Fig. 5: Physical solutions for profit distribution in a game of characteristic function form.
From: A balance for fairness: fair distribution utilising physics
a The taxi problem: Initial liquid levels were E1 = −9, E2 = −3, E3 = −3, C1 = −7, C2 = −7 and C3 = −7, respectively (see ‘Method’). Owing to gravity (stand it vertically), the system could reach equilibrium state E1 = E2 = E3 = −5. Therefore, the displacements from each initial liquid level were y1 = +4, y2 = −2 and y3 = −2, respectively. We obtained the allocations x1 = 11, x2 = 5 and x3 = 5 from xi = yi + 7. Person 1 would receive $11 and Persons 2 and 3 would both receive the allocation of $5 in this case. b The bankruptcy problem (M = 100 cases): initial liquid levels were E1 = E2 = E3 = − 200/3 and C1 = C2 = C3 = −100/3, respectively (see ‘Method’). Owing to the volume conservation of both liquids, the liquid levels of the cylinders stopped moving. Therefore, the displacements y1, y2 and y3 were all 0. We obtained the allocations x1 = x2 = x3 = 100/3 from xi = yi + 100/3. Every creditor would receive the same allocation of 100/3 (equal division). c The bankruptcy problem (M = 200 cases): initial liquid levels were E1 = −100/3, E2 = E3 = − 100 −100/3 and C1 = C2 = C3 = −200/3, respectively (see ‘Method’). Although the maximum complaint E1 gradually decreased, heading toward the equilibrium level of −100, it stopped at the level of −50 because C1 also reached −50 from the initial liquid level of −200/3 due to the interlocking between E1 and C1. Therefore, the displacements y1, y2 and y3 were −50/3, 25/3 and 25/3, respectively. We obtained the allocations x1 = 50, x2 = 75 and x3 = 75 from xi = yi + 200/3. Creditor 1 would receive 50 and Creditors 2 and 3 would both receive the allocation of 75. d The bankruptcy problem (M = 300 cases): Initial liquid levels were E1 = 0, E2 = 100, E3 = 200 and C1 = C2 = C3 = − 100, respectively (see ‘Method’). Although the maximum complaint E1 gradually decreased, heading toward the equilibrium level of −100, it stopped at the level of −50 because C1 also reached −50 from the initial level of −100 due to the interlocking between E1 and C1. Therefore, the displacements y1, y2 and y3 were −50, 0 and +50, respectively. As a result, we obtained the allocations x1 = 50, x2 = 100 and x3 = 150 from the equation xi = yi + 100. Creditors 1, 2 and 3 would receive 50, 100 and 150, respectively (proportional division).
