Table 1 Homogeneous cake-cutting using the Boltzmann division and proportionality (Egalitarian, Prop I (Capitalist), Prop II (Socialist)).

Player	E_j	D_j	Boltzmann (N_j)	Def	Utility	Egalitarian (N_j)	Def	Utility	Prop I (N_j)*	Def	Utility	Prop II (N_j)**	Def	Utility
1	5	4	12.17	8.17	0.9955	20	16	0.9999	5	1	0.8483	3.20	−0.80	0.6640
2	10	10	14.06	4.06	0.8866	20	10	0.9640	10	0	0.7616	8.00	−2.00	0.6640
3	20	24	18.75	−5.25	0.6535	20	−4	0.6823	20	−4	0.6823	19.20	−4.80	0.6640
4	25	34	21.66	−12.34	0.5628	20	−14	0.5286	25	−9	0.6263	27.20	−6.80	0.6640
5	40	53	33.36	−19.64	0.5576	20	−33	0.3604	40	−13	0.6380	42.40	−10.60	0.6640
Total utility			3.66/5.00			3.54/5.00			3.56/5.00			3.32/5.00

\(\mathrm{E}_{\mathrm{j}}\) = Contribution to cake production of player j
\(\mathrm{D}_{\mathrm{j}}\) = Need for cake of player j
Def (Deficiency) = Allocated share of cake \(\left( {\mathrm{N}_{\mathrm{j}} } \right)\)—Need \(\left( {\mathrm{D}_{\mathrm{j}} } \right)\)
*Capitalist: Players receive cake units linearly proportional to their contributions \(\left( {\mathrm{E}_{\mathrm{j}} } \right)\)
**Socialist: Players receive cake units linearly proportional to their needs (\(\mathrm{D}_{\mathrm{j}}\)).

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