Coordinating processes with an unreliable channel

Alice and Bob are generals in an army together. One night, their battalions are stationed respectively on two mountain peaks separated by a valley. The valley is occupied by the opposing army, whose forces are stronger than either of Alice or Bob's battalions alone. However, Alice and Bob are stronger if they join forces. Alice and Bob can only communicate by a carrier pigeon with a poor sense of direction.

Bob writes a letter to Alice. "Let us mount a joint attack on the enemy tomorrow. If you do not show up, my battalion will be decimated. Please reply to confirm you have received this letter; if you do not reply, I will not show up".

Alice receives the letter. She would like to show up to the joint attack. She responds, "I confirm that I received your last message. But suppose the pigeon gets lost and this letter is never delivered. You will not show up, and my battalion will be decimated. Please reply to confirm that you have received this letter; if you do not reply, I will not show up"

Bob receives the letter and responds, "I confirm that I received your last message. But suppose the pigeon gets lost and this letter is never delivered. You will not show up, and my battalion will be decimated. Please reply to confirm that you have received this letter; if you do not reply, I will not show up"

This back-and-forth continues long into the night, and neither general able to fully confirm both parties' commitment. They remain deadlocked this way until morning, when the opportunity to attack has passed.