"Live and Let Live" was a phenomenon in the early stages of World War I in which the trench warfare combatants tacitly agreed to a non-agression pact.
In essence, the grunt soldiers on both sides of the Western Front quickly realized three things: (1) The enemy can make life hell for us, (2) we can make life hell for the enemy, and (3) the war isn't ending anytime soon. Thus, soldiers took up a tit-for-tat strategy -- we'll only shoot at you if you shoot at us. This resulted in very few offensive hostilities, at least for a time.
To look at this in-depth, we need to use Game Theory. The actions on the Western Front represented a Prisoner's Dilemma in which two players can either cooperate with each other or defect. (In the classic Prisoner's Dilemma case, there's two prisoners who must decide whether to rat on their counterpart. If they both rat, they'll get medium-length sentences; if they both stay quiet, they'll get light sentences. However, if one stays quiet and the other testifies, the "informant" goes free, while the "sucker" gets a long prison sentence. What makes this interesting is that the pair ought to cooperate to get the light sentences, but they'll both have the tempation of ratting, and in the end, they with both likely rat on each other.)
Back to WWI. In this scenario, both the German and the Allied troops can either shoot or not shoot at their opponents. Here's the range of possibilities:
Germans shoot; Allieds shoot
Because of reasons too numerous to explain in this node (machine guns, stupid commanders, etc.), this combination would result in heavy losses on both sides, and neither would gain ground. For the grunts, this is bad.
Germans cease fire; Allieds cease fire
This would result in zero losses on both side, and neither would gain ground. Sure, you'd fire shots now and then so commanders didn't think you were treasonous, but you'd intentionally minimize damage to the enemy. The war would continue indefinitely, which wasn't good, but eventually you'd get discharged, right?
One side shoots; the other side ceases fire
The aggressive side would have few losses and gain ground, while the pacifist side would have heavy losses and lose ground. The temptation for the aggressive side is real; both the Allieds and the Germans did want to eventually win the war.
Spelling out the results:
(Gain ground, no losses) == TEMPT
(No losses, neither side gains ground) == PEACE
(Heavy losses, neither side gains ground) == DEADLOCK
(Heavy losses, lose ground) == SUCKER
What's important here is the order of prefences:
TEMPT > PEACE > DEADLOCK > SUCKER
That is, the Allied troops and the German troops would each prefer PEACE to DEADLOCK.
This fits the framework of the Prisoner's Dilemma. But how did "Live and Let Live" arise? You'll recall that the Prisoner's Dilemma resulted in both sides tempting into defecting from the cooperative pact. In other words, assuming both armies choose the PEACE action, then each army would be tempted into opening fire -- hoping that they could gain TEMPT, or at worst, not get the SUCKER.
This is true for one iteration of the Prisoner's Dilemma, as well as in trench warfare. But if the game is played over and over again -- as it was during each day of the trench warfare -- both participants realize that a stable PEACE was better than a stable DEADLOCK. The TEMPT and SUCKER aren't stable; whoever gets SUCKER sure won't get suckered again, and next time, it'll be DEADLOCKed.
So PEACE won out.
(A British staff officer was) astonished to observe German soldiers walking about within rifle range behind their own line. Our men appeared to take no notice. I privately made up my mind to do away with that sort of thing when we took over; such things should not be allowed. These people evidently did not know there was a war on. Both sides apparently believed in the police of "live and let live."
(Dugdale, p. 94)
The inability of the two sides to verbally communicate made little difference. They communicated through their artillery shells and sniper fire. Artillery guns would continue to fire, but at a consistent time and at a consistent place, serving a double purpose -- showing off your strength while minimizing enemy casualties. And, as new troops rotated into the front lines, the departing soldiers would remind the Greenhorns of the new rules: "Mr. Bosche ain't a bad fellow. You leave 'im alone; 'e'll leave you alone." (Gillon, p. 77)
Naturally, the cooperation did not take place immediately; after a bloody beginning to the war, a natural event would cause a lull in the fighting (a storm, for instance), and neither side would be anxious to resume the bombardment.
This went on throughout the first year of the war, from late 1914 until sometime in 1915. (A celebrated incident was the Christmas Truce of 1914, when both sides came out of their trenches singing Yuletide carols.) The troops' commanders had no idea what was going on -- they attributed the lack of fighting to a morale problem -- but they couldn't be blamed for their ignorance, as game theory wouldn't be invented for another few decades.
The Allied commanders ended up breaking the cooperation quite unintendedly. Britain entered the trench warfare, and they hoped to show off their fighting stuff by directing small raids. These couldn't be faked by the grunts like sniper fire could be, and the resulting uncertainty ended the unofficial "Live and Let Live" policy.
Sources:
R. Axelord, "The Evolution of Cooperation," 1984
G. Dugdale, "Langemarck and Cambrai," 1932
S. Gillon, "The Story of the 29th Division," no date.