1]. Point 1 roughly says that a rational agent would make no difference between portfolios in which a set on the list is replaced by a set of the same cardinality. Is that correct?
So, for example, if one portfolio P1 is a list composed of the first intervals (1/2^(m+1),1/2^m), for m between 0 and 1 billion, one can replace every single one of the intervals but the first with the first interval, yielding P2 which only has the first interval on the list, and those are rationally equivalent.
Alternative: One might replace each of the intervals in P1 with a Cantor set of (Lebesgue) measure zero contained in that interval, yielding P3, which is a list whose union has measure zero, and that is still the same rationally.
Am I getting this right?

Q2: Normally, in the case of a fair coin and independent events, one is rational in assigning 1/2 to each outcome. But that's due to limited information: an agent with more information about the coin, the surrounding environment, etc., might well be rational in assigning a different probability. But it's not clear what probability you're assigning in the infinite case.

Q3: Why exclude all tails or all heads in point 3?
After all, that would still be a great deal, it seems to me, as long as the win is certain.

Alexander R Pruss said...

Ad 1: Point 1, rather, says that it makes no difference that a portfolio is defined in terms of coins number 1,2,3,4,5,6,... rather than in terms of coins number 2,1,4,3,6,5,.... For instance, the portfolio could contain a set of all sequences with the property that all the even-numbered coins are heads. Applying the above permutation generates a portfolio with a set of all sequences with the property that all the odd-numbered coins are heads.

Likewise, the probability that there are exactly three heads among coins numbered 1,2,3,4 equals the probability that there are exactly three heads among coins numbered 3,4,5,6.

The Permutation property is a strengthening (by allowing permutations that affect infinitely many coins at a time) of de Finetti's exchangeability condition.

Ad 2: I am assuming that all the indeterministic coins are exactly alike.

Ad 3: The paradox requires that there be a non-zero probability that there are two coins with different outcomes. Suppose the coins are all linked (e.g., by a quantum entanglement) such that they are guaranteed to all come out the same way. Then Permutation holds trivially, and a Dutch Book can be avoided.

Angra Mainyu said...

Thanks for the clarification.

Ad 1: My bad, I misread that one.

Ad 2: Is the assumption that the coins are indeterministic required?
If so, the rational epistemic probabilistic assessment would seem to still depend on how the specific indeterministic universe works, so on the information available to the agent. I'm still not sure how you're picking your assignment in the premises, but if it leads to a paradox, it seems that's not a proper assignment after all.

Ad 3: I was asking for the justification for the exclusion. In other words, if the exclusion is needed for deriving the paradox, it might be objected that maybe it's not justified to say that a rational agent will make such exclusion.

Angra Mainyu said...

Alex, with respect to the solution you propose - namely, causal finitism -, it seems to me that the reasons you're giving in support of it may be used to derive further results, such as:

a. There aren't infinitely many galaxies.

This case seems relevantly similar, because:
a.1. In both cases, there are infinitely many concrete objects where stuff happens so to speak, so a limited agent might want to make a probabilistic assessment about some events that happened in them, like coin tosses or some other stuff.
a.2. While it might be suggested that perhaps no non-omniscient agent could know what happened in infinitely many places, if that objection works, it seems it may just as well work for the case of infinitely many coin tosses.

b. There aren't infinitely many parallel universes.

For similar reasons.

c. There aren't infinitely many agents. Also, that's for similar reasons.

I suppose you might think a relevant difference would be that in those cases, the coin tosses aren't causally connected, and you said that "the only known way to operationalize the use of the Axiom of Choice in the proof in a betting scenario also involves infinite causal dependencies.", but given that there are still infinite sequences and agents making probabilistic assessments, it's hard to see how causation would be the problem (still, if you think it is, I would like to ask why).

That aside, we can construct scenarios that do involve causal dependencies, and reckon that:

d. It's not the case that there are infinitely many future years and also that time is tenseless. [even on tensed time, I think, but I'll leave that one for later]
The reasons are similar: if time is tenseless and there are infinitely many future years, then it seems that two agents who live forever (which are possible) could decide to toss infinitely many coins, so there are possible worlds in which those sequences exist (at least if time is necessarily tenseless, or at least also tenseless in those worlds, but why not?).

e. Let's say that two agents, Gabriel and Uriel (maybe Morriston's angels) decide to toss the coins, and bet. Let's further say that a being who knows with certainty what will happen in the future (say, God) exists. Even if time is tensed, the fact remains that God knows the whole future, and knows which sequence will be the winner. So, how would Gabriel and Uriel bet? It might be true that God will not tell them which sequence will win, but that's not the issue: rather, the issue is that there is something to bet on. The bets do not have to involve actual money; it could be a game played just for fun by Gabriel and Uriel, and the "money" is imaginary. The relevant issue is that there is something to bet on, or just to assign probability - namely, what infinite sequence God knows will obtain.
If there are any moral objections (I don't see why, but just in case), we may replace Gabriel and Uriel with, say, Lucifer and Azazel.

So, a consequence here is that no being knows the future with certainty.

Do you think one should also accept those consequences, or is there a relevant difference?

Alexander R Pruss said...

I agree that there are far-reaching consequences for what is possible. But a-c aren't among them, since there are no infinite causal entanglements there.

Imaginary money perhaps isn't enough to generate a rationality paradox.

"a consequence here is that no being knows the future with certainty": Not quite. Rather, no being can act on the basis of knowledge of an infinite number of events, or something like that.

Angra Mainyu said...

I would argue in support of the parallel between infinite causal links and infinitely many galaxies as follows:

Either the paradox that you derive in the case of the infinite consecutive coin tosses can also be derived in the case of infinitely many galaxies (one coin toss per galaxy), or it cannot.
If the paradox can be derived in the case of the galaxies, then the same rationale supporting causal finitism supports finitism about galaxies just as much. A similar argument would apply to infinitely many planets, infinitely many people, etc.
On the other hand, if the paradox cannot be derived in the case of the galaxies, then there is the following solution in the case of infinite consecutive causal dependencies: a rational agent may simply consider the case in which the coins are tossed in separate galaxies, and bet in the same fashion. So, for example, let's say that there is a betting portfolio. The agent simply assumes that the coins have been tossed in different galaxies, and bets. No paradox there.

That aside, I don't think I agree with your assessment about knowledge of the future, because if there is such being with such knowledge (say, God), then a rational agent might assign probability to events like "God knows that the winning bet is such-and-such", and once you have a rational way of assigning probabilities, it seems you have a rational way of betting - unless you're making a distinction between a rational way of betting and a rational way of assigning probability?

If so, let's say for the sake of the argument that the conclusion is only that no being can act on the basis of knowledge of an infinite number of events. Then, why not draw the same conclusion from your paradox? In other words, instead of causal finitism, a conclusion might be that no being can act on the basis in question.

Alexander R Pruss said...

I do distinguish betting from probability assignments. That said, it's rather hard for an agent to make reference to the sets in question. It might be doable with the help of an omnipotent being, but even that isn't completely clear to me.
As to the question of why not go for the weaker conclusion, that's because the weaker conclusion doesn't resolve all the other paradoxes I will adduce in the book. :-)

Angra Mainyu said...

I was going by the arguments given in the OP and your replies; as I mentioned, I'm afraid I don't have time to read the whole book and study the arguments in detail :-), so I'll have to leave that case aside.

But regarding the cases of infinitely galaxies, parallel universes, and people, do you think there is a way out of the parallel argument I made above? (i.e., the one involving assuming it's one coin toss per galaxy).

Alexander R Pruss said...

To get the problem, the payoff would have to depend on what happened in infinitely many galaxies. And that violates causal finitism.