Fair Division of an Inheritance: A Case Study Cara Nickolaus CSE 544T Dr. R. Cytron Washington University in St. Louis Fall 2012 Summary: I conduct a case study of an inheritance problem in which the heirs designed an original fair- division procedure to allocate furniture and household items among 13 participants. I compare their procedure to Knaster’s Procedure of Sealed Bids, a First Price Auction, a Second Price Auction, and a variation of the Adjusted Winner Procedure using normative metrics of proportionality, envy-freeness, stability, equitability, and efficiency as well as positive metrics including manipulability, collusion, and financial fairness. I evaluate each mechanism in accordance with the traits most valued by the participants. Nickolaus 2 I. Introduction In their will, Ted and Frieda Pope (my great-grandparents) provided detailed instructions regarding the division of proceeds from their estate among their two surviving children and 13 grandchildren. They did not, however, provide any indication as to how furniture, household items, and items of sentimental value were to be divided or distributed. Their descendants developed a complex fair-division procedure (henceforth referred to as the “Pope Procedure”) to reach a suitable allocation. While the Pope Procedure shares commonalities with other frequently used procedures, it has distinct advantages and disadvantages. In Section II, I describe the Pope Procedure and the resulting allocation. For comparative purposes, Section III records the allocations obtained by Knaster’s Procedure of Sealed Bids, a First Price Auction with both truth-telling and strategic bid behavior, a Second Price Auction, and a version of the Adjusted Winner Procedure. Section IV compares these allocations with the allocation given by the Pope Procedure using the positive metrics of proportionality, envy-freeness, equitability, stability, and efficiency. I employ normative comparisons of manipulability, collusion, and financial fairness to provide further insight into the strengths and weaknesses associated with each mechanism in Section V. In Section VI, I evaluate the Pope descendants’ wisdom in designing their own procedure while Section VII concludes. All tables are located in the Appendix. II. The Pope Procedure and Allocation Enabling everyone to get an item or two of special significance was the primary criterion utilized in designing the Pope Procedure, which was created by grandchild and participant Lowell Nickolaus and adopted upon receiving general consent from the family. As the participants’ ages ranged from early 20’s to mid-60’s, their ability to pay was highly varied. In an attempt to prevent the wealthiest descendants from outbidding their less affluent relatives, each item was assigned a “fair market price” by a knowledgeable third party. Each participant then submitted ranked preferences for as many items as they wished. An item was allocated to the participant who had assigned it the best rank, who then paid the designated price and received the item. Ties in rank were handled last, with the overall number of items already assigned to each individual determining the outcome (in practice only one tie arose). The proceeds from the procedure were divided in accordance with the rule for dividing the cash value of the remainder of the estate: each of the two surviving children received one-third of the cash raised while the deceased child’s share was split equally among her nine children. Four grandchildren (the children of the surviving children) received no share in the cash proceeds. Note that although the Nickolaus 3 participants were not entitled to an equal share of cash, each had an equal claim on the furniture and possessions to be divided. Two of the grandchildren opted not to participate in the procedure, bringing the total number of participants to 13, each of whom was entitled to of the value of the items. The asymmetry between the share of items and cash each participant was entitled to receive complicates the analysis, particularly the assessment of envy-freeness, so I often evaluate the allocation both with and without the inclusion of cash transfers. Under the Pope Procedure, 28 items appraised at a total value of $4353 were allocated with individual prices ranging from $18 to $1000. Participants submitted anywhere from 1 to 10 bids, with an average of 3.8 bids per person. Each participant received between one and five items. The ranks submitted by each player, the cash values assigned to the items, and the resulting allocation are listed in Table 1a of the Appendix. Table 1b gives a more succinct description of the allocation and lists cash transfers and value received for three separate scenarios, which will also be utilized for the other mechanisms described. The first situation, “no cash”, describes each player’s valuation of the items they received plus any cash transfer prescribed by the mechanism (in the case of the Pope Procedure, participants paid the predetermined price for the item). The second situation, “direct cash” adds each participant’s share of any cash proceeds raised by the mechanism ( for Lavonne and Wilma; for Nancy, Steve, Dona, Kathy, Ted, Lisa, and Jim; and 0 for Lowell, Mark, Connie, and Jay). The final situation “indirect cash” takes into account the indirect inheritance that Lowell, Mark, Connie, and Jay could anticipate receiving eventually via their mothers (so for Lavonne and Wilma; for Nancy, Steve, Dona, Kathy, Ted, Lisa, and Jim; and for Lowell, Mark, Connie, and Jay). Note that the “no cash” treatment aligns with Brams and Taylors’ suggestion for handling envy-freeness with unequal entitlements (Brams and Taylor 152), in would each participant’s proportional entitlement would be defined as ( ) ( ). Since the cash is divided in an envy-free manner by definition, the non-cash allocation will be envy-free if and only if envy-freeness is achieved using Brams and Taylors’ method. By adopting ordinal ranks instead of valuations, the Pope Procedure is reminiscent of the Alternating Procedure in which participants take turns selecting items from the collection. In fact, due to the way in which the participants ranked the items in this instance, the Alternating Procedure (or Query Stop extension) would reach the same allocation regardless of the order in which participants were allowed to select items and this allocation agrees with that achieved by the Pope Procedure. Nickolaus 4 Obviously a substantial distinction arises in that the Alternating Procedure usually does not contain prices for the items at stake. III. Other Allocations While the Pope Family elected to design their own fair-division procedure, several prominent existing mechanisms would have also served this purpose. I compare the allocation achieved under the Pope Procedure to that attained by Knaster’s Sealed Bids Procedure, a First Price Auction, a Second Price Auction, and the Adjusted Winner Procedure. Knaster’s Sealed Bids In order to carry out this analysis, I collected preference data from each auction participant (Lavonne is deceased, I collected information on her preferences from her son, Lowell) indicating the maximum amount they would have been willing to pay for each item (see Table 2). Using this information, I conducted the traditional Knaster’s Sealed Bids procedure, whereby each item is assigned to the participant who values it most highly (let this be participant ), who pays ( ) , where represents the high bid and represents the average bid of all players. All other players ( ) are compensated with a transfer equal to ( ), where represents their valuation of the item (Corradi and Corradi, 2011). Under Knaster’s Procedure, the 28 items are awarded to 10 distinct recipients, so that 3 heirs receive no items. Five participants must pay into the system, with amounts ranging from $32 to $1428, while eight participants receive positive transfers ranging from $194 to $561. For the full allocation, see Table 3. First Price Auction Had the Pope Family adopted a First Price Auction framework, two strategic situations must be considered for later analysis, though both generate the same allocation of items. Under truthful bidding, each participant reveals his or her genuine valuation for each item. Under strategic bidding, the Nash Equilibrium predicts that agents will bid one half of their true value (Fudenberg and Tirole 1991, 288-294). In either case, as bids are a monotonic function of valuation, each item is given to the participant that values it most highly, resulting in an allocation of items that agrees with the Knaster Nickolaus 5 Procedure. Cash transfers are described in Table 4. This mechanism would generate $6175 under truthful bidding and $3087.5 under strategic bidding. Second Price Auction In a Second Price Auction, strategic and truthful behavior align as the Nash Equilibrium dictates truthful revelation (Fudenberg and Tirole 1991, 288-294). As a result, all items are allocated in a way identical to the Knaster Procedure and First Price Auction. This procedure raises $4273, and the resulting cash payments and receipts are listed in Table 5. Adjusted Winner In the Adjusted Winner Procedure, participants assign a fixed number of points (here normalized to 1) to items of interest and each item is initially given to the participant who assigns it the most points. Fairness is achieved by redistributing items to equalize the point totals, beginning with the items that the players value most similarly. While this procedure is straightforward for two players, the modification for many players is not entirely apparent and I performed two variations of the procedure. In the first variation, referred to as “Absolute Valuation” in the tables, each participant assigns an item points proportional to that item’s share of his or her total valuation. For instance, if Lavonne values item 1 at $100 and her total valuation for all items is $1000, she would assign 0.1 points to item 1. The point valuations given by this method are listed in Table 6a.