Assignment valuation

<h2 id="example">Example</h2>
<p>Suppose there are three items and two agents who value the items as follows:
</p>
<table><tbody><tr><th></th><th>x</th><th>y</th><th>z</th></tr><tr><td>Alice:</td><td>5</td><td>3</td><td>1</td></tr><tr><td>George:</td><td>6</td><td>2</td><td>4.5</td></tr></tbody></table>
<p>Then the assignment-valuation <i>v</i> corresponding to the group {Alice,George} assigns the following values:
</p>
<ul><li>
  
    
      
        v
        (
        {
        x
        }
        )
        =
        6
      
    
    {\displaystyle v(\{x\})=6}
  
 - since the maximum-weight matching assigns x to George.</li>
<li>
  
    
      
        v
        (
        {
        y
        }
        )
        =
        3
      
    
    {\displaystyle v(\{y\})=3}
  
 - since the maximum-weight matching assigns y to Alice.</li>
<li>
  
    
      
        v
        (
        {
        z
        }
        )
        =
        4.5
      
    
    {\displaystyle v(\{z\})=4.5}
  
 - since the maximum-weight matching assigns z to George.</li>
<li>
  
    
      
        v
        (
        {
        x
        ,
        y
        }
        )
        =
        9
      
    
    {\displaystyle v(\{x,y\})=9}
  
 - since the maximum-weight matching assigns x to George and y to Alice.</li>
<li>
  
    
      
        v
        (
        {
        x
        ,
        z
        }
        )
        =
        9.5
      
    
    {\displaystyle v(\{x,z\})=9.5}
  
 - since the maximum-weight matching assigns z to George and x to Alice.</li>
<li>
  
    
      
        v
        (
        {
        y
        ,
        z
        }
        )
        =
        7.5
      
    
    {\displaystyle v(\{y,z\})=7.5}
  
 - since the maximum-weight matching assigns z to George and y to Alice.</li>
<li>
  
    
      
        v
        (
        {
        x
        ,
        y
        ,
        z
        }
        )
        =
        9.5
      
    
    {\displaystyle v(\{x,y,z\})=9.5}
  
 - since the maximum-weight matching assigns z to George and x to Alice.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Shapley, Lloyd S. (1962). "Complements and substitutes in the opttmal assignment problem". Naval Research Logistics Quarterly. 9 (1): 45–48. doi:10.1002/nav.3800090106. <a href="https://ideas.repec.org/a/wly/navlog/v9y1962i1p45-48.html" target="_blank">https://ideas.repec.org/a/wly/navlog/v9y1962i1p45-48.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Lehmann, Benny; Lehmann, Daniel; Nisan, Noam (2006-05-01). "Combinatorial auctions with decreasing marginal utilities". 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. <a href="http://www.sciencedirect.com/science/article/pii/S089982560500028X" target="_blank">http://www.sciencedirect.com/science/article/pii/S089982560500028X</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Benabbou, Nawal; Chakraborty, Mithun; Elkind, Edith; Zick, Yair (2019-08-10). "Fairness Towards Groups of Agents in the Allocation of Indivisible Items". {{cite journal}}: Cite journal requires |journal= (help) <a href="https://hal.sorbonne-universite.fr/hal-02155024" target="_blank">https://hal.sorbonne-universite.fr/hal-02155024</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>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. Vol. 12283. pp. 32–46. arXiv:2003.07060. doi:10.1007/978-3-030-57980-7_3. ISBN 978-3-030-57979-1. S2CID 208328700. <a href="978-3-030-57979-1" target="_blank">978-3-030-57979-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

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Assignment valuation