# Assignment valuation

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In [economics](/source/Economics), **assignment valuation** is a kind of a [utility function](/source/Utility_function) on sets of items. It was introduced by [Shapley](/source/Lloyd_Shapley)[1] and further studied by Lehmann, Lehmann and Nisan,[2] who use the term **OXS valuation** (not to be confused with [XOS valuation](/source/Fractionally_subadditive_valuation)). [Fair item allocation](/source/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](/source/Maximum_weight_matching) of the items in *S* to the individuals in the group.

The assignment valuations are a subset of the [submodular valuations](/source/Submodular_valuation).

## 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"](https://ideas.repec.org/a/wly/navlog/v9y1962i1p45-48.html). *Naval Research Logistics Quarterly*. **9** (1): 45–48. [doi:10.1002/nav.3800090106](https://doi.org/10.1002/nav.3800090106)

1. Lehmann, Benny; Lehmann, Daniel; Nisan, Noam (2006-05-01). ["Combinatorial auctions with decreasing marginal utilities"](http://www.sciencedirect.com/science/article/pii/S089982560500028X). *Games and Economic Behavior*. **55** (2): 270–296. Mini Special Issue: Electronic Market Design. [doi:10.1016/j.geb.2005.02.006](https://doi.org/10.1016/j.geb.2005.02.006). [ISSN 0899-8256](https://www.worldcat.org/issn/0899-8256)

1. Benabbou, Nawal; Chakraborty, Mithun; Elkind, Edith; Zick, Yair (2019-08-10). ["Fairness Towards Groups of Agents in the Allocation of Indivisible Items"](https://hal.sorbonne-universite.fr/hal-02155024)

1. Benabbou, Nawal; Chakraborty, Mithun; Igarashi, Ayumi; Zick, Yair (2020). *Finding Fair and Efficient Allocations When Valuations Don't Add Up*. Vol. 12283. Lecture Notes in Computer Science. pp. 32–46. [arXiv:2003.07060](https://arxiv.org/abs/2003.07060). [doi:10.1007/978-3-030-57980-7_3](https://doi.org/10.1007/978-3-030-57980-7_3). ISBN 978-3-030-57979-1. [S2CID 208328700](https://api.semanticscholar.org/CorpusID:208328700)

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