Say $400 is to be divided among creditors who claim $100, $200, and $300. They are awarded $50, $125, and $225.

To see that this is consistent, we have to think about each pair of creditors separately and check that they are dividing their collective share appropriately.

The first and second creditors assert claims of $100 and $200, respectively, to a collective share of $175. The second claim is reduced to $175. Now $100 is disputed, and $75 is undisputed. The disputed $100 is divided equally. Thus the first creditor gets $50, and the second gets $125. This agrees with the prescribed awards.

The first and third creditors assert claims of $100 and $300, respectively, to a collective share of $275. The second claim is reduced to $275. Now $100 is disputed, and $175 is undisputed. The disputed $100 is divided equally. Thus, the first creditor gets $50, and the third gets $225. This agrees with the prescribed awards.

The second and third creditors assert claims of $200 and $300, respectively, to a collective share of $350. Thus the second concedes $150 to the third, the third concedes $50 to the second, and the remaining $150 is in dispute. The disputed $150 is divided equally ($75 to each), so the second creditor gets $50 + $75 = $125, and the third gets $150 + $75 = $225. Again, this agrees with the prescribed awards.

Because all three pairs divide their collective shares in accordance with the stated principles, the allocation is consistent. Because Aumann and Maschler have proved there can be only one consistent allocation, there can be no other.