This is the true story of the actual use of a formal, decentralized division procedure to allocate silver heirlooms among eight grandchildren fairly and efficiently without distasteful direct monetary payments. Each grandchild's stated preferences for objects in contention were roughly...

Call b your balance function at wealth W if you are indifferent between W and a 50–50 lottery with outcomes x and b(x). Given one b, u is arbitrary on one side of W but then determined on the other. Given two b‘s, u is arbitrary between the two W′ s but then determined elsewhere....

When rational risk-averse agents must choose among and share monetary risks, it is known that efficient sharing is typically nonlinear, even with common beliefs. Wherever it is, the sharing rule may affect the choice, randomized choice may allow everyone to gain, and indeed a randomized choice...