|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
- 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
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
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
They should agree on the
Individual bids. Each
player bids for items, specifying a
$ bid for each item.
Calculate individual valuation
of estate. After the bidding
bids. Player with high bid receives
item, subject to cash adjustment.
Calculate cash adjustment and
- Calculate cash adjustment (each player's
fair share minus player's winning $
- Calculate cash bonus (cash adjustments
/ # players - positive number)
Final settlement. Each
player pays (or is paid) cash adjustment
plus cash bonus
Value of estate
piano / scuba
- - -
= Fair share minus item received
[8,500 - 11,200]
[8,300 - 15,100]
[7,800 - 000]
(must be positive)
/3 = 1700/3]
You can plug your numbers into the attached
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
Cognitive bias in valuing goods
of Hypothetical Bias in Stated Preference
Valuation" JAMES J. MURPHY, P.
GEOFFREY ALLEN, THOMAS STEVENS, DARRYL
A. WEATHERHEAD SSRN
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