Theoretical Economics Letters, 2013, 3, 220-225
http://dx.doi.org/10.4236/tel.[phone] Published Online August 2013 (http://www.scirp.org/journal/tel)
Real Estate Pricing under Two-Sided Asymmetric
Information
Jeremy Sandford1, Paul Shea2
1University of Kentucky, Lexington, USA
2Bates College, Lewiston, USA
Received June 5, 2013; revised July 5, 2013; accepted July 15, 2013
Copyright © 2013 Jeremy Sandford, Paul Shea. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
What happens when a buyer and a seller each have private information about the value of an item for trade, as is par-
ticularly common in real estate? We solve for the equilibrium price under both public information, where the seller
shares his information with the buyer, and private information, where the seller is constrained to be unable to credibly
share. Our main results are 1) even under public information, the equilibrium price differs from the expected value of
the item, 2) under private information, prices follow a step function, with small changes in information generically hav-
ing no effect on price, and 3) equilibrium price is more sensitive to informational changes under private information
than public information. This under-studied game of 2-sided asymmetric information reasonably describes real estate
transactions.
Keywords: Asymmetric Information; Game Theory; Information
1. Introduction
A buyer and a seller bargain over an asset of uncertain
value, such as real estate. Both have private information
relevant to the value of the asset. The seller makes a take-
it-or-leave-it offer, which the buyer either accepts or re-
jects. We characterize the function mapping the seller’s
private information into his optimal offer, under both
public information (the seller credibly reveals his infor-
mation) and private information. We find that even under
public information the seller’s optimal offer is less sensi-
tive to variations in private information than is the ex-
pected value of the asset. Under private information, the
game’s only equilibrium is a step function, under which
the seller charges one of several discrete prices, depend-
ing on his information. Most surprisingly, the average
sensitivity of price to the seller’s information is greater
under private information than public information,
meaning that the seller charges a relatively higher price
given favorable information if his information is private.
There is some evidence that the seller’s profit is higher
under public information, meaning that he would be
willing to pay to credibly reveal his private information,
even given the risk that this information will be unfavor-
able.
Markets in which both sides have private information
have not been studied extensively. Some existing work
has studied the propensity to settle a lawsuit when both
sides have private information about their likelihood of
success (Friedman and Wittman (2007) [1], Daughety
and Reinganum (1994) [2]), bargaining over labor dis-
putes (Kennan and Wilson (1993) [3]), or the setting of
point spreads in gambling markets (Sandford and Shea
(2013 [4]), Ottaviani and Sorenson (2006) [5], and Steele
and Zidek (1980) [6]). This paper extends the framework
of Sandford and Shea (2013) [4], which finds the unex-
pected result that bookmakers do not optimally set gam-
bling lines so that each side is equally likely to win when
both the bookmaker and gambler have private informa-
tion, to a real estate market, in which buyers and sellers
negotiate over the price of an asset of uncertain value1.
Our results on the relationship between optimal price and
whether information is public or private, and on the elas-
ticity between price and information are novel to the lit-
erature.
2. Model
Seller and Buyer negotiate over an asset, such as real
1Previous papers on real estate markets do not consider the implications
of asymmetric information. See, for example, Yavas (1992) [7], and
Yavas and Yang (1995) [8].
C
opyright © 2013 SciRes. TEL

J. SANDFORD, P. SHEA 221
estate. The value of the asset to Buyer is X  [0,1].
Seller’s value of retaining the asset is

1
X
A, so the
efficient outcome is for Seller to sell to Buyer. Both agents
receive information relevant to X. For Seller, this signal
may represent the information that they have acquired
from having owned the property, or from getting a pro-
fessional appraisal. For Buyer, this signal may result
from his own appraisal, advice from his real estate agent,
or his own preferences over type of house.
Formally, suppose that Seller draws information z1
[‒1,1] while Buyer draws information z2  [‒1,1].
Conditional on both pieces of information, the true dis-
tribution and density of X are given by and

12
|,Gxz z

12
|,

g
xzz , respectively. We assume that G and g have
the following functional forms:


2
1212 12
1
,1
22
x
Gxzzxz zz z



 


1212 12
1
,1
2
g
xzzz zxz z 
For the remainder of the paper we consider the case
where Seller makes a take-it-or-leave-it offer to Buyer
consisting of price p, which Buyer will either accept or
reject, based on his private information. However, the
model’s results are qualitatively similar if we instead
assume that Buyer makes the offer.
Buyer will accept the offer only if his information is
sufficiently favorable, if 2
zz for some z which
depends on p and, if known to Seller, z1. Formally, pay-
offs are then as follows:
0
R
B
 
12
,
A
BEXzz p
  

12
,
1
R
S
EXzz z
A





A
Sp
 
Buyer’s strategy consists entirely of a choice of z,
conditioned on p and, if available, z1. Seller’s strategy
consists of a price p, conditioned on z1. We first consider
the case of public information, where z1 is public infor-
mation, known to both Buyer and Seller.
2.1. Public Information
Suppose Seller’s information z1 is known to Buyer, while
z2 is Buyer’s private information. This may represent a
case, for example, where Buyer’s appraisals are able to
successfully reveal all relevant information about the
property, where Seller can commit to truthfully revealing
his information, or it can emerge endogenously from a
model in which Seller chooses whether or not to disclose
and Buyer believes that any Seller who doesn’t disclose
z1 has very bad information (low z1), and so Seller is al-
ways better off by disclosing z1, regardless of its value.
Suppose that both z1 and z2 are independently uni-
formly distributed over [‒1,1], that is z1, z2 ~ U[‒1,1].
We analyze the game backwards, first determining how
Buyer sets


1
,zpz and then how Seller sets p(z1). First,
given z1, Buyer is better off accepting an offer of p if
AR
B
B

 . In the case of


21
Buyer is indif-
ferent between accepting and rejecting the offer, while he
accepts (rejects) for z2 greater than (less than)
,,zzpz


1
,zpz.
Therefore, z is defined implicitly by (1).


11
,,EXzzpzp



(1)
For very high (very low) prices p, Buyer always re-
jects (always accepts). For intermediate prices, Lemma 1
establishes a functional form for z and shows that z
is linearly increasing in p and linearly decreasing in z1.
Lemma 1: Under public information, given informa-
tion z1 Buyer optimally accepts an offer of p if and only if


2
,zzpz1 , where:

1
11 1
1
71
1if
12 12
51 71
,12 6if,
12 121212
51
1if
12 12
pz
zpzp zpzz
pz




 






1
Proof: Performing the integration in (1):



11
1
11
0
,,
1
1d
2
EXzzpz
x
zz xzzxp




 



1
222 33
11
0
244 33
xxx xx
zzzz 
11
111 11
244 33
zzzz  p


1
,12zpzp z
1
6 (2)
Equation (2) and the fact that z2 is bounded between
−1 and 1 establish the lemma. □
Seller takes Buyer’s optimal strategy


1
,zpz as
given in setting p. In maximizing his payoff, Seller con-
siders the gain in setting a higher price (higher profit in
the event Buyer accepts) against the cost (lower prob-
ability of acceptance). First, it is immediate that for suf-
ficiently low A, Seller will never sell—he will set a high
p so that Buyer rejects the offer. This is because for low
A, Buyer accepts only if his private information suggests
the value of the asset is worth more than p, in which case
Seller would be better off keeping the land for himself.
Second, it is clear that Seller’s optimal price must be

J. SANDFORD, P. SHEA 222
increasing in z1; as Seller has more favorable public in-
formation, Buyer’s willingness to pay and Seller’s op-
portunity cost of selling both increase. Lemma 2 solves
for Seller’s optimal strategy p(z1) and formalizes the
above claims.
Lemma 2: Under public information, Seller optimally
sets a price of p(z1) where:

1
1
7
12
24 1
z
pz
A



(3)
p is increasing in z and decreasing in A. As A ap-
proaches zero,
1




11
,zp zz
 approaches 1 for all z1.
Proof: Given z1, Seller chooses p to maximize ex-
pected profit,



 

 


212
21
21
11
12 22
10
12 2
1
2
11
2
11
,
Pr ,1
Pr ,
111
,ddd
22
1
111 d
1424 2
1
11 11111
14244848 242
1
111111
4(1)61212 62
z
z
z
z
EXzz z
zzpz A
zzpz p
xgx zzxzz
A
pz
zz z
A
pz
zzzzz
A
pz
zzz zz
A




 














 




(4)
Taking the derivative of (4) and setting to zero then
yields (3). That p(z1) is increasing in z1 and decreasing in
A follows from inspection of (3). That




11
,1xp zz


for sufficiently low A follows from inserting p from (3)
into (2). □
We analyze the public information equilibrium by
comparing p(z1) to two bench marks. First, we show that
p(z1) has no consistent relationship with E(X|z1). Second,
in the next section, we compare the public information
p(z1) with its counterpart when z1 is Seller’s private in-
formation, and cannot be credibly revealed.
First, a straightforward calculation shows that:
1
11
212
EXz z
 1
(5)
Consider the value of 1
p
EXz
 

, calculated from
(3) and (5):
 
11
15
12121212
AA
pEXz z
A
A

 
 
(6)
Equation (6) is decreasing in z1; Seller extracts a
higher price relative to the expected value of the asset
when his information is unfavorable. To put another way,
Seller’s optimal price is less elastic in his own informa-
tion than is the expected value of the asset. Figure 1
demonstrates an example where A = 0.2, implying that
Seller obtains 25% less value from the property than
Buyer, and that the average of 1
p
EXz
 

from (6)
is 0. That p(z1) is flatter than 1 EXz
 
reflects the
greater price sensitivity of the latter.
2.2. Private Information
We now consider the case of private information where
z1 is known only to Seller, and z2 only to Buyer. In this
case, the Seller’s private knowledge from owning the
property cannot be fully extracted by Buyer, and by as-
sumption, Seller cannot credibly reveal z1 to buyer.
We again solve the game backwards. Upon observing
a price set by Seller, Buyer forms beliefs over the distri-
bution of z1. Call this belief f(z1), and let denote
Buyer’s expected value of z1 based on a price of p. Again,
Buyer will optimally accept an offer if and only if his-
signal z2 is above some threshold,

1
e
zp


zp
, defined im-
plicitly by:


1,EXfz zp



(7)
Lemma 3 establishes an analogue of Lemma 1 under
private information, and describes the cutoff value of z2,
above which Buyer accepts and below which Buyer re-
jects. z is shown to depend positively on p and nega-
tively on .
1
Lemma 3. Under private information, Buyer optimally
accepts an offer of p if and only if
e
z


2,zzp where:



 

1
1
1
11
1
71
1if
12 12
12 6
,51 71
if ,
12 1212 12
51
1if
12 12
e
e
ee
e
pz
pzp
zpz pzpz
pz






p
p
p


 









Proof: Consider (7). Taking the expectation across
both X and z1, we get:
 
11
11
10
,dEXfzzfzxgxxz



 1
d
(8)
Given the result in the proof of Lemma 1, (8) reduces
to:



1
1
11
1
11
,
111d
212 12
11 1
212 12
e
EXfz z
1
f
zzz
zz







 
z


(9)
The lemma follows from setting (9) equal to the price

J. SANDFORD, P. SHEA 223
Figure 1. E[X|z1] and p(z1) for all possible z1.
set by Seller, p. □
Note from Lemma 3 that it is immaterial what the
perceived probability distribution over different values of
z1 is; only the expectation matters. This follows from
Seller’s assumed risk neutrality.
Given Buyer’s beliefs


1
e
zp
and strategy


zp,
Seller faces a trade off between a higher price and greater
profit from a sale, and lower price and greater likelihood
of sale. Formally, following (4) from the proof of Lemma
2, for each z1 Seller solves the following optimization
problem:

 

2
11
11
Max 1
4161212 6
1
2
p
z
zp zp
A
zp
p
 11 z

 
 







(10)
Unsurprisingly, Seller’s public information equilib-
rium strategy identified in Lemma 2 does not carry over
to the case of private information; the temptation for
Seller to shade his price to give Buyer a false impression
of his private information is too great. Indeed, we show
below that there is no equilibrium in which Seller’s price
is a linear function of his private information z1. We
cannot rule out exotic equilibrium functions, such as
nonlinear functions p(z1). There are step equilibria of the
following form:2





111
12 212
21
if 1,
if ,
if ,
NN
pz
pz pz
pz


N








... (11)
Lemma 4 proves that the equilibrium pricing function
under privateinformation is not a linear function of z1.
Lemma 4: Under private information, no linear func-
tion of the form


11
,pza bzb0

 solves Seller’s
optimization problem (10) over any subset of


11, 1z.
Proof: Suppose that there did exist some linear func-
tion


1
pz abz
1
giving Seller’s equilibrium price
under private information, for some numbers a and b.
Noting that
1
12
pb




the first order condition for (10) is:

1
11
12 112
416 6
11
12 0
2
zz1
A
bb
zpb

  
 
  
 



 





(12)
2Step equilibria occur in other settings with asymmetric information,
most notably Crawford and Sobel (1982) [9].

J. SANDFORD, P. SHEA 224
Equation (12) in turn implies that:

1
1
24 112
b
Ab


(13)
As (13) has no solution for b ≠ 0, there does not exist a
linear equilibrium pricing function for Seller, including
the public information equilibrium. □
Step equilibria always exist. In particular, if the num-
ber of steps is N = 1,


12 6xp p

 and there is an
equilibrium with trade under private information for any

77
,
12112
pA





. If 7
12
p, even for z2 = 1, Buyer
will optimally reject any offer, while if

7,
12 1
p
A

Seller will prefer to keep the item for z1
= 1. If, A = 0.5 there is an equilibrium with N = 2 where
Seller plays p1 = 0.4715 for all


11,0z and p2 = 0.59
for all


20,1z. In general, there are a multiplicity of
step equilibria for any number of steps N. However, for
any value of N, the price of any one step pi uniquely de-
termines the price at all other N ‒ 1 steps.
Table 1 gives example equilibria for


1, 3, 5, 7, 9N.
In each case, the equilibrium prices were chosen so that
1) the interval [‒1,1] is partitioned into segments of equal
size and 2) the interval centered on 0 is priced at the
same value. There does not appear to be a limit to the
number of steps in an equilibrium, and Seller’s profit
does not appear to fluctuate wildly in the number of steps.
All equilibria in the table are the unique equilibria with a
price of 0.546 at the interval centered on 0, but in each-
case a different price for this interval will produce a dif-
ferent equilibrium.
For any N, must be the case that at ςk Seller is indif-
ferent between pk and pk+1 (if not, then he would surely
also not be indifferent in some neighborhood


11
,

   . Seller’s optimality then requires that p2
be played for all z1 ≥ ς1 and p1 for all $z1 < ς1. Inspec-
tion of (4) gives us that this requires:
 
21
2121
12
ee
zz
xpxpp p  (14)
Table 1. Different symmetric step equilibria, where N is the
number ofsymmetric steps.
N p1 p
2 p3 p4 p5 p6 p7 p8 P9E[π]
1 0.55 0.37
3 0.47 0.55 0.61 0.37
5 0.45 0.50 0.55 0.59 0.63 0.37
7 0.44 0.48 0.52 0.55 0.580.60 0.63 0.37
9 0.44 0.47 0.50 0.52 0.550.57 0.59 0.61 0.630.37
Comparing Equations (14) and (15) leads to a surpris-
ing conclusion: under a step equilibrium, price is more
elastic with respect to Seller’s information when that
information is private. Lemma 5 formalizes.
Lemma 5: Under any step equilibria of the form (11)
on average the price charged by Seller increases more
quickly in z1 than the corresponding public information
pricing function.
Proof: Referring back to the public information equi-
librium price function, (3), we see that for any two prices
p2 > p1 charged in equilibrium,
21
21
12
24 1
zz
pp
A



(15)
From Equation (14) we see that the average rate of in-
crease in a step equilibrium is 1/12. The average rate of
increase under public information is
11
12 12
24 1A


. □
Lemma 5 tells us that Seller’s payoff from better in-
formation is higher if that information is private than if
he always shares his information with Seller.
Figure 2 demonstrates the result of lemma 5 under A =
0.5, comparing public and private equilibrium pricing
functions. Note the greater rate of increase under private
information. In relative terms, Seller’s profit increases by
more upon a high draw of z1 under private information.
In this case, the expected profit under private information
is 0.373, while under public information it is 0.419, sug-
gesting that if Seller has the ability to credibly reveal his
private information, he is better off doing so.
It is surprising that the case of private information,
where Seller is able to manipulate Buyer’s expectation,
may result on a worse outcome for Seller. This result
depends on our assumption that Seller cannot credibly
reveal z1.
Because the agents are bargaining over a surplus, they
Figure 2. E[X|z1] and p(z1) for all possible z1.

J. SANDFORD, P. SHEA 225
have a mutual interest in reaching an agreement. By be-
ing unable to observe z1, however, Buyer has worse in-
formation and it is more likely that a deal does not occur
which reduces the average welfare of both agents.
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