Toss one’s cake, and eat it too: partial divisions can improve social welfare in cake cutting

Abstract

We consider the problem of fairly dividing a heterogeneous good (a.k.a. “cake”) between a number of players with different tastes. In this setting, it is known that fairness requirements may result in a suboptimal division from the social welfare standpoint. Here we show that, in some cases, leaving some of the cake unallocated, and fairly dividing only the remainder of the cake may be socially preferable to any fair division of the entire cake. We study this phenomenon, providing asymptotically-tight bounds on the social improvement achievable by such partial divisions.

Appendices

Appendix A No PD-improvement with multiple intervals per player

Throughout the paper we considered the case where each player gets a single interval. Here we show that if each player can get multiple intervals then there is no PD-improvement. To formalize this, we first need to extend the model to this setting, and, in particular, extend the definitions of: pieces, valuations functions, and divisions. In the multiple interval setting a piece is element of the Borel $\sigma$-algebra over [0, 1]. Each valuation function $V_i$ is a non-atomic measure over this $\sigma$-algebra.⁷ In particular, the valuations functions are monotone (that is, $V_i(d)\le V_i(c)$ for pieces $d\subset c$). A division assigns disjoint pieces to the players.

Proposition 2

In the multiple interval setting, there can be no PD-improvement for any monotone welfare function. In addition, in this setting no envy-free partial division can Pareto dominate all envy-free complete divisions.

Proof

Fix some cake instance, and let x be a partial envy-free division. We construct a complete envy-free division $\hat{x}$, in which each player gets a piece that contains the piece it gets in x. Thus, by monotonicity of $V_i$, x does not Pareto dominate $\hat{x}$, and by monotonicity of the welfare function w, $w(\hat{x}) \ge w(x)$. Thus, there is no PD-improvement.

Let $d_i$ be the piece allotted to i in x, and set $D=\cup _{i=1}^n d_i$. Then, D is measurable, and hence $C=[0,1]\setminus D$ is also measurable. Thus, by  Dubins and Spanier (1961) there exists a complete envy-free division, y of C. Let $c_i$ be the piece (of C) allotted to i by y. Let $\hat{x}$ be the division in which player i gets $d_i\cup c_i$ (for all i). Then, $\hat{x}$ is a complete division of [0, 1], and for any i, j

$$\begin{aligned} V_i(d_i\cup c_i) = V_i(d_i)+ V_i(c_i) \ge V_i(d_j)+ V_i(c_j) = V_i(d_j\cup c_j), \end{aligned}$$

where the inequality is provided by the envy-freeness of x and y. Thus, $\hat{x}$ is envy-free.

We note that a similar statement also holds if we allow each player to get only a finite number of intervals (rather than any measurable set). The proof in this case would be similar, but invoking  Stromquist (1980) existence theorem rather than the one of  Dubins and Spanier (1961).

Appendix B Non-additive valuations

Throughout the paper, we have assumed that the players’ valuations are additive. In this appendix we study the extent to which this assumption can be weakened while maintaining the results. First note that most of the results in the paper are lower bound constructions. Thus, all of these results remain true for any class of valuation functions that generalize additivity (in particular—no additivity requirement).

As for the upper bounds, we now identify some natural weakened requirements under which the results still hold.

The first requirement is that the valuations are continuous, in the sense that the value of a piece changes “smoothly” with small additions or subtractions from the piece (this in particular implies the non-atomicity requirement). Clearly, without this assumption envy-free divisions may not even exist (e.g. if all players give value of 1 to any piece containing the point $\frac{1}{2}$ and 0 to any piece not containing it), and the PD-improvement is not well-defined. On the other hand, Stromquist (1980) shows that this requirement is sufficient to guarantee existence of envy-free divisions.

• Continuity For any piece $I\subseteq [0,1]$ and every $\epsilon >0$ there exists $\delta >0$ such that for any other piece J if $|I\bigtriangleup J| < \delta$ (where $|I\bigtriangleup J|$ is the total length of the symmetric difference between I and J) then $|V_i(I) - V_i(J)| < \epsilon$.

Our second requirement formalizes the idea that the cake is a “good”, and therefore getting less of it is never preferred.⁸

• Monotonicity For pieces I, J, if $I\subseteq J$ then $V_i(I)\le V_i(J)$.

These two assumptions turn out to be already sufficient for proving some of the results. In particular, no further assumptions are required for proving Proposition 2, showing that there is no PD-improvement with non-connected pieces, Proposition 1, showing the connection between the Price of Envy-Freeness and the PD-improvement, and the first part of Theorem 4, showing that no strict Pareto improvement exists when starting from a complete envy-free division.

For proving the remaining upper bounds, we also need to assume that the value of a piece is never greater than the sum of the value of its parts.

• Sub-additivity For any piece I and partitions $I=I_i\cup I_2$ (with $I_1, I_2$ disjoint pieces), $V_i(I) \le V_i(I_1) + V_i(I_2)$.

With this assumption, we can also prove the second part of Theorem 4, showing that one cannot increase the utility of half the players by more than a twofold simultaneously, and the upper bound on the utilitarian Price of Envy-Freeness used in Theorem 7. This assumption also allows us to show an upper bound of n on the egalitarian Price of Envy-Freeness, since it implies that any envy-free division must give each player a piece of value $\ge \frac{1}{n}$; thus, the upper bound in Theorem 6 can be replaced with n, which also asymptotically matches the lower bound of $\frac{n}{3}$.