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It is the acknowledgement of the receipt of the last message that is important. Wikipedia is not entirely clear on this one. The source of the problem are highe

It is the acknowledgement of the receipt of the last message that is important. Wikipedia is not entirely clear on this one. The source of the problem are higher-order statements: "A knows that B knows that A knows ...". Suppose A sends the message to B, B sends an ack to A but A's ack back to B is lost. Now both A and B know the time of the attack but B considers it possible that [A does not know that B knows]. B thinks if A really does not know that B knows then A may not attack out of the consideration of the possibility that B does not know. And this possibility may make B call off the attack even if B knows. Each round of successful exchange increase the order of statements that are known, but there is a statement of higher order which is still not known and which can cause the attack to unravell inductively. Look up 'Rubinstein's e-mail game' for a choice of payoffs that causes no attack to happen for any positive probability, howsoever small, of a message getting lost.

vacri 9y ago
There comes a point in the chain where both sides have got the initial message, and confirmation that the other side has confirmed the message. message 1 -> attack! message 2 -> sure! message 3 -> I got your sure! At this point, general A knows the message got through message 4 -> I got your confirmation! At this point, general B knows that general A got the reply message 5 -> I got your confirmation confirmation! At this point, general A is sure that both parties know when the attack is. General A knows that the confirmation got through, and Gen A is good to go. message 6 -> conf^n! At this point, General B knows that the confirmation got through, and Gen B is good to go. message 7 => icing Even if message 6 doesn't get through, message 2 and 4 have, so general A knows that both generals know. Message 3 tells general B that both generals know - all that's missing is the nice-to-have confirmation. You don't need to keep on confirming the last message ad infinitum, because that doesn't add new data about the original message and who knows it. If messenger 77 were to be captured by the city, it's not like the generals would call off the attack because they weren't sure of the timing info. Perhaps another way of putting it: this isn't a continuous string of information that gets added to. It's a set of discrete parcels of information. The additional messages only add information about the veracity of the later parcels, not the initial ones.
- maffydub 9y ago
  The point is that it's not just about knowledge - it's about agreement. A and B must not attack alone, i.e. A must not attack if B is not going to attack, and B must not attack if A is not going to attack. Now, message 1 (from A to B) says "attack!". So B knows that A wants B to attack. ...but A mustn't attack alone, so A won't attack unless A receives B's confirmation. So B sends message 2 ("sure!"). ...but B mustn't attack unless B knows that A is going to attack, and A will only attack if A has received message 2. So A has to send another message to confirm receipt of message 2. ...and so on. None of these confirmations are simply "nice-to-have" - if you need to be 100% sure that you're both going to attack, you need to ensure that every confirmation you send has been received by your peer.
  - vacri 9y ago
    "Dear diary. I have just sent the 537th messenger confirming that we are attacking together tomorrow morning at 9. I'm just waiting on confirmation." You have that confirmation of agreement in the third round of messages. Receipt of the fourth message proves that both sides have knowledge and have agreed, aware of the other's intent. That's the message you have to prove has arrived for surety. Once you've done enough messages to prove message #4, you're good to go.

vacri 9y ago

There comes a point in the chain where both sides have got the initial message, and confirmation that the other side has confirmed the message. message 1 -> attack! message 2 -> sure! message 3 -> I got your sure! At this point, general A knows the message got through message 4 -> I got your confirmation! At this point, general B knows that general A got the reply message 5 -> I got your confirmation confirmation! At this point, general A is sure that both parties know when the attack is. General A knows that the confirmation got through, and Gen A is good to go. message 6 -> conf^n! At this point, General B knows that the confirmation got through, and Gen B is good to go. message 7 => icing Even if message 6 doesn't get through, message 2 and 4 have, so general A knows that both generals know. Message 3 tells general B that both generals know - all that's missing is the nice-to-have confirmation. You don't need to keep on confirming the last message ad infinitum, because that doesn't add new data about the original message and who knows it. If messenger 77 were to be captured by the city, it's not like the generals would call off the attack because they weren't sure of the timing info. Perhaps another way of putting it: this isn't a continuous string of information that gets added to. It's a set of discrete parcels of information. The additional messages only add information about the veracity of the later parcels, not the initial ones.

maffydub 9y ago
The point is that it's not just about knowledge - it's about agreement. A and B must not attack alone, i.e. A must not attack if B is not going to attack, and B must not attack if A is not going to attack. Now, message 1 (from A to B) says "attack!". So B knows that A wants B to attack. ...but A mustn't attack alone, so A won't attack unless A receives B's confirmation. So B sends message 2 ("sure!"). ...but B mustn't attack unless B knows that A is going to attack, and A will only attack if A has received message 2. So A has to send another message to confirm receipt of message 2. ...and so on. None of these confirmations are simply "nice-to-have" - if you need to be 100% sure that you're both going to attack, you need to ensure that every confirmation you send has been received by your peer.
- vacri 9y ago
  "Dear diary. I have just sent the 537th messenger confirming that we are attacking together tomorrow morning at 9. I'm just waiting on confirmation." You have that confirmation of agreement in the third round of messages. Receipt of the fourth message proves that both sides have knowledge and have agreed, aware of the other's intent. That's the message you have to prove has arrived for surety. Once you've done enough messages to prove message #4, you're good to go.