Proportional rule (bankruptcy)

Examples

Examples with two claimants:

• ${\displaystyle PROP(60,90;100)=(40,60)}$. That is: if the estate is worth 100 and the claims are 60 and 90, then ${\displaystyle r=2/3}$, so the first claimant gets 40 and the second claimant gets 60.
• ${\displaystyle PROP(50,100;100)=(33.333,66.667)}$, and similarly ${\displaystyle PROP(40,80;100)=(33.333,66.667)}$.

Examples with three claimants:

• ${\displaystyle PROP(100,200,300;100)=(16.667,33.333,50)}$.
• ${\displaystyle PROP(100,200,300;200)=(33.333,66.667,100)}$.
• ${\displaystyle PROP(100,200,300;300)=(50,100,150)}$.

Characterizations

The proportional rule has several characterizations. It is the only rule satisfying the following sets of axioms:

• Self-duality and composition-up;
• Self-duality and composition-down;
• No advantageous transfer;
• Resource linearity;
• No advantageous merging and no advantageous splitting.

Truncated-proportional rule

There is a variant called truncated-claims proportional rule, in which each claim larger than E is truncated to E, and then the proportional rule is activated. That is, it equals ${\displaystyle PROP(c_1^{\prime },\dots ,c_n^{\prime },E)}$, where ${\displaystyle c_i^{\prime }:=min(c_i,E)}$. The results are the same for the two-claimant problems above, but for the three-claimant problems we get:

• ${\displaystyle TPROP(100,200,300;100)=(33.333,33.333,33.333)}$, since all claims are truncated to 100;
• ${\displaystyle TPROP(100,200,300;200)=(40,80,80)}$, since the claims vector is truncated to (100,200,200).
• ${\displaystyle TPROP(100,200,300;300)=(50,100,150)}$, since here the claims are not truncated.

Adjusted-proportional rule

The adjusted proportional rule first gives, to each agent i, their minimal right, which is the amount not claimed by the other agents. Formally, ${\displaystyle m_i:=max(0,E-\sum _{j\ne i}{c}_j)}$. Note that ${\displaystyle \sum _{i=1}^n{c}_i\ge E}$ implies ${\displaystyle m_i\le c_i}$.

Then, it revises the claim of agent i to ${\displaystyle c_i^{\prime }:=c_i-m_i}$, and the estate to ${\displaystyle E^{\prime }:=E-\sum _i{m}_i}$. Note that that ${\displaystyle E^{\prime }\ge 0}$.

Finally, it activates the truncated-claims proportional rule, that is, it returns ${\displaystyle TPROP(c_1,\dots ,c_n,E^{\prime })=PROP(c_1^{″},\dots ,c_n^{″},E^{\prime })}$, where ${\displaystyle c_i^{″}:=min(c_i^{\prime },E^{\prime })}$.

With two claimants, the revised claims are always equal, so the remainder is divided equally. Examples:

• ${\displaystyle APROP(60,90;100)=(35,65)}$. The minimal rights are ${\displaystyle (m_1,m_2)=(10,40)}$. The remaining claims are ${\displaystyle (c_1^{\prime },c_2^{\prime })=(50,50)}$ and the remaining estate is ${\displaystyle E^{\prime }=50}$; it is divided equally among the claimants.
• ${\displaystyle APROP(50,100;100)=(25,75)}$. The minimal rights are ${\displaystyle (m_1,m_2)=(0,50)}$. The remaining claims are ${\displaystyle (c_1^{\prime },c_2^{\prime })=(50,50)}$ and the remaining estate is ${\displaystyle E^{\prime }=50}$.
• ${\displaystyle APROP(40,80;100)=(30,70)}$. The minimal rights are ${\displaystyle (m_1,m_2)=(20,60)}$. The remaining claims are ${\displaystyle (c_1^{\prime },c_2^{\prime })=(20,20)}$ and the remaining estate is ${\displaystyle E^{\prime }=20}$.

With three or more claimants, the revised claims may be different. In all the above three-claimant examples, the minimal rights are ${\displaystyle (0,0,0)}$ and thus the outcome is equal to TPROP, for example, ${\displaystyle APROP(100,200,300;200)=TPROP(100,200,300;200)=(20,40,40)}$.