I’ve skimmed over the beginning of your paper, and I think there might be several problems with it.
I don’t see where it is explicitly stated, but I think information “seller’s prediction is accurate with probability 0,75” is supposed to be common knowledge. Is it even possible for a non-trivial probabilistic prediction to be a common knowledge? Like, not as in some real-life situation, but as in this condition not being logical contradiction? I am not a specialist on this subject, but it looks like a logical contradiction. And you can prove absolutely anything if your premise contains contradiction.
A minor nitpick compared to the previous one, but you don’t specify what you mean by “prediction is accurate with probability 0.75”. What kinds of mistakes does seller make? For example, if buyer is going to buy the B1, then with probability 0.75 the prediction will be “B1”. What about the 0.25? Will it be 0.125 for “none” and 0.125 for “B2”? Will it be 0.25 for “none” and 0 for “B2”? (And does buyer knows about that? What about seller knowing about buyer knowing...)
When you write “$1−P (money in Bi | buyer chooses Bi ) · $3 = $1 − 0.25 · $3 = $0.25.”, you assume that P(money in Bi | buyer chooses Bi )=0.75. That is, if buyer chooses the first box, seller can’t possibly think that buyer will choose none of the boxes. And the same for the case of buyer choosing the second box. You can easily fix it by writing “$1−P (money in Bi | buyer chooses Bi ) · $3 >= $1 − 0.25 · $3 = $0.25″ instead. It is possible that you make some other implicit assumptions about mistakes that seller can make, so you might want to check it.
>I think information “seller’s prediction is accurate with probability 0,75” is supposed to be common knowledge.
Yes, correct!
>Is it even possible for a non-trivial probabilistic prediction to be a common knowledge? Like, not as in some real-life situation, but as in this condition not being logical contradiction? I am not a specialist on this subject, but it looks like a logical contradiction. And you can prove absolutely anything if your premise contains contradiction.
Why would it be a logical contradiction? Do you think Newcomb’s problem also requires a logical contradiction? Note that in neither of these cases does the predictor tell the agent the result of a prediction about the agent.
>What kinds of mistakes does seller make?
For the purpose of the paper it doesn’t really matter what beliefs anyone has about how the errors are distributed. But you could imagine that the buyer is some piece of computer code and that the seller has an identical copy of that code. To make a prediction, the seller runs the code. Then she flips a coin twice. If the coin does not come up Tails twice, she just uses that prediction and fills the boxes accordingly. If the coin does come up Tails twice, she uses a third coin flip to determine whether to (falsely) predict one of the two other options that the agent can choose from. And then you get the 0.75, 0.125, 0.125 distribution you describe. And you could assume that this is common knowledge.
Of course, for the exact CDT expected utilities, it does matter how the errors are distributed. If the errors are primarily “None” predictions, then the boxes should be expected to contain more money and the CDT expected utilities of buying will be higher. But for the exploitation scheme, it’s enough to show that the CDT expected utilities of buying are strictly positive.
>When you write “$1−P (money in Bi | buyer chooses Bi ) · $3 = $1 − 0.25 · $3 = $0.25.”, you assume that P(money in Bi | buyer chooses Bi )=0.75.
I assume you mean that I assume P(money in Bi | buyer chooses Bi )=0.25? Yes, I assume this, although really I assume that the seller’s prediction is accurate with probability 0.75 and that she fills the boxes according to the specified procedure. From this, it then follows that P(money in Bi | buyer chooses Bi )=0.25.
>That is, if buyer chooses the first box, seller can’t possibly think that buyer will choose none of the boxes.
I don’t assume this / I don’t see how this would follow from anything I assume. Remember that if the seller predicts the buyer to choose no box, both boxes will be filled. So even if all false predictions would be “None” predictions (when the buyer buys a box), then it would still be P(money in Bi | buyer chooses Bi )=0.25.
I assume you mean that I assume P(money in Bi | buyer chooses Bi )=0.25? Yes, I assume this, although really I assume that the seller’s prediction is accurate with probability 0.75 and that she fills the boxes according to the specified procedure. From this, it then follows that P(money in Bi | buyer chooses Bi )=0.25.
Yes, you are right. Sorry.
Why would it be a logical contradiction? Do you think Newcomb’s problem also requires a logical contradiction?
Okay, it probably isn’t a contradiction, because the situation “Buyer writes his decision and it is common knowledge that an hour later Seller sneaks a peek into this decision (with probability 0.75) or into a random false decision (0.25). After that Seller places money according to the decision he saw.” seems similar enough and can probably be formalized into a model of this situation.
You might wonder why am I spouting a bunch of wrong things in an unsuccessful attempt to attack your paper. I do that because it looks really suspicious to me for the following reasons:
You don’t use language developed by logicians to avoid mistakes and paradoxes in similar situations.
Even for something written in more or less basic English, your paper doesn’t seem to be rigorous enough for the kinds of problems it tries to tackle. For example, you don’t specify what exactly is considered common knowledge, and that can probably be really important.
You result looks similar to something you will try to prove as a stepping stone to proving that this whole situation with boxes is impossible. “It follows that in this situation two perfectly rational agents with the same information would make different deterministic decisions. Thus we arrived at contradiction and this situation is impossible.” In your paper agents are rational in a different ways (I think), but it still looks similar enough for me to become suspicious.
So, while my previous attempts at finding error in your paper failed pathetically, I’m still suspicious, so I’ll give it another shot.
When you argue that Buyer should buy one of the boxes, you assume that Buyer knows the probabilities that Seller assigned to Buyer’s actions. Are those probabilities also a part of common knowledge? How is that possible? If you try to do the same in Newcomb’s problem, you will get something like “Omniscient predictor predicts that player will pick the box A (with probability 1); player knows about that; player is free to pick between A and both boxes”, which seem to be a paradox.
>You might wonder why am I spouting a bunch of wrong things in an unsuccessful attempt to attack your paper.
Nah, I’m a frequent spouter of wrong things myself, so I’m not too surprised when other people make errors, especially when the stakes are low, etc.
Re 1,2: I guess a lot of this comes down to convention. People have found that one can productively discuss these things without always giving the formal models (in part because people in the field know how to translate everything into formal models). That said, if you want mathematical models of CDT and Newcomb-like decision problems, you can check the Savage or Jeffrey-Bolker formalizations. See, for example, the first few chapters of Arif Ahmed’s book, “Evidence, Decision and Causality”. Similarly, people in decision theory (and game theory) usually don’t specify what is common knowledge, because usually it is assumed (implicitly) that the entire problem description is common knowledge / known to the agent (Buyer). (Since this is decision and not game theory, it’s not quite clear what “common knowledge” means. But presumably to achieve 75% accuracy on the prediction, the seller needs to know that the buyer understands the problem...)
3: Yeah, *there exist* agent models under which everything becomes inconsistent, though IMO this just shows these agent models to be unimplementable. For example, take the problem description from my previous reply (where Seller just runs an exact copy of Buyer’s source code). Now assume that Buyer knows his source code and is logically omniscient. Then Buyer knows what his source code chooses and therefore knows the option that Seller is 75% likely to predict. So he will take the other option. But of course, this is a contradiction. As you’ll know, this is a pretty typical logical paradox of self-reference. But to me it just says that this logical omniscience assumption about the buyer is implausible and that we should consider agents who aren’t logically omniscient. Fortunately, CDT doesn’t assume knowledge of its own source code and such.
Perhaps one thing to help sell the plausibility of this working: For the purpose of the paper, the assumption that Buyer uses CDT in this scenario is pretty weak, formally simple and doesn’t have much to do with logic. It just says that the Buyer assigns some probability distribution over box states (i.e., some distribution over the mutually exclusive and collectively exhaustive s1=”money only in box 1“, s2= “money only in box 2”, s3=”money in both boxes”); and that given such distribution, Buyer takes an action that maximizes (causal) expected utility. So you could forget agents for a second and just prove the formal claim that for all probability distributions over three states s1, s2, s3, it is for i=1 or i=2 (or both) the case that (P(si)+P(s3))*$3 - $1 > 0. I assume you don’t find this strange/risky in terms of contradictions, but mathematically speaking, nothing more is really going on in the basic scenario.
The idea is that everyone agrees (hopefully) that orthodox CDT satisfies the assumption. (I.e., assigns some unconditional distribution, etc.) Of course, many CDTers would claim that CDT satisfies some *additional* assumptions, such as the probabilities being calibrated or “correct” in some other sense. But of course, if “A=>B”, then “A and C ⇒ B”. So adding assumptions cannot help the CDTer avoid the loss of money conclusion if they also accept the more basic assumptions. Of course, *some* added assumptions lead to contradictions. But that just means that they cannot be satisfied in the circumstances of this scenario if the more basic assumption is satisfied and if the premises of the Adversarial Offer help. So they would have to either adopt some non-orthodox CDT that doesn’t satisfy the basic assumption or require that their agents cannot be copied/predicted. (Both of which I also discuss in the paper.)
>you assume that Buyer knows the probabilities that Seller assigned to Buyer’s actions.
No, if this were the case, then I think you would indeed get contradictions, as you outline. So Buyer does *not* know what Seller’s prediction is. (He only knows her prediction is 75% accurate.) If Buyer uses CDT, then of course he assigns some (unconditional) probabilities to what the predictions are, but of course the problem description implies that these predictions aren’t particularly good. (Like: if he assigns 90% to the money in box 1, then it immediately follows that *no* money is in box 1.)
I’ve skimmed over the beginning of your paper, and I think there might be several problems with it.
I don’t see where it is explicitly stated, but I think information “seller’s prediction is accurate with probability 0,75” is supposed to be common knowledge. Is it even possible for a non-trivial probabilistic prediction to be a common knowledge? Like, not as in some real-life situation, but as in this condition not being logical contradiction? I am not a specialist on this subject, but it looks like a logical contradiction. And you can prove absolutely anything if your premise contains contradiction.
A minor nitpick compared to the previous one, but you don’t specify what you mean by “prediction is accurate with probability 0.75”. What kinds of mistakes does seller make? For example, if buyer is going to buy the B1, then with probability 0.75 the prediction will be “B1”. What about the 0.25? Will it be 0.125 for “none” and 0.125 for “B2”? Will it be 0.25 for “none” and 0 for “B2”? (And does buyer knows about that? What about seller knowing about buyer knowing...)
When you write “$1−P (money in Bi | buyer chooses Bi ) · $3 = $1 − 0.25 · $3 = $0.25.”, you assume that P(money in Bi | buyer chooses Bi )=0.75. That is, if buyer chooses the first box, seller can’t possibly think that buyer will choose none of the boxes. And the same for the case of buyer choosing the second box. You can easily fix it by writing “$1−P (money in Bi | buyer chooses Bi ) · $3 >= $1 − 0.25 · $3 = $0.25″ instead. It is possible that you make some other implicit assumptions about mistakes that seller can make, so you might want to check it.
>I think information “seller’s prediction is accurate with probability 0,75” is supposed to be common knowledge.
Yes, correct!
>Is it even possible for a non-trivial probabilistic prediction to be a common knowledge? Like, not as in some real-life situation, but as in this condition not being logical contradiction? I am not a specialist on this subject, but it looks like a logical contradiction. And you can prove absolutely anything if your premise contains contradiction.
Why would it be a logical contradiction? Do you think Newcomb’s problem also requires a logical contradiction? Note that in neither of these cases does the predictor tell the agent the result of a prediction about the agent.
>What kinds of mistakes does seller make?
For the purpose of the paper it doesn’t really matter what beliefs anyone has about how the errors are distributed. But you could imagine that the buyer is some piece of computer code and that the seller has an identical copy of that code. To make a prediction, the seller runs the code. Then she flips a coin twice. If the coin does not come up Tails twice, she just uses that prediction and fills the boxes accordingly. If the coin does come up Tails twice, she uses a third coin flip to determine whether to (falsely) predict one of the two other options that the agent can choose from. And then you get the 0.75, 0.125, 0.125 distribution you describe. And you could assume that this is common knowledge.
Of course, for the exact CDT expected utilities, it does matter how the errors are distributed. If the errors are primarily “None” predictions, then the boxes should be expected to contain more money and the CDT expected utilities of buying will be higher. But for the exploitation scheme, it’s enough to show that the CDT expected utilities of buying are strictly positive.
>When you write “$1−P (money in Bi | buyer chooses Bi ) · $3 = $1 − 0.25 · $3 = $0.25.”, you assume that P(money in Bi | buyer chooses Bi )=0.75.
I assume you mean that I assume P(money in Bi | buyer chooses Bi )=0.25? Yes, I assume this, although really I assume that the seller’s prediction is accurate with probability 0.75 and that she fills the boxes according to the specified procedure. From this, it then follows that P(money in Bi | buyer chooses Bi )=0.25.
>That is, if buyer chooses the first box, seller can’t possibly think that buyer will choose none of the boxes.
I don’t assume this / I don’t see how this would follow from anything I assume. Remember that if the seller predicts the buyer to choose no box, both boxes will be filled. So even if all false predictions would be “None” predictions (when the buyer buys a box), then it would still be P(money in Bi | buyer chooses Bi )=0.25.
Yes, you are right. Sorry.
Okay, it probably isn’t a contradiction, because the situation “Buyer writes his decision and it is common knowledge that an hour later Seller sneaks a peek into this decision (with probability 0.75) or into a random false decision (0.25). After that Seller places money according to the decision he saw.” seems similar enough and can probably be formalized into a model of this situation.
You might wonder why am I spouting a bunch of wrong things in an unsuccessful attempt to attack your paper. I do that because it looks really suspicious to me for the following reasons:
You don’t use language developed by logicians to avoid mistakes and paradoxes in similar situations.
Even for something written in more or less basic English, your paper doesn’t seem to be rigorous enough for the kinds of problems it tries to tackle. For example, you don’t specify what exactly is considered common knowledge, and that can probably be really important.
You result looks similar to something you will try to prove as a stepping stone to proving that this whole situation with boxes is impossible. “It follows that in this situation two perfectly rational agents with the same information would make different deterministic decisions. Thus we arrived at contradiction and this situation is impossible.” In your paper agents are rational in a different ways (I think), but it still looks similar enough for me to become suspicious.
So, while my previous attempts at finding error in your paper failed pathetically, I’m still suspicious, so I’ll give it another shot.
When you argue that Buyer should buy one of the boxes, you assume that Buyer knows the probabilities that Seller assigned to Buyer’s actions. Are those probabilities also a part of common knowledge? How is that possible? If you try to do the same in Newcomb’s problem, you will get something like “Omniscient predictor predicts that player will pick the box A (with probability 1); player knows about that; player is free to pick between A and both boxes”, which seem to be a paradox.
Sorry for taking some time to reply!
>You might wonder why am I spouting a bunch of wrong things in an unsuccessful attempt to attack your paper.
Nah, I’m a frequent spouter of wrong things myself, so I’m not too surprised when other people make errors, especially when the stakes are low, etc.
Re 1,2: I guess a lot of this comes down to convention. People have found that one can productively discuss these things without always giving the formal models (in part because people in the field know how to translate everything into formal models). That said, if you want mathematical models of CDT and Newcomb-like decision problems, you can check the Savage or Jeffrey-Bolker formalizations. See, for example, the first few chapters of Arif Ahmed’s book, “Evidence, Decision and Causality”. Similarly, people in decision theory (and game theory) usually don’t specify what is common knowledge, because usually it is assumed (implicitly) that the entire problem description is common knowledge / known to the agent (Buyer). (Since this is decision and not game theory, it’s not quite clear what “common knowledge” means. But presumably to achieve 75% accuracy on the prediction, the seller needs to know that the buyer understands the problem...)
3: Yeah, *there exist* agent models under which everything becomes inconsistent, though IMO this just shows these agent models to be unimplementable. For example, take the problem description from my previous reply (where Seller just runs an exact copy of Buyer’s source code). Now assume that Buyer knows his source code and is logically omniscient. Then Buyer knows what his source code chooses and therefore knows the option that Seller is 75% likely to predict. So he will take the other option. But of course, this is a contradiction. As you’ll know, this is a pretty typical logical paradox of self-reference. But to me it just says that this logical omniscience assumption about the buyer is implausible and that we should consider agents who aren’t logically omniscient. Fortunately, CDT doesn’t assume knowledge of its own source code and such.
Perhaps one thing to help sell the plausibility of this working: For the purpose of the paper, the assumption that Buyer uses CDT in this scenario is pretty weak, formally simple and doesn’t have much to do with logic. It just says that the Buyer assigns some probability distribution over box states (i.e., some distribution over the mutually exclusive and collectively exhaustive s1=”money only in box 1“, s2= “money only in box 2”, s3=”money in both boxes”); and that given such distribution, Buyer takes an action that maximizes (causal) expected utility. So you could forget agents for a second and just prove the formal claim that for all probability distributions over three states s1, s2, s3, it is for i=1 or i=2 (or both) the case that
(P(si)+P(s3))*$3 - $1 > 0.
I assume you don’t find this strange/risky in terms of contradictions, but mathematically speaking, nothing more is really going on in the basic scenario.
The idea is that everyone agrees (hopefully) that orthodox CDT satisfies the assumption. (I.e., assigns some unconditional distribution, etc.) Of course, many CDTers would claim that CDT satisfies some *additional* assumptions, such as the probabilities being calibrated or “correct” in some other sense. But of course, if “A=>B”, then “A and C ⇒ B”. So adding assumptions cannot help the CDTer avoid the loss of money conclusion if they also accept the more basic assumptions. Of course, *some* added assumptions lead to contradictions. But that just means that they cannot be satisfied in the circumstances of this scenario if the more basic assumption is satisfied and if the premises of the Adversarial Offer help. So they would have to either adopt some non-orthodox CDT that doesn’t satisfy the basic assumption or require that their agents cannot be copied/predicted. (Both of which I also discuss in the paper.)
>you assume that Buyer knows the probabilities that Seller assigned to Buyer’s actions.
No, if this were the case, then I think you would indeed get contradictions, as you outline. So Buyer does *not* know what Seller’s prediction is. (He only knows her prediction is 75% accurate.) If Buyer uses CDT, then of course he assigns some (unconditional) probabilities to what the predictions are, but of course the problem description implies that these predictions aren’t particularly good. (Like: if he assigns 90% to the money in box 1, then it immediately follows that *no* money is in box 1.)