Table 2 Heterogeneous cake-cutting using the weighted Boltzmann division.

Player	\({\mathbf{E}}_{{\mathbf{j}}}\)	\({\mathbf{D}}_{{\mathbf{j}}}\)	Homogeneous \(\left( {{\mathbf{N}}_{{\mathbf{j}}} } \right)\)	Heterogeneous					Diff
Player	\({\mathbf{E}}_{{\mathbf{j}}}\)	\({\mathbf{D}}_{{\mathbf{j}}}\)		Vanilla \(\left( {{\mathbf{N}}_{{\mathbf{j}}}^{{\mathbf{1}}} } \right)\)	Chocolate \(\left( {{\mathbf{N}}_{{\mathbf{j}}}^{{\mathbf{2}}} } \right)\)	Strawberry \(\left( {{\mathbf{N}}_{{\mathbf{j}}}^{{\mathbf{3}}} } \right)\)	Broccoli \(\left( {{\mathbf{N}}_{{\mathbf{j}}}^{{\mathbf{4}}} } \right)\)	Total \(\left( {{\mathbf{N}}_{{\mathbf{j}}} } \right)\)	Diff
1	5	4	12.17	2.61	2.27	3.29	5.50	13.67	1.50
2	10	10	14.06	6.03	2.62	3.80	0.00	12.44	−1.62
3	20	24	18.75	0.00	13.94	0.00	0.00	13.94	−4.81
4	25	34	21.66	9.26	0.00	0.00	19.50	28.75	7.10
5	40	53	33.36	7.11	6.17	17.91	0.00	31.19	−2.17
Total	100	125	100	25	25	25	25	100	0

\(\mathrm{E}_{\mathrm{j}}\) = Contribution to cake production of player j
\(\mathrm{D}_{\mathrm{j}}\) = Need for cake of player j
\(\mathrm{N}_{\mathrm{j}}^{\mathrm{i}}\) = Cake share of flavor i allocated to player j
Diff (Difference) = Heterogeneous \(\left( {\mathrm{N}_{\mathrm{j}} } \right)\)—Homogeneous \(\left( {\mathrm{N}_{\mathrm{j}} } \right)\).

Quick links

Search