One of the least attractive features of Steven Landsburg’s column in Slate was always his habit of assuming that anyone who disagreed with him obviously did so out of ignorance, and indeed that appears to be his response to my post on the subject of quantum game theory and information. As a matter of fact, I do understand a bit (just a bit) about quantum probability and I understand a bit more after mugging up on the relevant chapter of David Williams excellent book on probability. Landsburg’s point appears to be that since no information is exchanged, there is no communication, but this won’t do. “Information” in the physical sense is not exchanged, but “quantum information” (not the same thing, but neither something completely different) is, and that is enough to turn it into a communication game. Let me elucidate with yet another variation on the cats/dogs game.
Assume that me and John Quiggin are playing the cats/dogs co-operation game for money (for details of this game, see here. Assume further that Quiggin is thoroughly bored with this game and has decided to stop playing it properly. Instead of adopting any strategy at all, he just flips a coin to decide whether he’s going to answer “yes” or “no”.
However, although he can’t be bothered playing properly himself, JQ realises that the game matters more to me, so he has decided to help me out. Since getting his Federation Fellowship, we assume that John now employs a butler, called Lurch. Every time the game is played, Lurch observes whether John was asked “do you like cats?” or “do you like dogs?”, and whether he answered (at random) “yes” or “no”? Armed with this information, Lurch, who is a kindly soul, walks round to my office, observes which question I have been asked, and tells me what to answer. He gives me the answer I need to win the money in all cases, unless JQ and I have both been asked “do you like dogs” and JQ has answered “no”. In this case, there is no win possible for me, so Lurch just tells me to answer “no”.
It can be seen that with this arrangement, JQ and I will win seven times out of eight (Lurch can always give me the right answer except in the one case where we are both asked about dogs (25% chance) and JQ answers “no” (50% chance)). But what information is being transferred to me?
Well, let’s assume that I have just been asked about cats. Lurch comes in and says “Say YES!”. I know that either JQ has been asked about cats and said “NO” or he has been asked about dogs and said “YES”. But I don’t know which of these situations has obtained. So, I am in no better a position than I was before to guess either which question JQ was asked, or what reply he made.
Similarly, if Lurch comes in and says “Say NO!”, I know either that JQ has been asked about cats and said “YES” or asked about dogs and said “NO” (in this second situation I’m about to lose, but there you go). I still don’t know which of these situations has obtained.
So, Lurch never gives me any information which tells me either which question was asked in JQ’s room, or what answer he gave. We can see this more clearly by generalising; if JQ was using a biased coin to decide on his answers and if the questioner was using a biased coin to decide what question to ask him, I would never be able to figure out the bias of either coin from the information Lurch gave me, no matter how many times we played the game; I wouldn’t know whether a preponderance of “Say YES” instructions was the result of JQ saying “NO” more often, or JQ being asked about dogs more often.
No information about the state of affairs in the other room is being transmitted here, so by the criterion Landsburg is using, I think he would have to call this a “non-communication game”. To which all I can say is that it is a pretty funny “non-communication game” where one party is sending his butler round to tell the other party which strategy to choose.
Furthermore, having looked at the Lurch game, we can begin to understand what is actually being transmitted in cases of quantum communication. Although I would never learn anything about the biases of the two coins in the other room, I would, over time, learn quite a lot about the correlation between the two coins. In this sense, information is being transmitted by Lurch between the two rooms; not in any particular case or about the average state of our rooms, but over a long run and about the relationship between them.
And this is exactly what happens in cases of quantum entanglement. Two entangled particles have their states commixed until you observe one of them, at which point you determine the state of them both. Because superposition isn’t the same as probabilistic mixture, this means that you can’t tell what the state of one is likely to be by conditioning (classically) on the state of the other. However, because their states were entangled, you can know that the measurement made of an observable on one particle will be correlated in a predictable way with the measurement made of the same observable on the other. And this correlation is a “proper” classical correlation; its frequentist interpretation makes use of the same Strong Law of Large Numbers that one uses when drawing balls out of a Polya urn, and its Bayesian interpretation makes use of the same Bayes’ Theorem that is used in WinBUGS. As Williams points out in his excellent book, if it was literally the case that you couldn’t tell anything at all about one particle by observing the other, we’d never have been able to verify propositions about quantum entanglement empirically, and we can (using normal classical statistics), so the implication that using entangled quanta isn’t cheating in a co-ordination game is clearly far too strong a claim. Like the original quanta or Landsburg’s sunglasses, Lurch is a magical device which allows me to choose a correlation between my observation and my partner’s which is different from 0. That’s cheating in what’s meant to be a noncommunication game.
Which is why I maintain my original proposition; that quantum game theory is interesting for what it tells us about quantum probability, but that its implications for the fundamentals of game theory aren’t that great. The rigorous definitions non-communication games used by the game theory nargs already use a definition of communication which rules out mobile phones, morse code, comedy butlers and anything else which allows one player to make his choice of strategy conditional on the other’s, so the only economically interesting implication of quantum game theory would be that we might want to start modelling quantum-probabilistic mixed strategies more often if we thought that particle-entangling devices were going to become as common and as portable as, say, mobile phones. Which, at present, they ain’t.
{ 63 comments }
Brian Weatherson 10.24.04 at 8:25 pm
I agree entirely with the spirit of these comments, but I couldn’t follow one detail.
He gives me the answer I need to win the money in all cases, unless JQ and I have both been asked “do you like dogs†and JQ has answered “noâ€. In this case, there is no win possible for me, so Lurch just tells me to answer “noâ€.
The game works like this
You and I are each asked a single question, either “Do you like cats?” or “Do you like dogs?”. Our questions are determined by independent coin flips. We both win if our answers differ, unless we’re both asked about dogs, in which case we both win if our answers match.
So if JQ was asked about dogs and said “No”, then you can win, by saying “No” if you are asked about dogs and by saying “Yes” if you are asked about cats.
In any case the point works if we change the game rules so that if it’s dogs-dogs the only win is yes-yes.
Matt McIrvin 10.24.04 at 8:30 pm
I think you’re right. But it is important to understand that saying “information is transmitted” by these correlations makes many physicists’ teeth itch, just because they’ve gotten into arguments with cranks who were trying to use this effect to build a faster-than-light radio.
dsquared 10.24.04 at 8:33 pm
Brian is right; for some reason, I believed that the rules of the game only gave you a win in dogs-dogs if you both said “yes”.
Glenn Bridgman 10.24.04 at 9:19 pm
One wonders where Lurch got his FTL warp drive. There is an element of causaulity that applys to your Lurch analogy which does not apply to quantum entanglement.
dsquared 10.24.04 at 9:23 pm
That’s not the point of the analogy, Glen. The point is that in this case, Lurch is not “transmitting information” in exactly the same way in which entangled quanta don’t transmit information, but it’s obvious that there is communication in the sense relevant for game theory.
Steven E. Landsburg 10.24.04 at 9:27 pm
*sigh* You really do insist on missing the point.
Our ability to employ a butler *does* change the game theoretic analysis. In situations where people can employ butlers to carry messages back and forth, models that assume the contrary are likely to be inappropriate models.
Likewise, our ability to employ quantum signals *does* change the game theoretic analysis. In situations where we can use quantum signals to coordinate our actions, models that assume the contrary are likely to be inappropriate models.
It is useful to work out a variety of models that are appropriate to a variety of situations. “Butler game theory” is a legitimate area of study, but there’s not much new to be said about it, because most of it’s already been worked out.
Quantum game theory is neither more nor less intrinsically interesting than butler game theory except that a) less of it has been worked out (though see, for example, my paper at http://www.landsburg.com/qgt.pdf for some recent results) and b) there are (admittedly speculative) reasons to think that future technology will make it more widely applicable than it is.
Glenn Bridgman 10.24.04 at 9:28 pm
“The point is that in this case, Lurch is not “transmitting information†in exactly the same way in which entangled quanta don’t transmit information”
No he isn’t. The information is being transmitted from point A to point B, he just isn’t telling you it. With the entanglement, the information from point A is never at point B.
Steven E. Landsburg 10.24.04 at 9:32 pm
Following up on Glenn Bridgman: The point here is that to have a butler walk back and forth, tell you some things, and not tell you others, is a perfectly fine theoretical assumption but also perfectly unlikely to apply to any real world situation.
When we have access to quantum entanglement, but not to a butler, we should model the situation accordingly. It is therefore worth studying such models. Why is that so hard to understand?
Keith M Ellis 10.24.04 at 9:38 pm
I should be very interested in this debate. But I find that the evenly-matched enormous arrogance of dsquared and landsburg rids me of all desire to follow their argument.
Glenn Bridgman 10.24.04 at 9:43 pm
Keith, I’m not sure if my omission from that list is a good thing or a bad thing:P
dsquared 10.24.04 at 9:43 pm
Our ability to employ a butler does change the game theoretic analysis.
Yes. Specifically, it changes it from the analysis of noncommunication games to the analysis of communication games. I seem to remember writing a post about that subject a couple of days ago, and looking back, I’m rather glad that I was rude to you when I wrote it. Since you have, seemingly, given up on the particular point of economics, I think that the only thing we are still arguing about is whether qubits are easier to get hold of than butlers. I think that this can probably be settled with a quick headcount.
Glen: since point A is John Quiggin and point B is me, Lurch isn’t transmitting information between those points. He could if he wanted to (which is disanalogous with the quantum case), but he doesn’t (hence the cases are analogous)
dsquared 10.24.04 at 9:47 pm
Keith: infuriating, isn’t it?
more generally, I would surmise that for any quantum game theoretic result, I can design a set of instructions for Lurch that would lead to the same relationship between my actions and JQ’s, so I guess (and would be interested if anyone could prove otherwise) that quantum game theory reduces to classical game theory for the fundamental reason that game theory and quantum probability make use of the same (classical) strong law.
Glenn Bridgman 10.24.04 at 9:50 pm
D^2, I’m not sure what you mean. Lurch listens to what JQ gets, then as a result, ambles over to you and tells you what to answer. Since he can’t know what the answer is unless he knows data from both positions, of course he has communicated it from A to B.
Steven E. Landsburg 10.24.04 at 9:51 pm
I think that the only thing we are still arguing about is whether qubits are easier to get hold of than butlers
No, I don’t think that’s the point of contention at all.
The point of contention is: Is it interesting to model a game in which players can use quantum entanglement but cannot use classical methocs of communication?
Your position, as I understand it, is that the answer is no, because we can always mimic these games in classical situations by employing a butler who tells us some things but doesn’t tell us others.
My response is that no butler would actually behave that way, so the potential existence of such a butler does not suffice to make this model interesting. Quantum entanglement *does* behave that way, so quantum entanglement—insofar as you believe it will someday be technologically relevant—does suffice to make this model interesting.
My position is: here’s a situation that could arise and for which the classical model is inadequate. Your position is (I think): but that’s not interesting
because I can imagine some other highly contrived situation in which the classical model is inadequate in exactly the same way.
I absolutely grant that the classical model is inadequate if you’ve got a butler going back and forth. Do you grant that the classical model is inadequate if you’ve got the use of quantum entanglement?
dsquared 10.24.04 at 9:55 pm
Glenn, can I reflect back one of your own questions at you; after Lurch has told me “say YES”, what do I now know about the state in John’s office?
abb1 10.24.04 at 9:58 pm
With the entanglement, the information from point A is never at point B.
I think one could argue that with the entanglement points A and B are actually the same point, because the two quanta together represent one single device. Both parties have access to the same device, as if each of them had a monitor connected to the same computer.
dsquared 10.24.04 at 10:00 pm
Steven: No, I don’t accept that at all. Classical game theory has already modelled situations with no talk, cheap talk, expensive talk and noisy talk. I’d be interested in situations in which quantum entanglement led to a case which couldn’t be simulated as one of these kinds of signal. But, as mentioned above, since quantum probability agrees with classical probability in all cases where a classical probability exists, and since any game which has payoffs that can be measured is going to have a classical mixed strategy equilibrium, I don’t believe that this can be the case. Hence, the point I’ve been making all along; quantum game theory is interesting to game theorists, but only important to quantum theorists.
Lindsay Beyerstein 10.24.04 at 10:04 pm
Are we getting hung up on the idea of transmission? Steven seems to assume that information must take the form of a physical signal relayed from one partner to another. Relativity says that can’t be any physical signal between the parties. Yet, we all agree that quantum theory supplies probabilistic evidence to one player about his partner’s state.
It seems that any evidence about your partner’s state vitiates the claim that you are playing a traditional coordination game.
It is interesting that quantum evidence is different from classical evidence–most people assume that evidence must have some causal connection to the state of affairs it evidences. Conceptual analysis only takes you so far, I guess.
Still, I don’t think this counts as an especially interesting solution to the coordination problem.
Steven E. Landsburg 10.24.04 at 10:17 pm
lindsay: Yet, we all agree that quantum theory supplies probabilistic evidence to one player about his partner’s state.
No, we do not all agree on this.
To help me decide whether I should start to agree with it, can you tell me what you mean by “state”?
Glenn Bridgman 10.24.04 at 10:22 pm
D^2, the set of possible partner states is {Cat, Dog} X {Yes, No}. So let’s say I get asked the dog question and am told to say yes. Your right, I do gain information: the set of possible states is restricted to {(Cat, No}, (Dog, Yes)}. The key point here is that your state has nothing to do with that restriction. Let’s say that you are on Earth and JQ is over having a party on Alpha Centuri. The reality at AC can have absolutely no effect on the reality of earth over a timescale of less then four years. It is forbidden. JQ’s actions/state have nothing to do with your gain of information. No communication has happened.
dsquared 10.24.04 at 10:29 pm
Glen, you’re correct on the physics. But note that nothing in the economics depends on whether or not the signal travels instantaneously, or whether it takes four years plus a couple of tea-breaks for Lurch. If there’s no communication taking place in the quantum situation, there’s no communication taking place in the butler situation, which is my point; although no information is transmitted, there is quanglement shared between the two points, and quanglement gives enough of a connection between the two strategies to make it a communication-game rather than a non-communication-game.
Glenn Bridgman 10.24.04 at 10:35 pm
Isn’t that the point? That quantum mechanics allows us to do things in a coordination game which classically would require turning it into communication game?
Steven E. Landsburg 10.24.04 at 10:38 pm
Glenn: Isn’t that the point? That quantum mechanics allows us to do things in a coordination game which classically would require turning it into communication game?
Yes, this is the point exactly.
Steven E. Landsburg 10.24.04 at 10:39 pm
Glenn: Isn’t that the point? That quantum mechanics allows us to do things in a coordination game which classically would require turning it into communication game?
Yes, this is the point exactly. The butler, as D^2 points out, does the same thing. The difference is that in the real world, butlers do not behave in this particular odd way, whereas entangled particles do.
Walt Pohl 10.24.04 at 10:39 pm
Like all truly passionate arguments on the internet, we have reached the point where both sides are arguing past each other, while insisting they are not.
1) Dsquared is right that you can simulate the result of quantum entanglement by classical means, ergo you can regard quantum entanglement games as a restricted form of games that allow communication.
2) Steven is right that given the peculiar restrictions imposed by resorting to quantum entanglement, it’s interesting to study these sorts of games in their own right.
Given that pointing out two people are arguing past each other never prevents them from continuing to argue past each other, we now return you to your previously scheduled argument.
Glenn Bridgman 10.24.04 at 10:46 pm
“1) Dsquared is right that you can simulate the result of quantum entanglement by classical means, ergo you can regard quantum entanglement games as a restricted form of games that allow communication.”
See, I’m not sure thats right. Indistinguishability aside, the fact that we can simulate it as a communication game doesn’t mean that it is neccesarily a subset of communication games. You could simulate the original strategy of always disagreeing with each other as a communication game as well(Lurch always gives you the oppisite answer of the other guy), but we aren’t treating that as a restricted form of a communication game.
Steven E. Landsburg 10.24.04 at 10:47 pm
Walt Pohl is right on all counts.
John Quiggin 10.24.04 at 10:59 pm
Lurch will be busy for the next hour, serving my morning coffee and crumpets, so please don’t ask him to transmit any information.
Lurch 10.24.04 at 11:02 pm
Brainnnnnnnnnnns…Braiinnnnnnns
junius ponds 10.24.04 at 11:22 pm
>thats right. Indistinguishability aside, the fact that we can simulate it as a communication game doesn’t mean that it is neccesarily a subset of communication games.< Yes; what distinguishes the classical and quantum games is that classically the correlation can only be obtained through the use of signals of speed c or less. AFAIK, practicing physicists don't consider the instantaneous, distant collapse of the wave function "communication."
junius ponds 10.24.04 at 11:28 pm
That is, even though you’ve cooked up a classical game that doesn’t involve information transfer, however you quantify it, it still involves a causal influence that is restricted to a velocity of c or less. This is barred in Landsburg’s game.
Lurch 10.24.04 at 11:56 pm
Bush tax cuts unfair to poor butlers…Bush tax cuts favor the rich…Steven Landsburg bad…
william 10.25.04 at 12:33 am
I sympathise with Walt Pohl’s even-handedness, but I think DD’s got the better of the argument here. The original (Landsburg) article demonstrates that there are, in the physical world, situations in which you can think you’ve set up a non-communicating game while in fact you have the properties of a noisily communicating game — even if what’s going on doesn’t count as information exchange in the standard physical sense. That’s an interesting observation, but it requires advanced technology or butlers to make it happen. I understand DD’s point to be that while this is true, there are still going to be non-communicating games. The existence of quantum entanglement doesn’t abolish all of that type of game, it just moves some game settings from non-communicating to some other group. And therefore it’s still worth considering non-communicating games as a category.
If this *is* DD’s argument, then I’m pretty certain that he’s right about the effects on game theory. In so far as Steven Landsburg’s argument is simply “Quantum entanglement is cool!” then I agree with him too, of course…
Steven E. Landsburg 10.25.04 at 1:18 am
william: The existence of quantum entanglement doesn’t abolish all of that type of game, it just moves some game settings from non-communicating to some other group. And therefore it’s still worth considering non-communicating games as a category.
Well, of course. None of this was ever at issue. I gave an example of a situation for which you wouldn’t want to use the classical model, and a specific alternative model that’s better in that situation.
I certainly did not mean to imply, and don’t believe I did imply, that this diminishes the interest of the classical analysis applied to classical situations.
Reimer Behrends 10.25.04 at 3:51 am
Daniel, regarding your response to Glenn: I do not see why there isn’t information transmitted by Lurch. Employing Lurch decreases information entropy. It is a bit messy, because you have left undefined how information is encoded in the signal (the signal being what Lurch says). But if, for example, you are told “answer: yes”, then you know that JQ has not both been asked about dogs and that his answer has been no. Thus, the state (dog, no) now has probability zero and information entropy drops.
Junius, I do not see how communication faster than c — while interesting in and of itself — is relevant with respect to either the game theory or the information theory being discussed; interesting information-theoretic aspects exist, such as the fact that you can use superdense codings over quantum channels (such as using n/2 quantum bits to encode n classical bits of information). But in the context given, using a quantum channel for communication does not appear to be different from using a suitably constrained classical channel; the speed of communication is never exploited.
Steven, I don’t think anybody ever said that it is not interesting to study the expressive power of quantum communication (which, after having had the opportunity to review some of the literature, seems to be an established term in the field) with respect to game theory. But the original article of yours and your initial responses seemed to imply that quantum channels could do things in principle that classical channels could not (quote: “Can we do any better? No, if we live in a world governed by classical physics. Yes, if we live in the world we actually inhabit—-the world of quantum mechanics.”); something that I have not seen backed up so far.
Tom West 10.25.04 at 4:53 am
I have to agree with Walt here, but I’m not impressed by either of the main participants.
Have I missed something here or this simply a long argument of the semantic meaning of the word “communication”?
Lurch 10.25.04 at 5:16 am
Reimer, the debate over speed is simply an example to illustrate why the quantanglment is not communication. Using the Alpha Centari example, it is physically impossible for any action taken on AC to have any effect whatsoever here on earth in less than four years. The quantum channel occurs instantaenously. Thus, their is no causual link between events on alpha centuri and whether my partical spits out a yes or a no. Since I imagine a causual relationship is implicit in communication, no communication occurs.
Tom, you are right, it is, fundamentally, a long debate on the semantics of communication, but it is nonetheless a fairly important question. The rigerous definition of things like communication is a foundational step to a lot of interesting and possibly useful things.
Glenn Bridgman 10.25.04 at 5:18 am
Err, that last lurch comment was actually me. However, I still want to eat your brains.
Randolph Fritz 10.25.04 at 7:34 am
Let me suggest that what quantum entanglment offers is not what we usually call communication but rather synchonicity. Thus, it can distort “identity”–apparently distinct particles can have invisible connections, though there is no mediating particle.
Does it matter? Perhaps. The plenum is, at base, quantum mechanical. So perhaps entanglement is already present and we just have not yet discovered it.
Zak Catem 10.25.04 at 11:53 am
Let me see if I can clear this up for everyone. Landsburg has figured out that quantum entanglement is like having two coins that almost always flip the same way. He’s figured out that having two coins that almost always flip the same way would enable people to coordinate their decisions based on the results flipping the magic coins. For some reason, he believes that this is non-obvious and worth bringing to the attention of libertarian bloggers. He also insists that communication doesn’t take place, based, apparently, on a peculiar definition of communication in which the use of a means which allows each of two people to infer information about what the other is doing doesn’t count as communication.
He also believes that the implications of these insights are worthy of deeper examination, despite the fact that the sole implication of this demonstration of quantum game theory is that two people with access to information about what each other is doing can generally do well at games whose objective is essentially to guess what each other is doing.
I can definitely see why someone would bother to spend two or three days defending that thesis. The most useful conclusion that I’ve been able to draw from all this is that it’s very hard to persuade a mother that her baby is ugly, particularly when she’s just published an article on the internet describing how beautiful her little treasure is.
Steven E. Landsburg 10.25.04 at 2:28 pm
Landsburg has figured out that quantum entanglement is like having two coins that almost always flip the same way.
No, it’s nothing at all like that.
He’s figured out that having two coins that almost always flip the same way would enable people to coordinate their decisions based on the results flipping the magic coins.
Completely wrong. The magic coins you describe would not allow you to beat the same 75% win rate that you can achieve without the coins.
For some reason, he believes that this is non-obvious
On the contrary, it’s completely obvious. The non-obvious (if you haven’t thought about it before) part is that with entanglement you can do better than you can with your “magic coins”.
He also insists that communication doesn’t take place, based, apparently, on a peculiar definition of communication in which the use of a means which allows each of two people to infer information about what the other is doing doesn’t count as communication.
No, neither of the people in this story can infer anything at all about what the other is doing.
There have been a lot of thoughtful posts to this thread, and a lot of semi-confused posts, but I believe this one takes the prize for most confusion per square inch.
Joshua W. Burton 10.25.04 at 2:40 pm
Daniel and John each flip a penny, in isolated rooms. Two bits of entropy, no correlation. No communication has occurred: do we all agree?
Daniel and John each carry a chess pawn, chosen from my hands (one black, one white) before they entered the isolated rooms. One bit of entropy now resides in each room, but there is one (negative) bit of correlation, so that _even though neither of them knows what pawn he has_ there is only one bit of entropy in the system. When Daniel and John look at their pawns, a past correlation is realized, but no new communication has occurred; this is a “prior agreement” game, not a “communication” game: do we all still agree?
Daniel and John each carry a qubit, correlated in a singlet state when I handed them out before they entered the isolated rooms. One bit of entropy in each room, and _two_ negative bits of correlation. That’s weird, and classically impossible, but it’s still just prior agreement; we’re not in any sort of a “communication” game.
Thanks to 200% correlation of their respective random bits (aka entanglement), D & J can designate _two_ questions to which they have “prior agreement” matched answers, in the event that those respective questions are asked. They could achieve exactly the same thing by simply carrying two classical correlated bits (use the matched pawns if the question is dogs; matched knights if it’s cats). However, in the quantum case, they get random results if the two questions don’t match.
So, we have prior agreement that is oddly incomplete from a classical perspective: D & J know “too much” for confederates who share a classical bit, and “not enough” for confederates who share _two_ classical bits.
What has the physicists stirred up (including lurkers like me who’ve been muttering “I will not post, I will not post!” in a valiant but doomed fight against human frailty) is the trope that runs, roughly: “Physicists have some halfwit technical notion of ‘information transfer’ or ‘communication’, and they’re so blinded by it that they can’t see that we mean ‘communication in the ordinary sense, thank you Lurch’.” That’s not the physicist’s hangup at all: nothing I’ve said above has anything to do with digging out of rooms or Alpha Centauri.
The physicists are merely pointing out that a qubit game is a form of “prior agreement” game, not a “communication” game. (And we mean it: the qubit game lies somewhere between the one-pawn and the two-pawn games, neither of which involves any robots at all.) Whether the qubit game is an interesting game of any sort, I couldn’t say.
Joshua W. Burton 10.25.04 at 2:50 pm
_…the qubit game lies somewhere between the one-pawn and the two-pawn games…_
Between the “matched pawns” and the “matched pawns, matched knights” games, to be more consistent.
Zak Catem 10.25.04 at 3:44 pm
Stephen, I should clarify. By “the same way, I meant each coin flips the same way as the other coin every time. This should have been fairly obvious despite my lack of clarity, but judging from your response to my second point, it wasn’t obvious enough.
Clearly, this doesn’t map directly onto your solution, but we can get very close. If we’re asked about dogs, we flip the coin and say yes if it’s heads, no if it’s tails. If we’re asked about cats, one of us agrees to answer no if it’s heads. Thus we have an imaginary mechanism which provides roughly as much information exchange as yours, but works one hundred percent of the time. The coin flips can match only 85% of the time if that makes you feel better.
As for your claim that neither party can infer what the other is doing, this is obviously untrue. If your quantum system worked one hundred percent of the time, then party A would be able to reduce the possible states in room B to 2, i.e. if A is being told to say yes, he knows either that B has been asked about dogs and answered yes, or about cats and answered no. This has halved the state space for the other room, thus party A has acquired information about the state of party B.
A final question, from sheer curiosity. Since the answers of both parties are now determined by the state of the particle, why are the particles’ states not considered information? It seems to me that what you’ve done is add an extra piece of information to the external data given to the two parties, so that they are now given their questions as usual, but are also provided with the answers. That the answers came from a different external source than the questions shouldn’t have any relevance from the perspective of game theory, should it?
Steven E. Landsburg 10.25.04 at 3:48 pm
Joshua Burton: They could achieve exactly the same thing by simply carrying two classical correlated bits (use the matched pawns if the question is dogs; matched knights if it’s cats).
I do not believe this. I don’t believe any number of classically correlated bits will allow them to beat the game more than 75% of the time. In particular, your “pawns if it’s cats, knights if it’s dogs” strategy quite clearly doesn’t work.
Steven E. Landsburg 10.25.04 at 3:55 pm
zak catem: Clearly, this doesn’t map directly onto your solution, but we can get very close. If we’re asked about dogs, we flip the coin and say yes if it’s heads, no if it’s tails. If we’re asked about cats, one of us agrees to answer no if it’s heads. Thus we have an imaginary mechanism which provides roughly as much information exchange as yours, but works one hundred percent of the time. The coin flips can match only 85% of the time if that makes you feel better.
If you think about it, you will see that this is quite entirely wrong. The strategy you’re suggesting will not allow us to win more than 75% of the time, even if the coins are perfectly matched.
I did understand what you meant, but I am quite certain you are mistaken. Magic coins don’t improve our performance. Entangled particles do.
Steven E. Landsburg 10.25.04 at 4:01 pm
Zak: Let me spell this out more clearly, in case I was too vague.
Suppose you’re agreed to say yes if you’re asked about cats. Now think about what happens if you’re asked about cats and I’m asked about dogs. Then we both say yes, and we lose. And that happens 25% of the time.
Reimer Behrends 10.25.04 at 4:15 pm
Glenn, causal relationship (as I think I explained before) is not necessary for communication to occur in the information-theoretic sense. That may be non-intuitive, but information theory specifically abstracts away the physical realities. Whether communication occurs through body language, smoke signals, ethernet, quantum entanglement, or magic rituals is irrelevant for information theory as long as it fits the mathematical model. (Note that use of quantum mechanics has consequences with respect to, say, channel capacity, but I don’t believe that is at issue here.)
Joshua, what Steven said. Any number of public coin tosses (which is what the pawn and knight sharing represents) cannot affect your information entropy regarding your partner’s state (which, as you recall, is drawn from { cat, dog } x { yes, no } and does not include or is correlated with the state of any chess piece).
Joshua W. Burton 10.25.04 at 4:16 pm
_I do not believe this. I don’t believe any number of classically correlated bits will allow them to beat the game more than 75% of the time. In particular, your “pawns if it’s cats, knights if it’s dogs†strategy quite clearly doesn’t work._
Ah, I see. You’ve set up the rules of the game cleverly, so that “winning” means a match in the dogs/dogs case only. I lost track of the original problem, in all the smoke.
Yes, in this game the prior arrangement of a singlet qubit state can beat any classical prior arrangement, still _without any communication_. This comes about because of the way the two negative bits of correlation (“pawns” and “knights”, or x- and z-spin) are related by a continuous rotation in the quantum singlet case. D can look at “pnights” and “kawns” while J looks at chessmen, and (by the ingeniously contrived win/loss criterion of this game) that’s checkmate.
Daniel is just wrong. Thanks for clarifying.
Glenn Bridgman 10.25.04 at 4:44 pm
Reimer, I guess the question then is whether the game forbids the gain of information or communication. I maintain that those are not bijective concepts.
Joshua said it better than I ever could. I really wish I had the physics/information theory toolkit to discuss this properly.
dsquared 10.25.04 at 5:48 pm
Just when I thought this had died down, it flares up again …
When I left for work this morning, we had agreed that I would never learn anything about John’s answers, or about the questions he was asked (in either case), and that therefore no “information” in any important sense was passing from room to room.
We had, however, also established that over time I would learn about the correlation between John’s answers and the questions he was asked. I could learn about this through my possession of a qubit which I and John were both measuring (because our measurements would be correlated in a known manner), or I could learn about it through the descriptions given by Lurch under the communication protocol I had described.
Because I learn (and fail to learn) exactly the same things in both cases, I established that a quantum game could be analysed as a classical game with cheap, noisy talk. Or in other words the physical properties of qubits do not introduce new issues to the game considered as a mathematical entity which cannot already modelled. I established this using the argument detailed above; that in cases where classical probability and quantum probability can both be calculated, they agree, and that for games with a definite payoff matrix, classical probabilities can always be calculated. I don’t think anyone has argued against this, so it has not been established that I am “just wrong”. In fact, AFAICS, I’m not.
For the physicists present, I’d note that one indicator that, in the game theoretic sense, the qubit channel is equivalent to cheap-talk communication is that it can be used to give me misinformation. For example, say that JQ takes against me and decides to cause trouble. Instead of making the measurements we agreed upon on his qubit, he measures a different observable. I am now certainly going to learn something worth knowing; as my losses build up, I will come to realise that John is no longer my friend. And I come to learn this without ever getting any information about what happened in the other room.
Mark Duty 10.25.04 at 7:35 pm
My understanding of this dispute:
It is possible to have two quantum-coins in separate locations that have different properties depending on if only one coin is flipped or if both coins are flipped.
This coordination is achieved faster than light so it is not considered “communication”
If the same coordination had been achieved by radio waves, it would be considered “communication”.
So as long as the rules say no “communication” you can beat the game by using this faster-than-light coordination.
If the rules say [no “communication” AND no faster-than-light coordination] then we are back to the original game, with these quantum-coins outlawed just like walkie-talkies.
The entire impact of quantum-coins on game theory is that non-communication games have to outlaw quantum-coins as well as slower than light communication.
It seems that D-squared is right as far as this is a pretty small impact. Certainly not a whole new world.
Steven E. Landsburg 10.25.04 at 7:43 pm
It is possible to have two quantum-coins in separate locations that have different properties depending on if only one coin is flipped or if both coins are flipped.
Wrong. No property of either coin is changed by flipping the other.
This coordination is achieved faster than light so it is not considered “communicationâ€
Wrong. The reason it is not considered communication is because no information is transmitted. The only relevance of the faster-than-light stuff is that it proves no information is transmitted.
If the same coordination had been achieved by radio waves, it would be considered “communicationâ€.
Wrong. No information is transferred whether you use radio waves or not.
Certainly not a whole new world. Well, of course not. I haven’t seen anyone in this long thread try to insinuate otherwise.
dsquared 10.25.04 at 7:58 pm
note of course, that although no information is transferred, the state of one particle is a correlation (by which I mean a proper correlation; you would expect that almost surely, in an arbitrarily long series of measurements, the pairwise results would not be the same as they would for two independent particles) between the measurement performed on one qubit and the measurement performed on the other. If this were not the case, then we would never have been able to verify the phenomenon experimentally.
Zak Catem 10.25.04 at 10:52 pm
Stephen: You’re quite correct, I apologise. It’s a side-point, though, as my main contention was that the results of particle observations do allow the participants to infer information about what the other party is doing, which I believe I’ve shown. I notice that you also haven’t addressed my final question, and I think it’s a good one. What’s the point of a game where you give the players all the information they need to win everytime? What can we possibly learn from this example?
Zak Catem 10.25.04 at 10:59 pm
Also, what happens in your quantum method when one party gets cat and one gets dog? What is the result of comparing a filtered measurement with an unfiltered one?
Steven E. Landsburg 10.25.04 at 11:03 pm
Zak: The point is that there are (speculative for sure, but not outrageously so) reasons to think that in the not too distant future, we might be communicating via channels that would make it easy for players to coordinate their actions in this particular way, as opposed to some other way. So it might be interesting to think about how the game turns out in those circumstances.
Let me point out that there is nothing intrinsically interesting about the dog/cat game itself. There is no plausible scenario, classical or quantum, in which anybody will play this game. But we still think it’s an interesting to work out optimal strategies in such games.
The quantum version of the dog/cat game is no more likely to ever be played than the classical version. But it’s still interesting to work out the results, just as in the classical case, because it might give us some insight into strategic behavior in more complicated situations that might (or might not) be likelier to arise in the future than they are right now.
Steven E. Landsburg 10.25.04 at 11:10 pm
zak: if the particles start in the state 0>1> + 1>0> (or more precisely, the state represented by that vector), and if the first party holds his measuring apparatus either vertical or at angle pi/4, while the other holds his measuring apparatus at angle either pi/8 or -pi/8, then they win a) 85% of the cat/cat trials, b) 85% of the cat/dog trials, c) 85% of the dot/cat trials and d) 85% of the dog/dog trials.
This is easy to verify if you’re comfortable with the yoga of quantum mechanics, and, of course, completely unverifiable if you’re not.
It is easy to prove that no particles obeying the laws of classical physics could ever give you that combination of correlations.
Zak Catem 10.25.04 at 11:15 pm
But it doesn’t, Steven. If we are given the answers along with the questions, the game is pointless. The only strategy is to answer as told. If by strategy you meant the way the particles are used, then that has nothing to do with game theory, and you’ve filed it under the wrong heading.
Steven E. Landsburg 10.25.04 at 11:29 pm
zak: i dont understand your last comment. we are not given the answers.
nobody disputes the (mild) interest of games like the dog/cat game as exercises in understanding, even though nobody actually ever plays them.
i put forward a different game: the dog/cat game with access to entangled particles, and i pointed out that you get a different (and perhaps unexpected) set of optimal strategies.
what’s the objection to doing that?
dsquared 10.25.04 at 11:54 pm
It is easy to prove that no particles obeying the laws of classical physics could ever give you that combination of correlations.
Just to keep this absolutely clear, this is not true without a bunch-load of other restrictions. Lurch is made entirely out of particles obeying the laws of classical physics and he could give you that combination of correlations if he wanted to.
The objection, Steven, is to publishing these things as if they were new economics, rather than interesting curiosa using old economics to do new quantum theory.
Backword Dave 10.26.04 at 10:50 am
I agree with dsquared (but then I’m guilty of the mobile phone parody, so that shouldn’t surprise anyone).
As Walt summed up Steve’s contribution:
I’ll take issue with the subordinate clause. Why is it interesting? And what does study mean here? If we were doing physics, we’d be working toward an experimental hypothesis. I don’t see one at the end of this. All we have is a model which shows off how well informed a few egg-heads who read this site and Marginal Revolution are.
Bringing in game theory may be a pedagogic tool for undergraduates, but it looks like a cul-de-sac to me.
And BTW, you may all be wrong on the FTL side of things too. There’s an explanation here (scroll down to “physics professor”). The FTL part isn’t necessary for the original set-up, but Glenn seems to be taking it as read, and I don’t think that he should. Lurch only has to move at c, like, for example, a mobile phone signal.
Mac Thomason 10.27.04 at 5:08 pm
That’s pretty unattractive. However, implying that John Edwards is the moral equivalent of David Duke, as Landsberg did yesterday, is utterly loathsome.
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