To simplify discussion it is best to think of the allocation problem as trying to match up supply and demand. Using this terminology, there are five properties that a good allocation formulation needs to have:
| 1. | Conservation of Matter: The amount received by the demanders (summed across all demanders) must be equal to the amount provided by the suppliers (summed across all suppliers). |
| 2. | Nonnegative: All quantities allocated must be positive or zero. |
| 3. | Conservation of Intent: No supplier shall provide more than it desired to supply and no demander shall receive more than it has chosen to demand. (Less, however, is possible). |
| 4. | No Loopholes: Under conditions of adequate supply, each demander should receive its stated demand. Under conditions of adequate demand, each supplier should supply its stated supply. |
| 5. | Clear Differentiation: When there is insufficient supply, extremely low priority demanders should receive little or nothing and extremely high priority suppliers should receive everything or almost everything. Conversely, when there is excess supply extremely unattractive suppliers should provide little or nothing and extremely attractive suppliers should provide the bulk of the demand. |
| 6. | Continuity: Small changes in priorities, supply and demand should cause small changes in the resulting allocations. Smoothness, which requires that small changes cause only small changes in the derivative of the allocations is also often desirable. |
With some adjustment and manipulation, the allocation logic in the previous section can be made to pass 1 through 4 above, but getting it to pass 5 is very hard. To come up with a formulation that satisfies all of these we need to move a little bit beyond simple algebra.
