Preference Uncertainty:
A Theory and Experimental Evidence∗
Tobias J. Klein†
September 2003
Abstract
I outline a theory of choice in situations in which the decisionmaker is
uncertain about his preferences. One prediction of this theory is that
his willingness to pay for a good is always lower than his valuation.
The outcome of an experiment, a repeated sealed-bid second price
auction, supports this finding.
Journal of Economic Literature Classification Numbers: D11, D80.
Key Words: Preferences, Valuation, Willingness to Pay, Experiment, SealedBid Second Price Auction.
∗
I would like to thank the students in my macro sections for participating in the experiment. I am grateful to Oliver Kirchkamp for an insightful discussion and to Stefanie
Brilon, Ossip Hühnerbein, and Kai Schwinger for helpful comments on an earlier version.
Finally, I would like to thank Kristina Bunjevac for her interest in my current research.
While talking about this paper, Mark Twain’s “Adventures of Tom Sawyer”, especially
the part that serves as the introductory quote, came into my mind.
†
University of Mannheim, Department of Economics, 68131 Mannheim, Germany. EMail: [email protected].
1
Preference Uncertainty: A Theory and Experimental Evidence
Tom appeared on the sidewalk with a bucket of whitewash and
a long-handled brush. He surveyed the fence, and all gladness
left him and a deep settled down upon his spirit. Thirty yards
of board fence nine feet high. Life to him seemed hollow, and
existence but a burden. [...]
Ben said:
“Hello, old chap, you got to work, hey?”
Tom wheeled suddenly and said:
“Why, it’s you, Ben! I warn’t noticing.”
“Say—I’m going in a-swimming, I am. Don’t you wish you could?
But of course you’d druther work —wouldn’t you? Course you
would!”
Tom contemplated the boy a bit, and said:
“What do you call work?”
“Why, ain’t that work?”
Tom resumed his whitewashing, and answered:
“Well, maybe it is, and maybe it ain’t. All I know, is, it suits
Tom Sawyer.”
“Oh come, now, you don’t mean to let on that you like it?”
The brush continued to move.
“Like it? Well, I don’t see why I oughtn’t to like it. Does a boy
get a chance to whitewash a fence every day?”
That put the thing in a new light. Ben stopped nibbling his
apple. Tom swept his brush daintily back and forth—stepped
back to note the effect—added a touch here and there—criticised
the effect again—Ben watching every move and getting more and
more interested, more and more absorbed. He said:
“Say, Tom, let me whitewash a little. [...] Say—I’ll give you the
core of my apple.” [...]
By the time Ben was fagged out, Tom had traded the next chance
to Billy Fisher for a kite, in good repair; and when he played out,
Johnny Miller bought in for a dead rat and a string to swing
it with—and so on, and so on, hour after hour. And when the
middle of the afternoon came, from being a poor poverty-stricken
boy in the morning, Tom was literally rolling in wealth.
Tom said to himself that it was not such a hollow world, after all.
He had discovered a great law of human action, without knowing
it—namely, that in order to make a man or a boy covet a thing,
it is only necessary to make the thing difficult to attain.
Mark Twain—The Adventures of Tom Sawyer
2
Preference Uncertainty: A Theory and Experimental Evidence
1.
3
Introduction
Economists usually assert that decisionmakers perfectly know their preferences so that they just choose the alternative among the feasible ones that
they like best. This assumption is the plinth for at least three building blocks
of economic theory: the foundations of microeconomic theory as we know it
today in Hicks and Allen (1937) as well as the foundations of general equilibrium theory in Debreu (1959) and expected utility theory in von Neumann
and Morgenstern (1944). Frequently, these normative theories are also used
in applied work as a descriptive model of economic behavior. In the context
of the exchange of commodities, in all those models, the willingness to pay
and the valuation for a good are essentially the same when actions are not
led by strategic deliberations. Moreover, both of them should not change
over time when no new information about the good is revealed.
This paper attempts to weaken this assumption of what we might call
perfect preference knowledge (as opposed to preference uncertainty). It is divided into two parts. In the first part, we outline how models of choice could
be constructed when decisionmakers are uncertain about their preferences.
We motivate why the willingness to pay for a good might be lower than
the (expected) valuation. In this context, changes of the valuation and the
willingness to pay over time, i.e. the ascertainment of preferences, become
meaningful. This in turn will allow us to interpret preferences in the classical
sense as a limit point of a complicated process of preference formation. In
the second part of the paper, we describe the outcome of an experiment, a
repeated sealed-bid second price auction, which was conducted at the University of Mannheim. We find that valuations might in fact be higher than
the willingness to pay—an outcome that can be reconciled with the proposed
theory of preference uncertainty.
There are numerous papers in economic theory in which decisionmakers
learn from prices or from the behavior of other players about the value of
some good or asset. In all those models, perfect preference knowledge and
uncertainty with respect to the value of the good or asset is assumed. Pollak
(1977) presents a model in which consumers judge the quality of a good by
its market price. Grossman and Stiglitz (1980, p. 393) investigate the role of
prices in financial markets and find that “prices perform a well-articulated
role in conveying information from the informed to the uninformed.” In the
literature on auctions, common value theory assumes that there is no difference in individual tastes among bidders. By observing the bidding behavior
of the others, which is a function of their value estimates, the observer can
learn about the value of the good. Here, of course, bidding behavior and not
Preference Uncertainty: A Theory and Experimental Evidence
4
market prices conveys information (Milgrom and Weber 1982).1
But what happens to Ben in the introductory quote? Presumably, he
perfectly knows what it means to whitewash a fence, perhaps because he has
done so before. This means that he knows which movements of hands he has
to undertake, how he has to use the brush, and so on: he knows how much
effort he has to spend in order to paint. This should enable him to judge
what whitewashing is worth to him and clearly, at the beginning, he feels
that he would consider this to be work. For him, work means that he would
only agree to whitewash if Tom compensated him, i.e. whitewashing has a
negative value to him. Later, during their conversation, he develops a desire
to whitewash himself, and even pays for this with the core of his apple. He
changes his mind—his preferences—even though no new information about
the properties of the action of painting a fence is revealed. Tom just convinces
him that it is worth liking to paint a fence.2
2.
Theory
The formal framework is as follows. There are two goods and I shall refer
to them as the numméraire and good 2. Moreover, I shall refer to the decisionmaker as Claire for reasons of brevity.3 Claire’s set X of alternatives
is R2+ , the nonnegative three-dimensional real space. She is endowed with
ω units of the numméraire and has the opportunity to trade p units of the
numméraire for one unit of good 2.
We will not only call one good the “numméraire” but we will also assume
that it is a “real numméraire”, an anchor, a reference good for which Claire
knows well—from trading it in the past and consuming the commodities she
has traded it for—how much benefit she may derive from it.4 She might
1
Another branch of the literature refers to the psychology of choice and in particular
to a process called mental accounting (Thaler 1999). According to this positive theory
of economic choice, decisionmakers evaluate choice alternatives using a system of mental
accounts. One implication of this theory is that “[m]oney in one mental account is not a
perfect substitute for money in another account (p. 185).” Consequently, the willingness
to pay or the valuation for a commodity depends on the mental account it is assigned
to. This in turn might depend on the price and the framing of an offer. Nevertheless,
preferences are exogenous.
2
This is along the lines of Stigler and Becker (1977). They explain differences in tastes
with differences in consumption experience. Here, talking with Tom about whitewashing
would correspond to such experience.
3
Actually, if this was a name, Un-Claire would be more appropriate.
4
We can think of the numméraire as being gold that serves as a substitute for money.
From a theoretical perspective, thinking of it as being money is problematic because money
Preference Uncertainty: A Theory and Experimental Evidence
units of 6
good 2
@
5
@
@
1–
A
d
@
B @C
t
d
@
@
@
@
@
@
ω
@t
-
units of the
numméraire
Figure 1: The Choice Problem
therefore have a good feeling for its value.
Figure 1 depicts an example for the tradeoff between good 2 and the
numméraire. Depending on the price, Claire chooses among her initial endowment ω 5 and offer A, B, and C. Observe that these offers differ only in
the price of good 2, not in its quantity—it is always one unit. The price for
good 2 is cheapest for offer C and most expensive for A and we denote by
pA the price that corresponds to A, etc. We have pA > pB > pC .
If we now think of good 2 as being traded in a market, we could make the
usual assumption that there is only one price. Claire now chooses between
buying good 2 at a price p and keeping her endowment ω. In Figure 1,
this corresponds to the choice between C and ω. The straight line is the
budget line and alternatives A and B are indicated with circles instead of
dots because they are not offered. Nevertheless, we could ask whether, if
prices had been different, she would have bought one unit of good 2 or not.
To move on, we assume Claire’s preferences to be strongly monotonic.
Assumption 1 (Strong Monotonicity): Preferences are strongly monotonic. That is, for x 6= y, x, y ∈ X, y ≥ x implies that y is preferred to x
and we write y  x.
Now suppose Claire reveals a preference for B over ω. Then, we could
is by itself useless and therefore, it is strictly speaking not a good.
5
In a slight abuse of notation, ω denotes both a point in Figure 1 and the amount of
the numméraire which amounts to her initial endowment.
Preference Uncertainty: A Theory and Experimental Evidence
units of 6
good 2
1–
6
BBB
BBB BBB
B
B
BBB
BBB BBB
AB B B BBB B B B C
B
d B dB B B d B B B B
B
B
BB BBBB BBBB
BBB BBBBB BBBB
BBB BBBBB
BBB
ω
t
-
units of the
numméraire
Figure 2: The Choice Problem When There is Preference Uncertainty
deduce that she would also have chosen C if it would have been offered to her.
On the other hand, we could not deduce anything about A. Alternatively, if
she reveals a preference for ω over B, we could deduce that she also prefers
ω over A but now, we cannot say anything about the decision between C
and ω. To summarize, in this simple example with only two goods and four
alternatives, a law of demand holds if preferences are monotonic.
Figure 2 is similar to Figure 1 except that we removed the budget line
and added a hatched area. This leads us to the crunshpoint of this paper.
Suppose Claire was offered C. While evaluating this alternative, it might be
that Claire does actually not know whether she is indifferent between B and
ω, or C and ω. Therefore, in this figure, I will refer to the hatched area as an
area of preference uncertainty. We assert that she assigns probability PA to
the event that she is indifferent between A and ω (this would consequently
be zero in Figure 2), PB to the event that she is indifferent between B and
ω, and so on.
Within this setup, we will be able to show that there might be constellations in which her willingness to pay for the good and her valuation do not
coincide.6 In order to do so, we have to impose more structure, though. But
let us first properly define the two concepts for our purposes. Recall that
PA , PB , and PC denote subjective probabilities for Claire being indifferent
between A and ω, etc. Then, we can define her valuation V to be
V ≡ PA · pA + PB · pB + PC · pC ,
i.e. the expected value of the price at which she is indifferent between buying
6
Note that she is a price taker so that there is no strategic component.
Preference Uncertainty: A Theory and Experimental Evidence
7
one unit of good 2 and not buying it.
For her willingness to pay, we are looking for a model which gives us the
maximum amount of money for which she would agree to buy one unit of
good 2. Coming back to Figure 2, this amount should lie somewhere in the
interval [pB , pC ]. Moreover, it should depend on the subjective probabilities
PB and PC . Observe that if Claire assigned probability one to being indifferent between C and ω then, by construction, both her valuation and her
willingness to pay would coincide and would be equal to pC . Of course, still,
she would be happier to get good 2 for a lower price. We can formalize this
by assuming that there is some real valued utility function uθ which is defined
on X and depends on a parameter θ. The parameter takes on the value C,
for example, if she is indifferent between C and ω. We normalize the utility
function such that uθ (ω) = 0. This in turn implies uC (C) = 0.7 Now, if she
is indifferent between buying and not buying C for pC , i.e. PC = 1, we could
write
uC (C) = uC (ω − pC , 1).
Observe that the assumption of strong monotonicity translates into uθ being
strictly increasing. This means that Claire likes low prices better than high
prices. Now, for instance, if pC is equal to 10 [units of the numméraire], if
PC = 1, both her willingness to pay and her valuation would be equal to 10.
Still, a little more structure is needed in order to say something about
the relationship between the willingness to pay and the valuation within this
framework. To motivate this, suppose that pC is equal to 10 and make the
thought experiment of comparing, for a given value of θ, Claire’s utility at a
price of 4 with Claire’s utility at a price of 2. We have already assumed that
her utility at a price of 2 is higher than at a price of 4 (strong monotonicity),
but we have not said by how much. We will assume that it is less than twice
as high which translates into the assumption of uθ being strictly concave
for all θ. Additionally, to keep things simple, we assume that the utility
functions are such that for all p ∈ R+
(1)
uA (ω − p, 1) = uB (ω − p + α, 1) = uC (ω − p + β, 1)
for some α, β > 0.8 If (1) holds, drawing uθ on the ordinate and ω − p on the
abscissa, the horizontal distance between uA and uB is α, between uA and
uC it is α + β, and between uB and uC it is β. To be explicit, we make
7
This is independent of her subjective probability measures, by construction of uC .
Recall that θ is equal to C whenever Claire is indifferent between C and ω. Hence,
uC (C) = uC (ω) = 0 by the normalization. Similarly, uA (A) = 0 and uB (B) = 0.
8
This will simplify the analysis substantially. We will argue later that a weaker condition is sufficient for our purposes.
Preference Uncertainty: A Theory and Experimental Evidence
uB 6
uC
uB
©©
z t©©
©©
0
8
uC
©©
©
©
ω − pB ©©
©
t©
t y t t©©
¢
¢ ω − pC
ω−p
V
¢
¢
¢
¢tx
¢
¢
¢
¢
¢
¢
¢¢
Figure 3: The Graphical Argument of the Proof
Assumption 2 (Shape of the Utility Function): Preferences are such
that uθ is strictly concave for all θ. Moreover, we assume that for all p ∈ R+ ,
(1) holds for some α, β > 0.
In Figure 2, both PB and PC are positive.9 Now, let us assert that Claire
maximizes expected utility which, by construction, depends on the price p of
one unit of good 2 and takes on the form
(
PB · uB (ω − p, 1) + PC · uC (ω − p, 1) if she buys 1 unit of good 2
U (p) =
0
otherwise,
where the second line follows from the normalization. Then, we can define
her willingness to pay for one unit of good two to be equal to the price p that
solves
(2)
PB · uB (ω − p, 1) + PC · uC (ω − p, 1) = 0.
At this price, she is indifferent between buying and not buying one unit
of good 2. It remains to show, for our example, that Assumption 1 and
Assumption 2 together imply that this price is lower than her valuation V .
Our proof will be based on graphical arguments and can be comprehended
using Figure 3 which is meant to suit Figure 2. The utility functions are
9
Otherwise, the hatched area would contain just one alternative. Since both probabilities add up to one, we have that each one is less than one.
Preference Uncertainty: A Theory and Experimental Evidence
9
drawn piecewise linear.10 We draw uB and uC for different values of ω − p
and for one unit of good 2 consumed. Observe that the horizontal distance
between the two is always equal. Then, uB is equal to zero if ω − pB units
of the numméraire are consumed and uC is equal to zero if ω − pC units
are consumed, by construction of the utility functions. We draw the picture
for PB = PC = 0.5 so that V , the “expected zero”, lies right in the middle
between the zeros of the two utility functions.
Now, we are looking for a solution to
0.5 · uB (ω − p, 1) + 0.5 · uC (ω − p, 1) = 0.
This is equivalent to
uB (ω − p, 1) = −uC (ω − p, 1),
that is, in our graph we are looking for a value of p so that uB = −uC . For
this value, in the graph, ω − p = y, uC = x, and uB = z. Now, one can see
that x must be to the right of V . Therefore, the willingness to pay for the
good must be smaller than its expected valuation, which is given by the value
of p that solves ω − p = V . This argument holds for every value of PB and
PC whenever the slope of uC is no less than the slope of uB and different for
some values between V and ω − pC . A sufficient condition for this to be the
case is given in Assumption 2 and therefore, we can argue that Assumption
2 is stronger than necessary. This completes the proof.11
There is a nice similarity between this theory of preference uncertainty
and von Neumann-Morgenstern expected utility theory. Here, concavity together with another condition on the shape of the utility functions implies
that the willingness to pay lies below the expected valuation. In expected
utility theory, we have that the willingness to pay for a risky asset lies below its (discounted) expected value whenever the utility function over final
outcomes is concave which is true whenever the decisionmaker is risk averse.
Therefore, one could say that this section’s model of preference uncertainty
incorporates risk aversion with respect to preference uncertainty as opposed
to uncertainty with respect to uncertainty about states of the world, or properties of the good.
To summarize our theoretical deliberations, once there are situations in
which decisionmakers are uncertain about their preferences, and once we
10
I must admit that I did not get my Latex picture environment to draw a nice strictly
convex function and I hope that the argument will nevertheless become clear. Of course,
they should be strictly concave and uB should be similar to uC , just shifted to the left.
11
If there were more than two values of θ for which Pθ > 0, a similar argument would
hold.
Preference Uncertainty: A Theory and Experimental Evidence
10
assume that they are in a way risk averse with respect to this uncertainty,
we should expect them to be willing to pay less for a good than their expected
valuation actually is.
3.
Experimental Evidence
In this section, we investigate the outcome of an experiment. It was conducted on July 22 and 23, 2003 at the University of Mannheim. The participants were first year undergraduate students in Business Administration and
Economics. The breadboard construction is rather simple: the experiment
was a two times repeated (hence, three times in total) sealed-bid second price
auction in which bidders were shown the bids from the previous round if there
was any. My first intention was to find out whether valuations change over
time. Secondly, I wanted to see whether there is any difference between the
valuation bidders state and their bids.
One could interpret such a second price auction as a so-called private value
auction.12 This means that bidders do precisely know their valuation for the
object but not the other bidders’. An advantage of the second price auction
is that they do not have to form expectations on the other bidders’ valuations
because it is a dominant strategy to bid one’s valuation even irrespective of
one’s attitude towards risk. A first price auction, for example, requires much
more sophistication since bidders do have to form expectations on the other
bidders’ behavior and especially the second highest bid.
Raw data can be found in the Appendix. There were two groups, Group
A (my Tuesday section) and Group B (my Wednesday section). On Tuesday,
the participants were told that only the third round would count and I kindly
asked them to act as if every round could count. On Wednesday, I did not
tell them how many times the auction will be conducted. The object to
be auctioned was an old English-German/German-English pocket dictionary
on Tuesday and a French-German/German-French dictionary on Wednesday.
By my count, they were from the early 20th century, maybe the 1920’s. Both
were similar and in good condition. The reason why I chose these books was
that it is very likely that nobody knew a market price for them so that
bids are unlikely to be biased towards market prices.13 Participants had the
opportunity to have a look at the book before the auction started. Before the
12
Cf. Kagel (1995) for a comprehensive survey on experiments on auctions.
After the experiment, I asked the winners for the reasoning behind their bids and
it turned out that neither of them had an idea what the market price for such a book
actually is.
13
Preference Uncertainty: A Theory and Experimental Evidence
11
10
valuation
bid
8
6
4
2
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Figure 4: Group A, First Round
first auction started, I took a decent amount of time to explain how a second
price auction works. It was my impression that this was well understood.
In every round, participants were asked to write on a piece of paper their
valuation for the book, a bid for the sealed-bid second price auction, and their
seat number on Tuesday, respectively a nickname on Wednesday since there
were no seat numbers in the lecture room. After every round, I wrote all bids
on the blackboard without sorting them. Figures 4 through 6 depict bids and
valuations for the Tuesday group in the first, second, and third round. On
the ordinate, we have Euro amounts, and on the abscissa, we count as if we
were looking at demand curves. First of all, it is conspicuous that the mean
valuation is much higher than the mean bid.14 The average spread, which
is defined as the difference between the valuation and the bid, is 1.33 in the
first round, 0.87 in the second, and 0.94 in the third. Unfortunately, the
groups were too small so that we cannot prove with our statistical toolbox
of hypothesis tests that they were different to zero. Spreads decline after
the first round which is an indication for a learning process of the bidders.
14
This is consistent with findings of Coppinger, Smith, and Titus (1980) and Schmith
(1980). Kagel, Harstad, and Levin (1987), however, find that bids are in general higher
than the bidders’ valuations.
Preference Uncertainty: A Theory and Experimental Evidence
12
10
valuation
bid
8
6
4
2
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Figure 5: Group A, Second Round
They are roughly the same in the second and third round.15 Clearly, more
repetitions are necessary to see whether spreads converge to zero or not.
If they would not, this could be reconciled with the theory of preference
uncertainty that was outlined above.
A more common explanation for this phenomenon would be that participants were not able to find the optimal bidding strategy even though they
perfectly knew their tastes. In principle, one could distinguish between these
two competing explanations by conducting another experiment. In this experiment, one would have to make sure that one group has an almost perfect
feeling for the value of an object, for example by telling participants that
they would not get the object itself but they would have to sell it back at a
certain price which could in principle differ across participants. This would
implant them a valuation for the object since it is reasonable to believe that
they have a good feeling for the value of money. Then, one could compare
the bidding behavior of the two groups.
Figures 7 and 8 show how the distribution of bids and valuations has
15
Even though average spreads did not decline from the second to the third round, the
average valuation and the average bid did go down by 0.57 and 0.31, respectively. One
could take this as an indication for the fact that it actually made a difference that the first
two rounds did not count. This is supported by the absence of such an effect in Group B.
Preference Uncertainty: A Theory and Experimental Evidence
13
10
valuation
bid
8
6
4
2
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Figure 6: Group A, Third Round
changed over time. In principle, there could be at least three reasons for
changes in either one—if there were any. The first one is introspection:
From round to round, bidders learn about themselves, their valuation for the
book and, hence, their putative optimal bidding strategy. The second one is
learning from other’s bidding behavior about the (market) value of the book.
This requires that bids convey information. A possible third explanation ist
that bidders did just not understand the game so that they learn how to
play it over time. In Figures 7 and 8 one can see that there are almost no
dynamics in both, bids and valuations, so that neither possible explanation
for dynamics seems to be relevant here. If this was robust, i.e. if there
were in fact no dynamics, this in turn would support the outlined theory of
preference uncertainty.16
Figures 9 and 10 show similar graphs for the Wednesday group except
that a logarithmic scale was used for the ordinate because bids were much
more dispersed. Patterns were, generally speaking, similar. Here, as for
the Tuesday group, over time, bids and valuations did not change either. It
16
To be precise, the theory of preference uncertainty implies that if the uncertainty remains, the willingness to pay and the valuation, as defined above, do not change over time.
Moreover, an ascertainment of preferences implies that the willingness to pay converges
towards the valuation.
Preference Uncertainty: A Theory and Experimental Evidence
14
5
1st round
2nd round
3rd round
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Figure 7: Group A, Bids in First, Second and Third Round
seems to be an artefact that average bids and valuations rose over time. This
could be explained by three (in the first two rounds) and two (in the third
round) bidders trying to outbid each other. Interestingly, even though the
two books were similar, the first one was sold for 5.10 Euros on Tuesday17
and the second one for 31.01 Euros on Wednesday. Again, there are at least
two competing explanations for this. One possible explanation is that in
one group, the information about the market value of the book was better
than in the other group. Another explanation would be that participants
were uncertain about their preferences so that no limit or reference price
actually exists towards which prices could converge. Then, prices would
arise at random or at least, they would be influenced by the interaction
between bidders through repetitions of the auction. Then, an ascertainment
of preferences might disembogue in an equilibrium. As I have argued before,
conducting another experiment as the one I have described above would help
to distinguish between these two competing explanations.
17
On Tuesday, in the third round, three participants were bidding 5 Euros. Only one of
them was actually willing to pay 5.10 when I asked them. The book was sold to him.
Preference Uncertainty: A Theory and Experimental Evidence
15
10
1st round
2nd round
3rd round
8
6
4
2
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Figure 8: Group A, Valuations in First, Second and Third Round
4.
Concluding Remarks
The common point of view is that preferences are given whereas information that is available to decisionmakers might be scarce and incomplete. In
this paper, we have proposed a theory which describes choice the other way
around: Not the properties of the good are unknown but the decisionmaker’s
preferences. Then, it becomes meaningful to think of preferences as depending on prices which is common practice in Marketing and which is of course
meaningless when preferences are exogenous.
Nevertheless, we shall not keep secret that we face a fundamental identification problem if we were keen to test this theory: it is hard to find experiments that distinguish between a lack of information with respect to the
properties of the good, which underlies the common point of view, a lack of
knowledge of the decisionmakers’ preferences, which underlies the proposed
theory, and, thirdly, a lack of rationality.
Nonetheless, an implication of the theory that was outlined in this paper
is that the willingness to pay for a good is always below the (expected) valuation. This is consistent with the finding that bids in a sealed-bid second price
auction are often lower than the bidders’ valuation which was the outcome
of our experiment.
Preference Uncertainty: A Theory and Experimental Evidence
16
100
1st round
2nd round
3rd round
10
1
0,1
1
2
3
4
5
6
7
8
9
10
11
12
13
Figure 9: Group B, Bids in First, Second and Third Round
In my opinion, we lack a formal theory of preference formation.18 I would
claim that when we were born, nobody was endowed with complete and
transitive preferences. If we claim that in our models, preferences of decisionmakers are given, then I find it natural to ask how these preferences came
about and how we could know that the process that forms preferences has
already come to an end.
So, does Tom trick Ben? But, if this was the case, why did Ben even offer
the core of his apple to Tom? Isn’t Ben happy at the end of the day?
18
Cf. Bowles (1998) for an interesting unformal paper on the evolution of preferences.
Preference Uncertainty: A Theory and Experimental Evidence
100
1st round
2nd round
3rd round
10
1
0,1
1
2
3
4
5
6
7
8
9
10
11
12
13
Figure 10: Group B, Valuations in First, Second and Third Round
17
seat
3
4
5
6
12
13
16
17
18
19
21
22
23
24
25
26
27
28
29
30
32
round 1
valuation bid
10
3
5
2.5
1
5
NA
NA
1
1
1
0.5
5
1
2
2
5
3
8
3
0.5
1
2.5
3
5
5
2
2
0
0
5
5
5
1
5
2
4
3.5
6
3
0
0
round 2
spread valuation bid
7
4
3.5
2.5
4
3
-4
1
1.01
NA
1
0
0
2
1.5
0.5
2
1
4
5
2.5
0
2
2
2
5
3
5
4
3
-0.5
0.5
1
-0.5
3.5
4.5
0
5
5
0
2
2
0
0
0
0
5
4.5
4
5
1
3
5
4.9
0.5
5
2
3
6
3.5
0
1
1
continued on next page
Appendix: Data
spread
0.5
1
-0.01
1
0.5
1
2.5
0
2
1
-0.5
-1
0
0
0
0.5
4
0.1
3
2.5
0
round 3
valuation bid
4
0.5
0.5
0.5
1
0
NA
NA
1
0.98
2
1.5
5
2.1
2
2
5
5
3
5
1.5
1
4.5
3.5
5
5
2
2
0
0
3
3
5
1
5
2.1
5
2
6
4
0
0
spread
3.5
0
1
NA
0.02
0.5
2.9
0
0
-2
0.5
1
0
0
0
0
4
2.9
3
2
0
Preference Uncertainty: A Theory and Experimental Evidence
18
seat
33
34
36
37
39
42
45
47
49
52
56
58
61
64
65
mean
std.
10% quantile
20% quantile
30% quantile
40% quantile
median
continued from previous page
round 1
round 2
valuation bid spread valuation bid spread
0
0
0
0
0
0
3
1
2
2.5
1.25 1.25
2
2
0
10
NA 0*
5
0.5 4.5
5
NA 0*
0
0
0
0
0
0
1
0.5 0.5
1
1
0
0
0
0
0
0
0
0.5
0.5 0
2
1
1
8
2
6
8
2.5 5.5
0.1
0.1 0
0.5
0.1 0.4
5
0.5 4.5
3
0.2 2.8
1
1.01 -0.01
NA
NA NA
1
0.1 0.9
1
0.1 0.9
2
1
1
2
2
0
1
0.5 0.5
1.5
1
0.5
2.93
1.61 1.33
2.99
1.79 0.87
2.67
1.50 2.25
2.40
1.54 1.36
0
0
-0.006
0.2
0
0
0.5
0.42 0
1
0.14 0
1
0.5 0
1.1
1
0
1
1
0
2
1
0
2.00
1.00 0.50
2.00
1.25 0.50
continued on next page
round 3
valuation bid
0
0
2
1.6
0
0
2
2
0
0
0.8
0.8
0
0
2
1
8
2.3
1
0.5
3
0.3
1
1.01
1
0.1
2
0
1.5
1
2.42
1.48
2.07
1.51
0
0
0.74
0
1
0.5
1.5
0.908
2.00
1.00
spread
0
0.4
0
0
0
0
0
1
5.7
0.5
2.7
-0.01
0.9
2
0.5
0.94
1.53
0
0
0
0
0.40
Preference Uncertainty: A Theory and Experimental Evidence
19
seat
Bart Simpson
Bernd
Christoph
David
DP
FB
Frederik
Hans
Marcel
MK
Susi
uncle tom
round 1
valuation bid
15
1
1
0.1
0.5
0.6
0.3
0.4
3
2
20
25
0.37
0.32
0.5
0.1
2
1
0.95
0.4
50
20
1
1
Table A: Group A
round 2
spread valuation bid
14
31
30.99
0.9
1
0.35
-0.1
NA
0.84
-0.1
0.5
0.8
1
1.5
1
-5
20
30
0.05
0.37
0.36
0.4
0.5
0.5
1
2
2
0.55
0.99
0.7
30
50
2.2
0
1
1
continued on next page
Numbers with an asterisk were set to zero.
seat
60% quantile
70% quantile
80% quantile
90% quantile
spread
0.01
0.65
0*
-0.3
0.5
-10
0.01
0
0
0.29
47.8
0
continued from previous page
round 1
round 2
valuation bid spread valuation bid spread
3.4
2
0.66
3.7
2
0.66
5
2
2
4.8
2.5 1
5
3
3.2
5
3
1.4
5.6
3.3 4.5
5
4.3 2.68
round 3
valuation bid
41
39.98
1
0.48
NA
1.02
0.5
0.7
1.3
0.9
20
31.01
0.37
0.5
0.74
2
2
0.99
0.99
50
2.2
1.01
1.01
round 3
valuation bid
2
1.54
3
2
5
2.14
5
3.8
spread
1.02
0.52
0*
-0.2
0.4
-11.01
-0.37
-0.24
0
0
47.8
0
spread
0.5
1
2.14
2.96
Preference Uncertainty: A Theory and Experimental Evidence
20
round 1
valuation bid
1
0.49
7.36
4.03
14.26
8.28
0.396
0.144
0.5
0.352
0.77
0.4
0.99
0.472
1.00
0.60
1.2
1
2.4
1
10.2
1.6
19
16.4
Numbers with an asterisk were set to zero.
seat
xxxx
mean
std.
10% quantile
20% quantile
30% quantile
40% quantile
median
60% quantile
70% quantile
80% quantile
90% quantile
Table B: Group B
continued from previous page
round 2
spread valuation bid
spread
0.51
1
0.1
0.9
3.32
9.16
5.45 3.07
9.04
16.14
11.13 13.74
-0.1
0.5
0.352 0
-0.06
0.598
0.416 0
0.03
0.993
0.62 0
0.33
1
0.78 0
0.51
1.00
0.84 0.01
0.62
1.3
1
0.066
0.94
1.85
1.4
0.374
1
16.4
2.12 0.59
11.4
29.9
24.44 0.85
round 3
valuation bid
1
0.09
10.85
6.27
18.16
13.11
0.5
0.392
0.99
0.568
1
0.724
1
0.868
1.01
0.99
1.3
1.012
2
1.412
20
2.12
41
25.248
spread
0.91
2.99
13.82
-0.344
-0.224
-0.216
-0.008
0.00
0.08
0.448
0.754
0.998
Preference Uncertainty: A Theory and Experimental Evidence
21
Preference Uncertainty: A Theory and Experimental Evidence
22
References
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