One argument for the common prior assumption (an assumption which underpins the Aumann Agreement Theorem, and is closely related to modest-epistemology arguments) is that a bookie can Dutch Book any group of agents who do not have a common prior, via performing arbitrage on their various beliefs.
There exists an agent H that believes with certainty that all coins only ever land on heads as its prior.
There also exists an agent T that is equally confident in tails. (Exists in the mathematical sense that there is some pattern of code that would consist of such an agent, not in the sense that these agents have been built)
Lets say that H and T, by construction will always take any bet that they would profit from if all involved coins come up heads or tails respectively.
Consider a bet on a single coin flip that costs 2 if a coin comes up heads, and pays X if the coin lands on tails. (X>>2) If you would be prepared to take that bet for sufficiently large X, then a bookie can offer you this bet, and offer H a bet that wins 1 if the coin lands heads, and looses X+1 if the coin lands tails.H will take this bet. So the bookie has Dutch booked the pair of you.
If you ever bet at all and your betting decisions don’t depend on who you are sitting next to, then you can be part of a group that is Dutch booked.
If you want to avoid ever betting as part of a group that is being Dutch booked, then if you are in the presense of H and T, you can’t bet at all, even about things that have nothing to do with coin flips.
If you bet 5 against 5 that gravity won’t suddenly disappear, then the bookie can Dutch book H and T for 100, and the 3 of you have been Dutch booked for at least 95 as a group.
If you have some reason to suspect that a mind isn’t stupid, maybe you know that they won a lot of money in a prediction market, maybe you know that they were selected by evolution, ect then you have reason to take what the mind says seriously. If you have strong reasons to think that Alice is a nearly perfect reasoner, then getting Dutch booked when you are grouped with Alice indicates you are probably making a mistake.
Why should I care if I together with other people are jointly getting Dutch booked, if I myself am not? If my neighbour loses money but I do not, I do not care that “we” lost money, if his affairs have no connection with mine.
First off, I am quite sympathetic, and by no means would argue that the Dutch Book for the common prior assumption is as convincing as other Dutch Book Arguments.
However, it’s still intriguing.
If you and your neighbor are game-theoretic partners who sometimes cooperate in prisoner’s-dilemma like situations, then you might consider this kind of joint Dutch Book concerning. A coalition which does not manage to jointly coordinate to act as one agent is a weaker coalition.
If the collision is able to negotiate bets internally, then for any instance of the coalition getting Dutch booked, they can just agree to run that bet without the bookie, and split the bookies cut.
If everyone else changes to my prior, that’s great. But if I change from my prior to their prior, I am just (from the point of view of someone with my prior, which obviously includes myself) making myself vulnerable to be beaten in ordinary betting by other agents that have my prior.
Let’s say your prior is P and mine is Q. I take your argument to be that P always prefers bets made according to P (bets made according to Q are at best just as good). But this is only true if P thinks P knows better than Q.
It’s perfectly possible for P to think Q knows better. For example, P might think Q just knows all the facts. Then it must be that P doesn’t know Q (or else P would also know all the facts.) But given the opportunity to learn Q, P would prefer to do so; whereupon, the updated P would be equal to Q.
Similar things can happen in less extreme circumstances, where Q is merely expected to know some things that P doesn’t. P could still prefer to switch entirely over to Q’s beliefs, because they have a higher expected value. It’s also possible that P trusts Q only to an extent, so P moves closer to Q but does not move all the way. This can even be true in the Aumann agreement setting: P and Q can both move to a new distribution R, because P has some new information for Q, but Q also has some new information for P. (In general, R need not even be a ‘compromise’ between P and Q; it could be something totally different.)
So it isn’t crazy at all for rational agents to prefer each other’s beliefs.
A weaker form of the common prior assumption could assert that this is always the case: two rational agents need not have the same priors, but upon learning each other’s priors, would then come to agree. (Either P updates to Q, or Q updates to P, or P and Q together update to some R.)
I don’t think H and T are rational agents in the first place, since they violate non-dogmatism: they place zero probability on non-tautologous propositions.
The common prior assumption, if true, is only supposed to apply amongst rational agents.
I would further point out that although I can’t use a classic Dutch Book to show H and T are irrational, I can use the relaxed Dutch Books of the sort used in the definition of logical induction—H and T are irrational because they expose themselves to unbounded exploitation. So I’m broadly using the same rationality framework to rule out H and T, as I am to argue for the common prior assumption.
The claim is more like: two TDT agents should never knowingly disagree about probabilities.
Here’s an intuition-pump. If I am sitting next to Alice and we disagree, we should have already bet with each other. Any bookie who comes along and tries to profit off of our disagreement should be unable to, because we’ve already made all the profitable exchanges we can. We’ve formed a Critch coalition, in order to coordinate rationally. So our apparent beliefs, going forward, will be a Bayesian mixture of our (would-be) individual beliefs. We will apparently have a common prior, when betting behavior is examined.
Sure, you can fix unbounded downside risk by giving H a finite budget. You can fix the dogmatism by making H have an ϵ=1/3↑↑↑3 probability of tails.
If you and H have a chance to bet with each other before going to the bookies, then the bookie won’t be able to Dutch book the two of you because you will have already separated H and H’s money.
If you can’t bet with H directly for some reason, then a bookie can Dutch book you and H, by acting as a middle man and skimming off some money.
Lets say I am in the same betting hall as a man called “The Idiot”. The Idiot has access to some undepletable fortune and will accept any bet (with any odds) that any person proposes to them. Now, whenever I bet on anything at all the Dutch bookmaker can protect themselves from a loss by making a compensating bet with the Idiot. (Although there are clearly simpler ways of getting money off the Idiot). Why should I feel worried that the category {me and the idiot} can be Dutch-booked? It doesn’t mean I am loosing or being foolish in my bets. At most it just means that I should be betting against the idiot not the Dutch bookmaker.
There exists an agent H that believes with certainty that all coins only ever land on heads as its prior.
There also exists an agent T that is equally confident in tails. (Exists in the mathematical sense that there is some pattern of code that would consist of such an agent, not in the sense that these agents have been built)
Lets say that H and T, by construction will always take any bet that they would profit from if all involved coins come up heads or tails respectively.
Consider a bet on a single coin flip that costs 2 if a coin comes up heads, and pays X if the coin lands on tails. (X>>2) If you would be prepared to take that bet for sufficiently large X, then a bookie can offer you this bet, and offer H a bet that wins 1 if the coin lands heads, and looses X+1 if the coin lands tails.H will take this bet. So the bookie has Dutch booked the pair of you.
If you ever bet at all and your betting decisions don’t depend on who you are sitting next to, then you can be part of a group that is Dutch booked.
If you want to avoid ever betting as part of a group that is being Dutch booked, then if you are in the presense of H and T, you can’t bet at all, even about things that have nothing to do with coin flips.
If you bet 5 against 5 that gravity won’t suddenly disappear, then the bookie can Dutch book H and T for 100, and the 3 of you have been Dutch booked for at least 95 as a group.
If you have some reason to suspect that a mind isn’t stupid, maybe you know that they won a lot of money in a prediction market, maybe you know that they were selected by evolution, ect then you have reason to take what the mind says seriously. If you have strong reasons to think that Alice is a nearly perfect reasoner, then getting Dutch booked when you are grouped with Alice indicates you are probably making a mistake.
Why should I care if I together with other people are jointly getting Dutch booked, if I myself am not? If my neighbour loses money but I do not, I do not care that “we” lost money, if his affairs have no connection with mine.
First off, I am quite sympathetic, and by no means would argue that the Dutch Book for the common prior assumption is as convincing as other Dutch Book Arguments.
However, it’s still intriguing.
If you and your neighbor are game-theoretic partners who sometimes cooperate in prisoner’s-dilemma like situations, then you might consider this kind of joint Dutch Book concerning. A coalition which does not manage to jointly coordinate to act as one agent is a weaker coalition.
If the collision is able to negotiate bets internally, then for any instance of the coalition getting Dutch booked, they can just agree to run that bet without the bookie, and split the bookies cut.
If everyone else changes to my prior, that’s great. But if I change from my prior to their prior, I am just (from the point of view of someone with my prior, which obviously includes myself) making myself vulnerable to be beaten in ordinary betting by other agents that have my prior.
This isn’t always true!
Let’s say your prior is P and mine is Q. I take your argument to be that P always prefers bets made according to P (bets made according to Q are at best just as good). But this is only true if P thinks P knows better than Q.
It’s perfectly possible for P to think Q knows better. For example, P might think Q just knows all the facts. Then it must be that P doesn’t know Q (or else P would also know all the facts.) But given the opportunity to learn Q, P would prefer to do so; whereupon, the updated P would be equal to Q.
Similar things can happen in less extreme circumstances, where Q is merely expected to know some things that P doesn’t. P could still prefer to switch entirely over to Q’s beliefs, because they have a higher expected value. It’s also possible that P trusts Q only to an extent, so P moves closer to Q but does not move all the way. This can even be true in the Aumann agreement setting: P and Q can both move to a new distribution R, because P has some new information for Q, but Q also has some new information for P. (In general, R need not even be a ‘compromise’ between P and Q; it could be something totally different.)
So it isn’t crazy at all for rational agents to prefer each other’s beliefs.
A weaker form of the common prior assumption could assert that this is always the case: two rational agents need not have the same priors, but upon learning each other’s priors, would then come to agree. (Either P updates to Q, or Q updates to P, or P and Q together update to some R.)
I don’t think H and T are rational agents in the first place, since they violate non-dogmatism: they place zero probability on non-tautologous propositions.
The common prior assumption, if true, is only supposed to apply amongst rational agents.
I would further point out that although I can’t use a classic Dutch Book to show H and T are irrational, I can use the relaxed Dutch Books of the sort used in the definition of logical induction—H and T are irrational because they expose themselves to unbounded exploitation. So I’m broadly using the same rationality framework to rule out H and T, as I am to argue for the common prior assumption.
The claim is more like: two TDT agents should never knowingly disagree about probabilities.
Here’s an intuition-pump. If I am sitting next to Alice and we disagree, we should have already bet with each other. Any bookie who comes along and tries to profit off of our disagreement should be unable to, because we’ve already made all the profitable exchanges we can. We’ve formed a Critch coalition, in order to coordinate rationally. So our apparent beliefs, going forward, will be a Bayesian mixture of our (would-be) individual beliefs. We will apparently have a common prior, when betting behavior is examined.
Sure, you can fix unbounded downside risk by giving H a finite budget. You can fix the dogmatism by making H have an ϵ=1/3↑↑↑3 probability of tails.
If you and H have a chance to bet with each other before going to the bookies, then the bookie won’t be able to Dutch book the two of you because you will have already separated H and H’s money.
If you can’t bet with H directly for some reason, then a bookie can Dutch book you and H, by acting as a middle man and skimming off some money.
I don’t think this really matters though.
Lets say I am in the same betting hall as a man called “The Idiot”. The Idiot has access to some undepletable fortune and will accept any bet (with any odds) that any person proposes to them. Now, whenever I bet on anything at all the Dutch bookmaker can protect themselves from a loss by making a compensating bet with the Idiot. (Although there are clearly simpler ways of getting money off the Idiot). Why should I feel worried that the category {me and the idiot} can be Dutch-booked? It doesn’t mean I am loosing or being foolish in my bets. At most it just means that I should be betting against the idiot not the Dutch bookmaker.
Correct. This was basically my point.
Oops. Yes, I have just re-read your comment. I somehow didn’t absorb that all-important “if you have reason to think they are at all sensible” clause!