2.4 Nash Demand Game

Bargaining

In startegic form theory, a Nash demand game is one of infinite strategies where each player $i\in N$ has a demand $d_i\in \mathbb{R}$ for an aggregate quantity of a good $Q$, such that
$u_i(d_i,d_{-i})=\{\begin{array}{ll}{d}_i& \text{if }{\sum }_{i=1}^{|N|}{d}_i\le Q\\ 0& \text{if }{\sum }_{|N|}^{i=1}{d}_i>Q\end{array}.$
In a Nash demand game, Nash equilibrium occurs when aggregate demand is met exactly with the aggregate quantity of the good, such that ${\sum }_{i=1}^{|N|}{d}_i=Q$.

Examples

Pizza Sharing

A / B	1	2	3	4	5	6	7
1	3.5, 3.5	1, 6	1.5, 5.5	2, 5	2.5, 4.5	3, 4	3.5, 3.5
2	6, 1	3.5, 3.5	2, 5	3, 4	3, 4	3.5, 3.5
3	5.5, 1.5	5, 2	3.5, 3.5	3, 4	3.5, 3.5
4	5, 2	4, 3	4, 3	3.5, 3.5	4, 3	4.5 ,3.5	5, 2
5	4.5, 2.5	4, 3	3.5, 3.5	4, 3	3.5, 3.5
6	4, 3	3.5, 3.5	3, 4	3.5, 4.5		3.5, 3.5
7	3.5, 3.5						3.5, 3.5