Assignment valuation

In economics, assignment valuation is a kind of a utility function on sets of items. It was introduced by Shapley[1] and further studied by Lehmann, Lehmann and Nisan,[2] who use the term OXS valuation (not to be confused with XOS valuation). Fair item allocation in this setting was studied by Benabbou, Chakraborty, Elkind, Zick and Igarashi.[3][4]

Assignment valuations correspond to preferences of groups. In each group, there are several individuals; each individual attributes a certain numeric value to each item. The assignment-valuation of the group to a set of items S is the value of the maximum weight matching of the items in S to the individuals in the group.

The assignment valuations are a subset of the submodular valuations.

## Example

Suppose there are three items and two agents who value the items as follows:

| x  | y  | z

|
|
|

Alice:  | 5  | 3  | 1
George:  | 6  | 2  | 4.5

Then the assignment-valuation v corresponding to the group {Alice,George} assigns the following values:

* v({x})=6 \- since the maximum-weight matching assigns x to George.
* v({y})=3 \- since the maximum-weight matching assigns y to Alice.
* v({z})=4.5 \- since the maximum-weight matching assigns z to George.
* v({x,y})=9 \- since the maximum-weight matching assigns x to George and y to Alice.
* v({x,z})=9.5 \- since the maximum-weight matching assigns z to George and x to Alice.
* v({y,z})=7.5 \- since the maximum-weight matching assigns z to George and y to Alice.
* v({x,y,z})=9.5 \- since the maximum-weight matching assigns z to George and x to Alice.

## References

1. ↑ Shapley, Lloyd S. (1962). "Complements and substitutes in the opttmal assignment problem" (in en). Naval Research Logistics Quarterly 9 (1): 45–48. doi:10.1002/nav.3800090106. https://ideas.repec.org/a/wly/navlog/v9y1962i1p45-48.html.
2. ↑ Lehmann, Benny; Lehmann, Daniel; Nisan, Noam (2006-05-01). "Combinatorial auctions with decreasing marginal utilities" (in en). Games and Economic Behavior. Mini Special Issue: Electronic Market Design 55 (2): 270–296. doi:10.1016/j.geb.2005.02.006. ISSN 0899-8256. http://www.sciencedirect.com/science/article/pii/S089982560500028X.
3. ↑ Benabbou, Nawal; Chakraborty, Mithun; Elkind, Edith; Zick, Yair (2019-08-10) (in en). Fairness Towards Groups of Agents in the Allocation of Indivisible Items. https://hal.sorbonne-universite.fr/hal-02155024.
4. ↑ Benabbou, Nawal; Chakraborty, Mithun; Igarashi, Ayumi; Zick, Yair (2020). Finding Fair and Efficient Allocations When Valuations Don't Add Up. Lecture Notes in Computer Science. 12283. pp. 32–46. doi:10.1007/978-3-030-57980-7_3. ISBN 978-3-030-57979-1.

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