WFU Law School
Law & Valuation
2.1.3 Risk aversion

2.1.4 Fair Division Procedure

We all value things differently. For example, some of us prefer a house with a view, others of us prefer one with modern plumbing, yet others one with a good location. Typically, markets allocate assets to those who value them most. But sometimes markets are unavailable or are expensive to replicate.

Can procedures be constructed to value non-financial assets (and the heterogeneous "returns" they promise) when we don't have agreement on values? That is, can procedures be used as a substitute for markets? For example, suppose Mom has just baked a cake for Jane and Dick -- equally. What if Jane likes frosting and Dick prefers quantity, what is the best way to value the cake and divide it between them?
  • Market valuation. One possibility would be to sell the cake (at fair market value) and divide the cash proceeds. Then Jane and Dick could each buy a smaller cake with lots of frosting and Dick a fruitcake without frosting, or even something else. But market transactions are not free. Even selling through E-bay has its risks and costs.
  • Authority valuation. The two could submit their distribution problem to an authority figure - perhaps Mom or even a judge. But even an impartial authority may not know about their individual preferences (frosting versus quantity, smooth versus crunchy) and make mistakes.
  • Fair division. The two could use a method to divide the cake so that each considers the result to be fair. One age-old method is for Jane to cut and Dick to choose -- or the other way around and, of course, under Mom's watchful eye. Their method becomes a "market" for their expression of their individual preferences.

Game theory has developed procedures (algorithms) for the division among n-players of a divisible object (a cake) or a set of objects (property in divorce) so that each player considers the division "fair" according to his or her value system. For example, how might neighbors solve the problem that one consider the other to be a nuisance. See Meurer, "Fair Division" 47 Buff. L. Rev. 937 (1999).


Here are some algorithms for solving Mom's cake problem:

Cut and choose method. This method works for two players, as in the case of Jane and Dick. (Although Mom often takes credit for the method, it was first described in Hesiod's Theogeny where Prometheus placed meat in two piles and Zeus got to choose.)

Moving knife procedure. Suppose there are three kids who are to split the cake. Can this be done fairly? One strategy is for Mom to place a knife over one corner of the cake and begin to move it slowly across the cake. When any of the kids says "stop," that kid (K1 let's say) gets the piece. Presumably K1 thinks she got at least 1/3, and K2 and K3 (who did not speak up) believe the remainder is at least 2/3. Mom keeps on moving the knife, until K2 says "stop." K2 is happy, because he thinks he got at least 1/3 -- and the same for K3! All thought their piece was "fair" at the time they chose. But there's a problem. K1 might be envious. For example, K1 might think K3 got more than 1/3 when K2 said "stop" too early. Although each was happy with her or his decision when made, there might not be a perception of equal sharing -- in our hierarchical world, a condition of complete fairness. [Saul Levmore and Elizabeth Cook - continuous envy-free method / Theodore Hill - n-countries divided contiguous disputed territory, all perceive get 1/n th].

Trimming method. Suppose there are three players who want to divide indivisible goods, such as heirs of an estate entitled to an equal share of cash and goods. Is there a procedure that will recognize their individual assessments of the goods? A procedure known since the 1940s, called Knaster's Inheritance Procedure, purports to fairly divide a set of indivisible goods. Here's how it works:

  • Each player values the various items according to how much she would pay for each one -- the players must be honest because they'll have to "purchase" items with a high valuation and will reduce their "cash settlement" with a low valuation.
  • Using these valuations, each player's own "fair share" of the estate is computed -- that is, how much the player believes she should receive in goods and cash.
  • The players then must "purchase" the goods for which they had the highest bids -- this cash can either come from the cash portion of the estate or their own pockets.
  • The players then are subject to a "cash adjustment" -- a positive amount if their "purchases" are less than their "fair share" and a negative amount if their "purchases" exceed their "fair share."
  • Each player then receives a "cash bonus" equal to the total cash adjustments divided equally among all the players -- this cash bonus, if the players disagree on valuation, is stated as a positive number.
  • In a final settlement the players receive their goods after paying (or being paid) their cash adjustment plus cash bonus.

For additional valuation methods that use mathematical models and game theory, see "Merger Simulations."

Example

An example illustrates the trimming method:

Aunt Estelle dies, and her last will and testament leaves "all my worldly possessions to Peter, Paul and Mary - to be divided equally among them." Excitedly, the three beneficiaries scour Estelle's rented beachside apartment to discover she owns only a scuba outfit, a sailboat, and a piano. What should the three do to divide these worldly goods?


Answer

They should agree on the following procedure:

Individual bids. Each player bids for items, specifying a $ bid for each item.

Calculate individual valuation of estate. After the bidding --

  • Calculate each player's total valuation (total $ bids by player)
  • Calculate each player's fair share ($ bids by player / # players
Identify winning bids. Player with high bid receives item, subject to cash adjustment.

Calculate cash adjustment and cash bonus

  • Calculate cash adjustment (each player's fair share minus player's winning $ bids)
  • Calculate cash bonus (cash adjustments / # players - positive number)

Final settlement. Each player pays (or is paid) cash adjustment plus cash bonus

Knaster's Inheritance Division

Peter
Paul
Mary
Sailboat
14,300
15,100
13,200
Piano
8,200
7,300
7,300
Scuba gear
3,000
2,500
2,900
Value of estate
25,500
24,900
23,400
Fair share
8,500
8,300
7,800
Items received

piano / scuba
[8,200+ 3,000]

sailboat
[15,100]

- - -
Cash adjustment = Fair share minus item received
-2,700
[8,500 - 11,200]
-6,800
[8,300 - 15,100]
+7,800
[7,800 - 000]
Cash bonus
(must be positive)
[2,700+6,800-7,800 /3 = 1700/3]
567
567
567
Final settlement
piano / scuba
-2,133
sailboat
-6,233
- - -
+8,366

You can plug your numbers into the attached spreadsheet.

Why does each player perceive that he or she received at least 1/3 of the estate -- fair division? Consider Peter who received goods he valued at $11,200 minus a cash payment of $2,133. In his mind his "net" take was $9,067 -- which is more than the $8,500, the value he placed on his 1/3 interest in the estate. You can confirm that the same is true for Paul and Mary.

Why does each perceive that she did as well or better than the others -- division without envy? Consider Peter, who sees Paul as netting $8,067 ($14,300 boat minus $6,233 payment) and Mary as netting $8,366 -- both less than his net of $9,067. By the same token Paul is envy-free, since from his perspective he netted $8,867 (a $15,100 boat minus $6,233 payment) -- more than Peter's net of $8,667 (goods worth $9,800 minus $2,133) and Mary's $8,366. Mary, with her cash, does not envy the others' goods..

Curiously, fairness is assured even though the players disagree on value. In fact, envy is prevented precisely because they disagree.


Student paper

Valuation techniques often have to be applied in different and difficult circumstances. For example, how should the value of frozen fertilized embryos be determined in the context of equitable distribution? See Catherine Pappas, The Price of Potential: Classification and Valuation of Cryogenically Frozen Embryos in Equitable Distribution.

Cognitive bias in valuing goods

"A Meta-Analysis of Hypothetical Bias in Stated Preference Valuation" JAMES J. MURPHY, P. GEOFFREY ALLEN, THOMAS STEVENS, DARRYL A. WEATHERHEAD SSRN 437620

Individuals are widely believed to overstate their economic valuation of a good by a factor of two or three. This paper reports the results of a meta-analysis of hypothetical bias in 28 stated preference valuation studies that report monetary willingness-to-pay and that used the same mechanism for eliciting both hypothetical and actual values. The papers generated 83 observations with a median value of the ratio of hypothetical to actual value of 1.35, and the distribution has severe positive skewness.

 

2.1.3 Risk aversion

©2003 Professor Alan R. Palmiter

This page was last updated on: March 21, 2005