3 Persons, 2 Cuts: A Maximin Envy-Free and a Maximally Equitable Cake-Cutting Algorithm

Steven J. Brams, Peter S. Landweber
2018 Social Science Research Network  
We describe a 3-person, 2-cut envy-free cake-cutting algorithm, inspired by a continuous moving-knife procedure, that does not require that the players continuously move knifes across the cake. By having the players submit their value functions over the cake to a referee-rather than move knives according to these functions-the referee can ensure that the division is not only envy-free but also maximin. In addition, the referee can use the value functions to find a maximally equitable division,
more » ... hereby the players receive equally valued shares that are maximal, but this allocation may not be envy-free. 11 cuts that an earlier 4-person moving-knife algorithm of Brams, Taylor, and Zwicker (1997) required. Besides 3-person and 4-person algorithms that use relatively few cuts, envy-free algorithms that work for any number n of players have been developed. These include an n-person moving-knife algorithm (Brams and Taylor, 1995) , which uses a finite number of cuts, but this number is unbounded-an upper bound cannot be specified independently of the preferences of the players-and an algorithm based on Sperner's Lemma (Su, 1999) , which requires convergence to an exact division so also is unbounded. 2 Aziz and Mackenzie ( 2017 ) provided an algorithm that has two advantages: It does not depend on continuously moving knives or convergence, and it is finite and bounded. Its disadvantage is that its bounds are presently extremely large, making it, like Brams and Taylor's algorithm, computationally complex; in particular, it is not solvable in polynomial time. Recently, it was modified and extended to chore division (Dehghani et al., 2018) , and a simplified 4-person discrete cake-cutting algorithm was found for four players (Amanatidis et al., 2018) . In this paper, we return to the 3-person cake-cutting problem but provide an envyfree algorithm that • does not require moving knives (unlike Stromquist's, Brams, Taylor, and Zwicker's, and Brams and Barbanel's algorithms);
doi:10.2139/ssrn.3126935 fatcat:ulwsj2esw5exdfknbanvlw3liq