Lower bounds on the number of envy-free divisions

We analyze lower bounds for the number of envy-free divisions, in the classical Woodall-Stormquist setting and in a non-classical case, when envy-freeness is combined with the equipartition of a measure. 1. In the first scenario, there are $r$ hungry players, and the cake (that is, the segment $[0,1]$) is cut into $r$ pieces. Then there exist at least two different envy-free divisions. This bound is sharp: for each $r$, we present an example of preferences such that there are exactly two envy-free divisions. 2. In the second (hybrid) scenario, there are $p$ not necessarily hungry players ($p$ is a prime) and a continuous measure $\mu$ on $[0,1]$. The cake is cut into $2p-1$ pieces, the pieces are allocated to $p$ boxes (with some restrictions) and the players choose the boxes. Then there exists at least $\binom{2p-1}{p-1} \cdot 2^{2-p}$ envy-free divisions such that the measure $\mu$ is equidistributed among the players.