Theoretical Economics Letters, 2013, 3, 220225 http://dx.doi.org/10.4236/tel.2013.34037 Published Online August 2013 (http://www.scirp.org/journal/tel) Real Estate Pricing under TwoSided Asymmetric Information Jeremy Sandford1, Paul Shea2 1University of Kentucky, Lexington, USA 2Bates College, Lewiston, USA Email: jeremy.sandford@uky.edu, pshea@bates.edu 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 understudied game of 2sided 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 itorleaveit 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 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 , 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 xzzz zxz z For the remainder of the paper we consider the case where Seller makes a takeitorleaveit 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 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 . 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 ,zzpz1 , 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 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 Copyright © 2013 SciRes. TEL
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 (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(Xz1). 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 EXz , calculated from (3) and (5): 11 15 12121212 AA pEXz z 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 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 zzz zz z (9) The lemma follows from setting (9) equal to the price Copyright © 2013 SciRes. TEL
J. SANDFORD, P. SHEA Copyright © 2013 SciRes. TEL 223 Figure 1. E[Xz1] 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 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 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 (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[Xz1] and p(z1) for all possible z1. Copyright © 2013 SciRes. TEL
J. SANDFORD, P. SHEA Copyright © 2013 SciRes. TEL 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. REFERENCES [1] D. Friedman and D. Wittman, “Litigation with Symmet ric Bargaining and TwoSided Incomplete Information,” Journal of Law, Economics, and Organization, Vol. 23, No. 1, 2007, pp. 98126. doi:10.1093/jleo/ewm004 [2] A. Daughety and J. Reinganum, “Settlement Negotiations with TwoSided Asymmetric Information: Model Duality, Information Distribution, and Efficiency,” International Review of Law and Economics, Vol. 14, No. 3, 1994, pp. 283298. doi:10.1016/01448188(94)900442 [3] J. Kennan and R. Wilson, “Bargaining with Private In formation,” Journal of Economic Literature, Vol. 31, No. 1, 1993, pp. 45104. [4] J. Sandford and P. Shea, “Optimal Setting of Point Spreads,” Economica, Vol. 80, No. 317, 2013, pp. 149 170. doi:10.1111/j.14680335.2012.00939.x [5] M. Ottaviani and P. Sorenson, “The Strategy of Profes sional Forecasting,” Journal of Financial Economics, Vol. 81, No. 2, 2006, pp. 441466. doi:10.1016/j.jfineco.2005.08.002 [6] J. M. Steele and J. Zidek, “Optimal Strategies for Second Guessers,” Journal of the American Statistical Associa tion, Vol. 75, No. 371, 1980, pp. 596601. doi:10.1080/01621459.1980.10477519 [7] A. Yavas, “A Simple Search and Bargaining Model of Real Estate Markets,” Real Estate Economics, Vol. 20, No. 4, 1992, pp. 533548. doi:10.1111/15406229.00595 [8] A. Yavas and S. Yang, “The Strategic Role of Listing Price in Marketing Real Estate: Theory and Evidence,” Real Estate Economics, Vol. 23, No. 3, 1995, pp. 374368. doi:10.1111/15406229.00668 [9] V. Crawford and J. Sobel, “Strategic Information Trans action,” Econometrica, Vol. 50, No. 6, 1982, pp. 1431 1451. doi:10.2307/1913390
