Fair Division of an Inheritance: A Case Study

Transcription

1 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 fairdivision 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.

2 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

3 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.

4 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, ). 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

5 Nickolaus 5 Procedure. Cash transfers are described in Table 4. This mechanism would generate $6175 under truthful bidding and $ 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, ). 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. After assigning points, I constructed the initial allocation by giving each item to the participant that assigned it the most points. Then, to increase fairness, I redistributed items by identifying the participant with the lowest point total and transferring the item with the highest value ratio ( ) to that player. I repeated this transfer process repeatedly, each time identifying the lowest player and making one transfer. While the two-player version of Adjusted Winner redistributes items until total equitability is reached, I terminated the process when no whole items could be transferred to yield a more equitable point distribution. For this version of the procedure, I conducted 11 transfers, which are indicated by the color-coding in Table 6a. The initial and final allocations and valuations are described in Table 6c. While the method just described seems the most obvious extension of Adjusted Winner to many players, an obvious drawback is the lack of revenue raised. I developed a modification to incorporate the prices used by the Popes, altering this mechanism to raise revenue equal to the Pope Procedure. In this version, which I refer to as Surplus Valuation in the Tables, I assigned points to items based on the

6 Nickolaus 6 share of surplus value associated with each item. For instance, if Lavonne valued item 1 at $100, the price of item 1 was $80 and Lavonne s total surplus from all items was $200 ( ( ), she would assign 0.1 points to item 1 ( ). The point totals given by this method are listed in Table 6b. Given these point assignments, I repeated the procedure as described above. The initial and final allocations and valuations are listed in Table 6c, which includes no cash, direct cash, and indirect cash treatments. IV. Positive Comparison Proportionality is the most intuitive fair-division concept. Each member of the Pope Family was entitled to of his or her valuation of the collection of items. Of the procedures considered, only Knaster s Sealed Bids and the Absolute Valuation version of Adjusted Winner achieve proportionality given the preferences of the Pope Family. In particular, Connie and Jay received no items under a First Price or Second Price Auction and they were not entitled to any share in the cash raised, so they received 0 from these procedures, making proportionality impossible. In the Pope Procedure and Surplus Valuation version of Adjusted Winner, Connie and Jay each receive at least one item, but value their items at the asking price so gain no surplus from receiving it. Therefore, Connie and Jay also received 0 from these procedures, once again condemning any hope of proportionality. Table 7 collects all proportionality data from each mechanism. For quick reference, a succinct comparison of the mechanisms involving all of the positive and normative measures is given in Table 15. Envy-freeness is a more subtle equity requirement, in which no participant may value another participant s allocation more highly than his or her own. Tables 8-14 list each participant s valuation of everyone else s allocation under each mechanism considered and each cash scenario. Without cash transfers, envy-freeness is achieved using a First Price Auction, the Pope Procedure, or the Surplus Valuation version of Adjusted Winner, and is nearly achieved in a Second Price Auction (one agent is envious). When cash transfers of either type are added, no mechanism is envy-free. This is unsurprising as agents were entitled to differing shares of the cash raised. Corradi and Corradi (2011) suggest a modification of the Knaster Procedure which achieves envy-freeness by giving each non-winning agent an equal side-payment equal to a weighted average of the first and second highest bids. While this modification makes the Knaster Procedure envy-free, in the context of the Pope estate this procedure will be ruled out for other reasons (to be discussed later), so I do not extend the analysis of such a modification. Additionally, utilizing the modification would require participants to agree on fair weights

7 Nickolaus 7 for the first and second highest bids, introducing a conflict between those players entitled to a large share of the cash proceeds (who would likely want to heavily weight the highest bid) and those entitled to a small or no share (who would want to weight the lower bid more heavily), although assigning each bid equal weight may be an obvious solution to this dilemma. Equitability would require that each participant felt they received an equal share of their overall valuation. The proportionality table (Table 7) shows that equitability is not achieved by any mechanism except for the First Price Auction with Truth-Telling and no cash transfers. In this scenario, each individual receives 0, hardly a desirable outcome. Equitability is achieved by the two-player version of Adjusted Winner, and could be achieved in the many-player modification if items were sold and the cash redistributed. Since the Popes wanted the items at stake to remain in the family, I elected to sacrifice complete equitability. With the Absolute Valuation modification, players receive anywhere from 0.14 to 1.0 of their total points. While this does not come close to equitability, it does attain strong proportionality and a more equal distribution of points than most of the other mechanisms. Stability is achieved if no two agents would mutually agree to switch allocations. It can be viewed as a baseline measure which any mechanism should satisfy in order to be considered. Luckily, all procedures are stable both with and without cash transfers. In some contexts, efficiency is a primary goal since it maximizes overall welfare by assigning each item to the participant who values it most highly. Both with and without cash transfers, all mechanisms other than the Pope Procedure and Adjusted Winner attain efficiency (and consequently allocate the items in an identical way). V. Normative Comparison Manipulability The Pope Procedure, Knaster s Sealed Bids, and Adjusted Winner are all vulnerable to strategic manipulation by the participants, while First and Second Price Auctions are immune to manipulation if players follow Nash Equilibrium strategies. In general, a participant s ability to misrepresent his or her own preferences in order to improve their allocation is dependent on each participant s information about the other participants. In particular, under the Pope Procedure, a savvy player with extensive information about others could potentially better their allocation by assigning low ranks to the most contested items and not ranking popular items that he or she was unable to win (because these were

8 Nickolaus 8 essentially wasted ranks). The gains from manipulation can be significant. Lowell s stated ranks afforded him 3 items and a surplus of $275, but he could have won 5 items and increased his valuation to $340 through optimal manipulation. Similarly, Nancy s ranks gave her 5 items which she valued at $350, but by manipulating she could have attained 7 items worth $405. It is difficult to gauge the extent to which manipulation actually occurred under the Pope Procedure, but there are some indications that it was present. Of the thirteen participants, ten followed monotonic ranking strategies in which they assigned the highest rank to the item yielding the greatest surplus ( ). This intuitive and straightforward ranking method suggests that most players were not attempting to manipulate the mechanism. On the other hand, Lowell, Nancy, and Lisa used non-monotonic ranking strategies. As the two players who ranked the highest number of total items, Lowell and Nancy stood to gain the most through manipulation. Both players submitted ranks in which the surplus granted by an item is generally decreasing with rank (as in a monotonic bidding strategy), with a few inconsistencies. For instance, the item Lowell ranked third granted him only $25 in surplus value, while his fourth ranked item gave $55. In actuality, the competition for his third ranked item was closer than that of the fourth ranked item, suggesting he anticipated the need to assign a better rank to this item. Lowell s high ranks also indicate strategic behavior. While he would derive relatively high surplus value from his 9 th and 10 th ranks, these ranks were given to very popular items, indicating that he realized he was unlikely to win these items and simply gave them high ranks in the unlikely event that he was mistaken about their popularity and won anyway. Nancy and Lisa followed similar strategies, assigning lower ranks to highly contested items than under a monotonic strategy. Knaster s Sealed Bids Procedure is open to strategic manipulation by not only the winning bidder for each item, but by all participants. Because of the side-payment feature of Knaster s Procedure, a participant who does not expect to win an item would optimally value it at just under the winning bid. A participant who does expect to win should assign a value just above the next-highest bid. Although the procedure is open to manipulation, Van Essen (2012) shows that the optimal Nash Equilibrium bid strategy rises monotonically with a participant s valuation, meaning that items are still distributed efficiently. Furthermore, in expectation each participant receives value equal to that obtained under truthful bidding, although there may be winners and losers in any specific instance of the procedure.

9 Nickolaus 9 If the Knaster Procedure had been utilized instead of the Pope Procedure, evidence of manipulation might include higher valuations assigned to the Pipe Organ. The participants valuations of this item ranged from $0 to $1200, providing participants with low valuations the potential for substantial side-payments by inflating their bids. The potential gains from manipulation under this procedure are also large. If Jay followed the optimal strategy, he could have improved his payoff from $ to $ Similarly, Lowell could have reduced the amount he had to pay from $ to $ In the Adjusted Winner Procedure, a participant can manipulate by assigning a favored item just enough points to acquire it, while assigning spare points to other items in order to increase the redistribution. Had Lowell pursued optimal manipulation under the Surplus Valuation version, he could have obtained 0.86 points instead of However, given the complexity of the transfers occurring in the modification of Adjusted Winner, it would have been extraordinarily difficult for a player to foresee the optimal manipulation, perhaps prompting players to report honestly in practice. Collusion All the mechanisms considered are intended for use by participants acting individually to maximize their own payoff. Consequently, if some players share information and work together, they may improve the outcome for themselves at the expense of others. While most mechanisms considered are open to some level of collusion, they exhibit differing degrees of vulnerability. When players wish to rank differing numbers of items, a coalition of players may benefit by colluding under the Pope Procedure. If participant is interested in many items and can collaborate with participant, who is interested in few items, could effectively utilize some of the extra, lower bids belonging to to increase the probability that obtains these items. For example, Lowell ranked 10 items while Jay ranked only one. If Jay allowed Lowell to use his extra ranks, Lowell could have obtained five items instead of three and increased his surplus value from $275 to $340 (giving him the same allocation as with optimal strategic manipulation). Although the Pope Procedure admits the possibility of collusion, evidence for collusion is limited. Collusion would seem most likely among each of the three families represented: the Nickolaus, Volkmers, and Bentz. As no member of the Bentz Family was interested in more than two items, they had no opportunity for collusion. The data shows a wide differential in the number of items ranked by the Volkmers and Nickolaus, creating the possibility of collusive activity. Yet, the presence of this

10 Nickolaus 10 differential in the data shows that these participants did not collude in the manner described. It is possible that some collusion did occur among the nine Volkmers, but since most ranked either a very high or very low number of items, the data provides no strong justification for such activity. In the Knaster Procedure, the potential for collusion is limited by the structure of the mechanism, but collusion is still possible. Participants could best work together simply by sharing their planned bids with one another. This would give players the needed information to adopt manipulative strategies as described in the previous section. However, the Knaster Procedure does contain a built-in protection against this type of cooperation. If a participant is interested in an item, sharing his or her valuation with other players so that they can inflate their bids accordingly raises the average bid on the item, thus raising the amount that the winning participant ultimately has to pay. For this reason, players might be hesitant to reveal which items they were interested in. Nonetheless, a subset of players can improve their expected pay-off by synchronizing their bids to maximize their side-payments when someone outside the coalition wins (Fragnelli and Marina 2009). However, since this is only an expected gain, risk-averse participants might be reluctant to gamble. Additionally, if players have some information about the preferences of other players, forming a coalition to share this information could enable a subset of participants to increase their side-payments without increasing risk. The only collusion possible in a First Price Auction arises if some participants have information about others. In this case, by pooling this information they may be able to infer the likely maximum bid and avoid over-bidding for items in which they are interested. However, if each participant has no information regarding others preferences, collusion cannot be successfully employed. The same type of information-sharing possible in Knaster s Procedure and a First Price Auction can be applied to a Second Price Auction, but additional collusive behavior may also be possible. For instance, if two players determine that both are interested in the same item, the player who values it most highly should bid their true valuation, while the other player should bid 0 instead of their true valuation. Since the winner has to pay the second highest price, this could reduce the price paid if the second player had the second highest valuation. In the Adjusted Winner Procedure, collusion is again possible if players within a coalition share inside information about outsiders valuations. A type of cooperation similar to that arising in a Second Price Auction may also occur. If two players discover that they are interested in the same item, it may be preferable for the player valuing it less to assign no points to the item and save his or her points for

11 Nickolaus 11 other items that he or she has the potential to win outright. However, the redistribution step of the procedure opens the possibility that the player with the lower valuation will ultimately attain the item, so it is not obvious whether this strategy would always be favorable. The complexity of this procedure makes both manipulation and collusion fairly risky for relatively uninformed players. Financial Fairness In recognition of the differing financial situations facing the family members involved, the Popes wished to level the playing field and allow the less affluent to receive some items. Because the distribution of items among the family members occurred prior to the division of the remaining estate, those relatives entitled to cash inheritance could not use these funds to purchase items. The Pope Procedure carefully balances several opposing forces in an attempt to achieve financial fairness. By setting a fair price for each item instead of allowing family members to bid the price up, even cash constrained family members could afford many of the items. On the other hand, the family wanted to set prices high enough to generate some cash to be divided among those entitled to receive it. The amount of revenue raised, $4273, falls in the middle of the spectrum, which ranges from $0 under Knaster s Procedure to $6175 under a First Price Auction. This feature suggests a balance between maximizing revenue to satisfy the relatives entitled to high shares of the cash proceeds and keeping prices reasonable to make the items accessible to all. The Pope Procedure also strives to reduce the financial uncertainty associated with a more traditional auction. By stating prices ahead of time, participants could carefully consider their financial situations and rank items accordingly. The parallel nature of the process was also attractive relative to the serial form of an auction. In an ordinary auction, a cash constrained participant intent on obtaining a specific item might refrain from bidding on earlier items only to be outbid and walk away empty-handed. By handling all items concurrently, the Pope Procedure attempted to resolve this problem. The Knaster Procedure is less sensitive to potential cash constraints. At the time of bidding, participants do not know the magnitude of the side-payments they will receive or pay out. Uncertainty is high and the wealthiest participants could easily dominate the procedure. Additionally, the procedure raises no revenue, slighting the relatives entitled to receive the cash proceeds. The procedure does contain the desirable parallel structure in which all items are considered simultaneously, however since prices are not set ahead of time, this does little to reduce uncertainty.

12 Nickolaus 12 First and Second Price Auctions suffer from similar weaknesses, in addition to problems arising from the order of the auctioned items described above. These procedures favor those with ready access to cash, though the First Price Auction with truth bids and the Second Price Auction do raise more revenue than the Pope Procedure. With the Surplus Valuation modification, the Adjusted Winner Procedure incorporates the fair price used in the Pope Procedure and raises the same level of revenue, striking a balance between the participants entitled to cash transfers and those receiving no cash. Similarly, it mimics the parallel structure of the Pope Procedure to lessen uncertainty and allow participants to consider their budgets in advance. The Absolute Valuation version does not raise revenue, so disadvantages participants entitled to cash shares. However, neither version requires side-payments, making this procedure sensitive to cash-constrained participants. VI. Did the Popes choose wisely? In order to determine the success of the Pope Procedure, I conducted interviews with several participants to gauge the importance the family attached to each of the attributes considered. Since the choice of any mechanism requires a tradeoff between various desirable features and the Popes placed differing levels of emphasis on these features, the Pope Procedure was a wise choice if it achieves the features they valued most highly. I examined the weight attached to each positive and normative fairness metric and then evaluate the ability of each mechanism considered to match the Popes criteria. The primary consideration for the Pope Family was insuring that each participant received some minimal level of satisfaction. In short, if the family adopted a Social Welfare Function to describe their satisfaction with an allocation, it would be more Rawlsian in form than Utilitarian. (In a Rawlsian Social Welfare Function, ( ) ( ) where ( ) represents participant individual utility from receiving allocation and ( ) represents the family s satisfaction with the mechanism; In a Utilitarian Social Welfare Function ( ) ( ) (Berliant 2011). The Pope Family wanted each participant to receive some level of satisfaction from the allocation, but attached no special significance to achieving a proportional allocation. They wanted to be fair to the younger, less affluent members of the family, but felt a truly proportional allocation would be difficult to attain, especially given the unequal cash transfers involved. They viewed equitability

13 Nickolaus 13 similarly, noting that absolute fairness among all participants was secondary in importance to giving each family member some positive payoff. Given the close family ties between the participants, envy-freeness was a more important consideration for the Popes. In particular, they strove to give every participant his or her first choice item in an attempt to guarantee that no one would walk away feeling slighted in comparison to someone else. Stability was also important, although they felt that any unstable allocation could be remedied after the fact if family members mutually agreed to trade. While efficiency is frequently one of the primary objectives sought by a division procedure, the Popes flatly rejected the idea of maximizing the amount of cash raised or insuring that each item went to the person who desired it most. Although not directly involved in the distribution, Lavonne s husband did express the opinion that too many items were being reserved for the family members instead of being placed on the general auction (which was open to the public), where family members could still bid if they chose. This viewpoint represents a stronger desire for efficiency, although as the proceeds raised from the auction would be divided unequally according to the rules governing cash and Lavonne was entitled to a large share of these proceeds, it is difficult to determine if this perspective was motivated by a desire for efficiency or an attempt to maximize personal pay-off. The family was largely unconcerned with the potential for manipulation or collusion. They felt that each participant was unlikely to have detailed information about the preferences of others and so any manipulation attempt was as likely to fail as to succeed, leading most family members to report their ranks honestly. They felt that collusion would be limited by a built-in enforcement mechanism: as the participants would be interacting in the future, it would be obvious if items were re-shuffled after the initial distribution. If one family member received an item only to have it appear in someone else s house, the others would be suspicious of the situation. Family respect and the desire to preserve a good reputation among the relatives also limited the possibilities for collusion or manipulation. The relatives were highly concerned with financial fairness, and the Pope Procedure was specifically designed with this in mind. In particular, the family wanted to avoid bidding wars for specific items, guarantee that some items were affordable to all, and avoid the uncertainty associated with dividing items in a serial fashion. In light of their priorities, the Pope Procedure seems to do a relatively good job of matching the criteria most valued by the Pope Family. Without cash transfers, it achieves both of the positive

14 Nickolaus 14 properties most esteemed by the relatives, envy-freeness and stability. Though it fails to be equitable, proportional, or efficient, these qualities were less important to the relatives. While the Pope Procedure is relatively open to manipulation and collusion, the family was similarly unconcerned about these weaknesses. Additionally, the Popes favored this procedure because it reduces uncertainty and is financially fair. One limitation of the Pope Procedure is its reliance on ordinal rather than cardinal valuations. While each family member did receive his or her first choice under this mechanism, the preference data reveals that Lavonne enjoyed $300 in surplus value from her first choice, while Lisa, Jim, Wilma, Connie, and Jay received no surplus from their top-ranked items. Given that the Pope Procedure allocates items in the same way as the Alternating Procedure with any possible selection order, it is not immediately clear that the Pope Procedure confers any benefits unattainable with the simpler Alternating Procedure. However, while the Alternating Procedure happens to perform identically no matter the selection order, this is a coincidence contingent on the ranks submitted by the participants. Ex ante, it would have been impossible for the relatives to ascertain the degree to which selection order would impact the distribution, making this procedure appear unfair. Even if the selection order was randomly determined and carried out, it would not have been obvious that the allocation was independent of the ordering, and some participants might have mistakenly felt disadvantaged by a poor random draw. Additionally, the Alternating Procedure does not include prices or raise revenue, although these components could be added to the mechanism in a straightforward fashion. When introduced to the Knaster Procedure, the family members I spoke to expressed concern about the complexity and uncertainty involved with this procedure. They felt family members would struggle to understand the side-payment system and be hesitant to assign values without knowing how much they could be paying. Although the maximum amount any participant could be asked to pay for an item is of their reported valuation, this upper-bound would probably not be obvious to the average player. The unequal cash pay-outs would also have been difficult for some to accept, especially in light of the family s concern with achieving an envy-free allocation. Perhaps most importantly, all cash proceeds were legally required to be handled through the estate and be divided in the prescribed ratio, raising doubt as to whether the cash component of the Knaster Procedure would even be legal. While the Knaster Procedure attains the highest number of positive and normative benchmarks, including envy-freeness if the modification suggested by Corradi and Corradi is implemented, it still lacks the financial fairness valued by the Pope Family, making this procedure a poor choice.

15 Nickolaus 15 Unlike the Knaster Procedure, the family considered employing a First or Second Price Auction to divide the items. Ultimately, they rejected these procedures due to their uncertainty and serial nature. While these procedures perform well in terms of envy-freeness with the added benefit of efficiency, they lack the financial fairness prized by the Popes. The Adjusted Winner Procedure can attain the aspects of Financial Fairness incorporated into the Pope Procedure if surplus valuations are used, but it fails to be envy-free. While the family members I spoke with felt this mechanism would have been an acceptable alternative, the complexity of the procedure concerned them. While assigning point totals is not difficult, actually finding the allocation requires dozens of calculations for each item transferred. As Table 16 shows, the valuations achieved by the Pope Procedure and surplus valuation version of Adjusted Winner are closely aligned. Given the full range of positive and normative considerations, it seems the Pope Family chose wisely in electing to design their own procedure. Their design best satisfies their two primary goals of envy-freeness and financial fairness, while raising revenue comparable to the other procedures. While it attains the same allocation as the Alternating Procedure in this setting, the Pope Procedure resolves the potential unfairness arising from selection order, making it more likely that the relatives would agree to abide by the procedure. VII. Conclusion While the procedure adopted by the Pope Family does not maximize average surplus value or minimize its standard deviation (see Table 16), fails to achieve proportionality, and gives an inefficient allocation which is susceptible to manipulation and collusion, the relatives were satisfied that the outcome was fair. The allocation was envy-free, gave each participant his or her first choice item, and respected the financial constraints affecting some family members. While not suited to every situation, the Pope Procedure appears to be a good alternative to the Knaster s Procedure of Sealed Bids, an auction mechanism, or Adjusted Winner when financial considerations and envy-freeness are of paramount importance. Of course, the most important consideration in selecting a fair-division procedure is attaining the general consent of the participants prior to performing the algorithm. If the process is unanimously accepted as fair, the resulting allocation can scarcely be questioned.

16 Nickolaus 16 VIII. Appendix Family Ranks Nickolaus Family Volkmer Family Bentz Family Lavonne Lowell Mark Nancy Steve Dona Kathy Ted Lisa Jim Wilma Connie Jay Item # Item Price 1 Organ Oa k Table & Chairs Couch & Chairs Piano Tall Oak Dresser Dres s er/mirror Buffet John Deere Ice Box Oak Table-east Walnut Table Spool Cabinet Pattern Rocker Wardrobe Des k Cupboard-Porch Plain Rocker Clock Pink Lamp Corner Shelf Painted Dres er Badges Tredle Machine Sheep Picture Wicker Rocker Chain in Bath Black Trunk Butter Crock 18 3 Table 1a: Allocation using the Pope Procedure. This table shows each participant s ranks for each item. Red indicates the recipient of the item. Pope Procedure No Cash Direct Cash Nickolaus Family Volkmer Family Bentz Family Lavonne Lowell Mark Nancy Steve Dona Kathy Ted Lisa Jim Wilma Connie Jay Items Received 8,24 1,5, ,7,14,25,27 12,13 20,21 9,11,18 3,4,16 15,26, Cash Received Value Received Cash Received Value Reived Indirect Cash Received Cash Value Received Table 1b: Cash transfers and value received under the Pope Procedure.

17 Nickolaus 17 Family Monetary Valuations Nickolaus Family Volkmer Family Bentz Family Item # Item Lavonne Lowell Mark Nancy Steve Dona Kathy Ted Lisa Jim Wilma Connie Jay 1 Organ Oa k Table & Chairs Couch & Chairs Piano Tall Oa k Dres s er Dres s er/mirror Buffet John Deere Ice Box Oa k Table-eas t Walnut Table Spool Cabinet Pattern Rocker Wardrobe Des k Cupboard-Porch Plain Rocker Clock Pink Lamp Corner Shelf Painted Dres er Badges Tredle Machine Sheep Picture Wicker Rocker Chain in Bath Black Trunk Butter Crock Total Valuation Table 2: Participants preferences. This table shows the monetary valuation of each participant for each item. Red indicates the highest valuation for that item, green indicates second-highest.

18 Nash Equilibrium Truth Telling Nickolaus 18 Knaster's Sealed Bids Nickolaus Family Volkmer Family Bentz Family Lavonne Lowell Mark Nancy Steve Dona Kathy Ted Lisa Jim Wilma Connie Jay Total Valuation Items Received 8,24 1,5,9,10, 11,13 22,23 2,6,7,14,25, ,21 16,17 3,4,18 15,26,28 19 Initial Value Received Initial Fair Share Difference Share of Surplus Adj Fair Share Transfer Final Valuation Table 3: Allocation given by Knaster s Procedure. First Price Auction Nickolaus Family Volkmer Family Bentz Family Lavonne Lowell Mark Nancy Steve Dona Kathy Ted Lisa Jim Wilma Connie Jay 6,10,14,1 7,11,12,1 20,31, Items Received 13,29 5,16,18 27,28 9,30, ,26 21,22 8,9, No Cash Direct Cash Indirect Cash No Cash Direct Cash Cash Received Value Received Cash Received Value Reived Cash Received Value Received Cash Received Value Received Cash Received Value Reived Indirect Cash Received Cash Value Received Table 4: Allocation, cash transfers, and final valuations given by a First Price Auction. Second Price Auction No Cash Direct Cash Nickolaus Family Volkmer Family Bentz Family Lavonne Lowell Mark Nancy Steve Dona Kathy Ted Lisa Jim Wilma Connie Jay 6,10,14, 7,11,12,19 Items Received 13,29 15,16,18 27,28,30, ,26 21,22 8,9,23 20,31,33 24 Cash Received Value Received Cash Received Value Reived Indirect Cash Received Cash Value Received Table 5: Allocation, cash transfers, and final valuations given by a Second Price Auction.

19 Nickolaus 19 Absolute Valuation Points Nickolaus Family Volkmer Family Bentz Family Item # Item Lavonne Lowell Mark Nancy Steve Dona Kathy Ted Lisa Jim Wilma Connie Jay 1 Organ Oa k Table & Chairs Couch & Chairs Piano Tall Oa k Dres s er Dres s er/mirror Buffet John Deere Ice Box Oa k Table-eas t Walnut Table Spool Cabinet Pattern Rocker Wardrobe Des k Cupboard-Porch Plain Rocker Clock Pink Lamp Corner Shelf Painted Dres er Badges Tredle Machine Sheep Picture Wicker Rocker Chain in Bath Black Trunk Butter Crock Table 6a: Point allocations for Adjusted Winner based on absolute valuations. Red values indicate that the participant initially received the item and it was not transferred to another player. Green values show that a participant initially received the item, but it was transferred. Blue items were received as a transfer.

20 Nickolaus 20 Surplus Valuation Points Nickolaus Family Volkmer Family Bentz Family Item # Item Lavonne Lowell Mark Nancy Steve Dona Kathy Ted Lisa Jim Wilma Connie Jay 1 Organ Oa k Table & Chairs Couch & Chairs Piano Tall Oa k Dres s er Dres s er/mirror Buffet John Deere Ice Box Oa k Table-eas t Walnut Table Spool Cabinet Pattern Rocker Wardrobe Des k Cupboard-Porch Plain Rocker Clock Pink Lamp Corner Shelf Painted Dres er Badges Tredle Machine Sheep Picture Wicker Rocker Chain in Bath Black Trunk Butter Crock Table 6b: Point allocations for Adjusted Winner based on surplus valuations. Red values indicate that the participant initially received the item and it was not transferred to another player. Green values show that a participant initially received the item, but it was transferred. Blue items were received as a transfer.