A division procedure F is called symmetric if, for any permutation p of (1,...,n), and for every i:Vi(F(V1,...,Vn; i)) = Vi(F(Vp(1),...,Vp(n); p−1(i)))In particular, when n=2, a procedure is symmetric if:V1(F(V1,V2; 1)) = V1(F(V2,V1; 2)) and V2(F(V1,V2; 2)) = V2(F(V2,V1; 1)) This means that agent 1 gets the same value whether he plays first or second, and the same is true for agent 2.
But the opposite is not true: it is possible that a permutation gives an agent different pieces with equal value.
The following chart summarizes the relations between the criteria: Every procedure can be made "symmetric ex-ante" by randomization.
Manabe and Okamoto[1] presented symmetric and envy-free ("meta-envy-free") deterministic procedures for two and three agents.
Nicolo and Yu[2] presented an anonymous, envy-free and Pareto-efficient division protocol for two agents.
The symmetric cut and choose procedure for two agents was studied empirically in a lab experiment.
[4] Alternative symmetric fair cake-cutting procedures for two agents are rightmost mark[5] and leftmost leaves.