Playing Divide-and-Choose Given Uncertain Preferences

Abstract

We study the classic divide-and-choose method for equitably allocating divisible goods between two players who are rational, self-interested Bayesian agents. The players have additive values for the goods. The prior distributions on those values are common knowledge. We consider both the cases of independent values and values that are correlated across players (as occurs when there is a common-value component). We describe the structure of optimal divisions in the divide-and-choose game and identify several cases where it is possible to efficiently compute equilibria. An approximation algorithm is presented for the case when the distribution over the chooser's value for each good follows a normal distribution, along with a randomized approximation algorithm for the case of uniform distributions over intervals. A mixture of analytic results and computational simulations illuminates several striking differences between optimal strategies in the cases of known versus unknown preferences. Most notably, given unknown preferences, the divider has a compelling "diversification" incentive in creating the chooser's two options. This incentive leads to multiple goods being divided at equilibrium, quite contrary to the divider's optimal strategy when preferences are known. In many contexts, such as buy-and-sell provisions between partners, or in judging fairness, it is important to assess the relative expected utilities of the divider and chooser. Those utilities, we show, depend on the players' levels of knowledge about each other's values, the correlations between the players' values, and the number of goods being divided. Under fairly mild assumptions, we show that the chooser is strictly better off for a small number of goods, while the divider is strictly better off for a large number of goods.

Figures (10)

• Figure 1: An instance with four goods where there are four locally-optimal divisions with a variety of different values of $P$. The globally optimal value of $P$ is indicated as $P^*$. The divider values are $g^D_1 = 3$, $g^D_2 = 2$, $g^D_3 = 1$, $g^D_4 = 1.2$, and corresponding chooser priors are $\overline{\mathcal{G}}^C_1 = \mathcal{N}\left(5, 1\right)$, $\overline{\mathcal{G}}^C_2 = \mathcal{N}\left(9.5, 1\right)$, $\overline{\mathcal{G}}^C_3 = \mathcal{N}\left(13.6, 96.04\right)$, $\overline{\mathcal{G}}^C_4 = \mathcal{N}\left(95, 28561\right)$.
• Figure 2: An example with six goods where it is optimal for the divider to split five of the goods between the two piles. Here $g^D_1 = 9.8$, $g^D_2 = 9.9$, $g^D_3 = 10$, $g^D_4 = 10.1$, $g^D_5 = 10.2$, $g^D_6 = 15$, and the chooser's prior for each good is $\overline{\mathcal{G}}^C_i = \mathcal{N}\left(10, 1\right)$. The optimal value of $P$ is 0.078.
• Figure 3: Optimal divisions as a function of the variance $s$ of the chooser's value for each good. This is the same setup as in Figure \ref{['figManyGoodsDivided']}, but instead of chooser values being drawn from $\mathcal{N}\left(10, 1\right)$, they are drawn from $\mathcal{N}\left(10, s\right)$, for $s$ starting at one and approaching zero. Thus the first (dark blue) bars are the same division as before.
• Figure 4: An instance with three goods where, despite all prior variances being the same, the $p_i$ are not monotone in the critical ratios. Here $g^D_1 = 1$, $g^D_2 = 2$, $g^D_3 = 3$, and corresponding chooser priors are $\overline{\mathcal{G}}^C_1 = \mathcal{N}\left(100, 25\right)$, $\overline{\mathcal{G}}^C_2 = \mathcal{N}\left(198, 25\right)$, $\overline{\mathcal{G}}^C_3 = \mathcal{N}\left(100, 25\right)$. The optimal value of $P$ is 0.005.
• Figure 5: Optimal divisions as a function of the correlation parameter $t$. As in Figure \ref{['figManyGoodsDivided']}, each value is is drawn from $\mathcal{N}\left(10, 1\right)$, and the divider's values happen to be $g^D = (9.8, 9.9, 10, 10.1, 10.2, 15)$. The difference is that now, for $t > 0$, the divider infers that the chooser has similar values to his own. The first (dark blue) bars corresponding to $t = 0$ are the same division as before. As $t$ approaches 1, the optimal division changes little from the final (red) bars at $t = 7/8$.

Theorems & Definitions (23)

• Lemma 3.1
• Lemma 3.2
• Proposition 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• Theorem 3.6
• proof
• Proposition 3.7