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In general: Suppose the fallback positions are x slices for Alice, and y slices for Bob, and they get n slices total if they reach a deal. Then we can figure out the split by defining the gain, g, as the additional pieces they get if they reach a deal. So g=n-(x+y). Then the fair split should be x+g/2 for Alice and y+g/2 for Bob. Another kind of situation is if there are 3 people, Alice, Bob, and Charlie, with fallbacks x, y, z respectively. Suppose they all need to reach an agreement to get n slices. Then g=n-(x+y+z) and Alice, Bob, and Charlie should get x+g/3, y+g/3, z+g/3 respectively. Another possible situation is if there are 3 people, but they have unequal roles. Alice only needs to make a deal with at least one of Bob or Charlie, to get n slices. If Bob and Charlie both agree to the deal, then the total is still n slices. Intuitively, this would make Bob and Charlie less important, so they should get less of the gain. The split here should be x+2g/3 for Alice, y+g/6 for Bob and z+g/6 for Charlie, if they're all in on the deal. The reasoning there is that if we're adding games together, then we should also add the payoffs together. We can make a Alice&(Bob|Charlie) game by adding together Alice&Bob + Alice&Charlie - Alice&Bob&Charlie. Similarly, the payoffs should add like this: (g/2, g/2, 0) + (g/2, 0, g/2) - (g/3, g/3, g/3) = (2g/3, g/6, g/6) |