Bidding in real estate: A game theoretic approach

Journal of Real Estate Finance and Economics, 2:239-251 (1989) �9 1989 Kluwer Academic Publishers

Bidding in Real Estate: A Game Theoretic Approach

TAEWON KIM Department of Finance and Law, School of Business and Economic, s, California State University, Los Angeles, 5151 State University Drive, Los Angeles, CA 90032

Abstract

Taking advantage of recent developments in bargaining theory, we examine the bidding process for residential real estate. Within this game theoretic context, we consider whether the phenomenon of buyer's remorse is compatible with the rationality required of participants.

Key words: game theory, winner's curse, buyer's remorse

1. Bargaining theory

Since as early as Edgeworth's (1881) analysis of bilateral monopoly, the conven- tional wisdom in economics has been that bargaining processes lead to indeter- minate outcomes. The consensus view, refined into Coase's theorem, has been that the outcome must be efficient, but that which efficient outcome along the contract curve is chosen must depend on delicate considerations of bargaining power, for which little of substance can be said.

Abandoning attempts to model the actual process of negotiation, the now classi- cal bargaining literature in game theory (Nash, 1950) has taken the cooperative approach, whereby it is presumed that bargaining must end in an outcome satisfying various desirable criteria, such as efficiency. This literature implicitly or explicitly assumes the ability of bargainers to jointly commit themselves to an outcome after an unspecified process of communication has inevitably led them to the indicated outcome.

Cooperative solutions are, however, somewhat foreign to economists' typical reasoning, which favors the noncooperative approach. Here, agents remain free to

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act in their individual self-interests, so that cooperative outcomes are liable to collapse, due to the familiar free-rider problem associated with a lack of commitment.

The standard solution concept in noncooperative game theory has been the Nash equilibrium, familiar to economists since at least the time of Cournot's (1838) work in duopoly theory. The Nash equilibrium is essentially a static concept. Games are reduced to their strategic form, where each agent picks an action from his/her set of opportunities, and the interactions of the agents' choices immediately determine an outcome. The outcome is a Nash equilibrium only when each agent's choice is his/her best one, given the other agents' choices.

The traditional difficulty in making much progress in noncooperative bargaining theory has been that virtually any outcome is supportable as a Nash equilibrium. However, by returning to an explicit consideration of the sequential nature of many dynamic gaming problems, game theorists have introduced compelling new restrictions on what is a reasonable equilibrium concept, leading to the so- called subgame perfection criterion for a sequential equilibrium (Selten, 1975). Starting with the seminal work of Rubinstein (1982), application of this concept to bargaining has led to much recent progress in isolating the rational outcome of explicitly modelled negotiation procedures under complete information (Binmore and Dasgupta, 1987). l

In a second major advance in game theory, Harsanyi (1967) has introduced the notion of a Bayesian equilibrium, to allow treatment of games of incomplete information. Here, players lack full knowledge of the environment in which they par- ticipate. In bargaining over purchase of a good, for instance, the buyer may not know the exact character of either the good being offered or the seller making the offer.

The two parallel developments in game theory just discussed have recently been united into the notion of a sequential Bayesian equilibrium, applicable to dynamic games of incomplete information (Kreps and Wilson, 1982). Interest has naturally turned to application of this concept to noncooperative negotiation procedures, while allowing for the fact that a bargainer's characteristics may be unknown to the other party (Wilson, 1987). The resulting equilibrium outcomes seem to bear a closer resemblance to actually observed bargaining conduct than those predicted in the previous literature under complete information. Standard con- clusions, such as the inevitable efficiency of the bargaining outcome, have been seen to no longer necessarily hold once incomplete information is allowed for.

2. Buyer's remorse and the winner's curse

Taking advantage of the game theoretic tools now made available, we exhibit a simple model of bargaining, with the intent of capturing certain interesting phenomena associated with bidding for residential real estate. Of particular interest to us is the problem of "buyer's remorse," commonly observed among new

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home buyers. By this we mean the phenomenon where, upon successfully winning the bid, a buyer--in this case, a woman--immediately expresses regret or at least consternation over her acquisition. (Note." For the purposes of this article, we dis- tinguish the seller as male and the buyer as female, as a means of showing that American women have earned major economic buying power within the system.)

Such buyer's remorse could be taken as prima facie evidence for the irrationality of home buyers, and hence require the abandonment of the entire game theoretic approach with its heavy emphasis on rationality. After all, why would a rational person ever regret an action which by assumption was the best under the circumstances?

If we admit the possibility that the buyer takes a suboptimal action, then buyer's remorse is naturally associated with the celebrated "winner's curse" (Milgrom and Weber, 1982). This can arise in principle when the value of a good to the buyer is correlated with other people's evaluations. In the case of real estate, this is presumably because the home buyer herself may eventually wish to sell the house. The winner's curse arises if the buyer fails to take into account the fact that her successful purchase indicates that others value the house less than she originally anticipated.

This potential problem of a winner's curse arises most prominently in the case where there are multiple buyers, in which case it is advisable for the seller to run an auction (Milgrom and Weber, 1982). The successful buyer must consider why the other potential buyers have not outbid her. The same phenomenon, however, arises in the case of one-on-one bargaining, since the seller's reservation price for the house is an important piece of information to the buyer, particularly if the seller is well informed and his reservation price reflects his ability to sell the house to others, and so presumably, the buyer's ability as well. Rational behavior requires the buyer to escape the winner's curse by incorporating the information to be inferred from a successful purchase into her estimate of the house's value, when deciding before the fact on how much to bid.

We examine the winner's curse within our bargining model. While such a phenomenon requires naive behavior on the buyer's part, we contend that this does not in fact call for the abandonment of the game theoretic approach, since it is only in setting up the game theoretic machinery and deriving the equilibrium outcome that we obtain a context within which to consider the winner's curse and a benchmark against which to judge the failure of the naive bidder.

However, even with the just given rationale for admitting suboptimal behavior in a game theoretic context, we are not prepared so easily to abandon rationality in the conduct of the bargaining game. This may be for no other reason than the fact that without rationality it seems hopeless to predict the outcome of bargaining and so make meaningful assessments. One ought in any case to make the distinction between rationality in the actual conduct of the negotiations and rationality in players' reviews of their own performance. For instance, we offer the view that buyer's remorse may be attributed to the regret the buyer has that she did not stand firm in her offer, when instead allowed herself to raise her bid. If in fact she did

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have the ability to commit herself to a single bid once and for all, then this remorse is well founded, since commitment is the preferable strategy. On the other hand, in the more likely event that she had no such ability to commit herself, any remorse that afterward arises is due to the illusion that she could have taken a stance that was not in fact credible.

One final interpretation of buyer's remorse we offer is that the buyer is neither irrational in the conduct of the game nor in her ex post evaluation. Instead, buyer's remorse may just reflect subsequent confirmation that had she possessed more complete information she could have done better for herself. This confirmation could be either the extreme discovery that the house is after all worth less to her than she paid for it, or simply the milder discovery that the house could have been had for less, had the buyer been better informed. This realization might, of course, arise from increased familiarity with the house itself, the neighborhood, and the market, as the process of appraisal, escrow, and eventual occupancy proceeds. However, buyer's remorse is usually thought of as swiftly following the seller's acceptance of the final bid, and so it is natural to ascribe it to the information gleaned from the bidding process itself. Such remorse is apparently pointless, in that no learning is likely to ensue, given that the buyer performed as well as any- one in her position could have. On the other hand, regret for what might have been had one only known remains an easily appreciated phenomenon. It might serve to remind us that our state of knowledge and the choice to proceed can themselves be endogenous decisions in a yet larger but less well-defined problem.

3. Description of model and solution technique

It is natural to assume that the buyer is incompletely informed, since in practice a buyer is liable to be uncertain both about the house's exact value to herself and about the reservation price of the seller. Note that since we will assume that both buyer and seller are risk-neutral, we can include both kinds of buyer's uncertainty in just her uncertainty about the seller's type. This is so since the buyer's subjective probability distribution as to the quality of a house belonging to a given type seller can be replaced by the expected quality for that type of individual.

It may also be argued that the seller is uninformed, since he may in turn not know the buyer's reservation price. However, the solution of models with two- sided uncertainty remains unresolved. 2 In such models, rationality requires the seller's beliefs to depend on the buyer's offers. In particular, if a buyer makes an offer that has zero probability for the given equilibrium, Bayes' rule becomes inap- plicable and no single inference can be made. This arbitrariness in conjectures permits a multitude of possible equilibria. 3 The same difficulty arises if there is only one-sided uncertainty, but it is the informed seller that makes the offers. Thus, in order to keep our model tractable, we must have both single-sided uncertainty and single-sided offers, with the offers being made by the uninformed party. Since the object is to examine acquisition of information by the buyer, she must be the uninformed party and so be making the offers.


Given that only one party is to make the offers, it is perhaps natural that it be the buyer, who is typically seen to take the more aggressive role in negotiations toward purchase of a house. Although the limitation to one-sided offers is clearly less than fully satisfactory, it should be pointed out that any equilibrium of our one-sided bargaining procedure can be made into an equilibrium of the corresponding two- sided bargaining process, by the device of having the seller always make non- serious and hence uninformative offers (Fundenberg et al., 1985). However, as just discussed, there will also be a multitude of other equilibria in this game of two- sided offers, hence our restriction to the one-sided procedure.

Noncooperative game theory excludes the ability to commit, except insofar as it is explicitly specified in the game form itself. It is this position that has led to the criterion of subgame perfection. Many Nash equilibria are supported by threats: you do not take an otherwise attractive action because my equilibrium strategy specifies that if you do I will respond with an action to your detriment. Without the ability to commit, many of these threats and their associated Nash equilibria are no longer credible. Without prior commitment, my supposed rataliatory action must be in my own best interests at that point in time; otherwise, you will call my bluff and I will abandon my threat. More formally, to be credible, an overall equilibrium must also be an equilibrium of every subgame of the overall game, hence the so-called subgame perfection criterion for sequential games (Selten, 1975). 4

The equilibrium concept we require is slightly more complicated due to incomplete information. Both parties to the bargaining form beliefs about the opposite party. In a Bayesian sequential equilibrium, strategies are sequentially optimal (subgame perfect) given beliefs, and in addition, these beliefs are modified accord- ing to Bayes' rule, based on the observed actions and the equilibrium strategies (Kreps and Wilson, 1982).

In models of the type we consider, the incentives to reach agreement in a timely fashion are provided by the parties' discount rates. For simplicity, we assume that both bargainers have a common, constant discount rate. With perfect capital markets, consumers will have adjusted their streams of consumption so that their discount rates on the margin are equal to the market discount rate, which we may suppose is constant over the period of bargaining.

Given the emphasis on the inability to commit, it is natural to allow games of arbitrary length. It is known that Bayesian sequential equilibria of bargaining games with incomplete information allow inefficient outcomes, unlike the efficient outcomes of the corresponding games under complete information (Crampton, 1984). This inefficiency often takes the form of delay in reaching an agreement. In a game of fixed length, this may entail the parties leaving the negotiations without having struck a deal, despite the common knowledge that there are gains from trade to be had. This may be regarded as an artifact of the imposed finite length. With knowledge of gains from trade and no credible commitment to ending the game at a particular time, the parties would never halt negotiations without an agreement being reached, and so games of infinite length must be allowed. Although we find this argument persuasive, nonetheless, in the interests of sim-

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plicity, we will restrict consideration to a two-period game. We feel justified in doing so, since it can be shown that games of arbitrary fixed length behave in the manner that would be anticipated from looking at the two-period game, and further, the infinite-horizon game is the natural limit of these games of finite length (Fudenberg et al., 1985; Gul et al., 1986). Indeed, as long as the buyer's valuation exceeds the seller's, the infinite-horizon game will in fact end in a finite number of periods.

Also in the interest of simplicity, we restrict attention to incomplete information in the form of a two-point distribution. That is, the buyer assumes that the seller is one of two possible types. Analysis shows that there is little loss of generality between a two-point distribution and, say, a continuous distribution over types (Fudenberg et al., 1985; Gul et al., 1986). As with all models of this sort, the pro- babilities the buyer attaches to either of the seller's types are also known to the seller; that is, they are common knowledge. The basic type of model we employ was first introduced by Sobel and Takahashi (1983) and Fudenberg and Tirole (1983) in the context, symmetrical to ours, where an uninformed seller makes all offers.

To introduce the notation to be used, we review the model. There is a single buyer and a single seller, who may be one of the two types. The house of the high- type seller has a value b to the buyer and reservation price ~ to the seller, whereas the house of the low-type seller is worth b to the buyer but _s to the seller. We assume b >/2, and ~ > s, consistent with our high/low terminology. Note that the house's value to the buyer is positively correlated with that of the seller. We also assume b > ~ and b > s, so that there are gains from trade to be had. We denote as ~1 the prior the buyer has as to the likelihood the seller is of the low type.

The buyer and seller are both risk-neutral, so that if the buyer buys the house in the first period at pricep, his payoffis b - p, while that of the seller isp - s. On the other hand, if the same transaction is concluded in the second period, then due to a common degree of impatience 8, the discounted direct payoffs will be 8(b - p) and 8(p - s), respectively. Since a house is a consumer durable yielding an explicit rental flow, it is natural to assume that the reservation price of a house worth s in the first period will remain s in the second period, with present value ~s, since this present value together with the present value (1 - 8)s of the rental flow yields the house's value in the present. Similarly, ifa buyer is denied the services of the house for a period, then with a first period valuation b, the second period present value of the house would only be 8b, giving it again the nominal value b in the second period. 5

4. The equilibrium solution

We now examine the noncommitment solution, as a function of the buyer's prior, 91. We restrict attention to offers in [s, ~], since there is never any reason to make offers outside this range.

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We speak of there being high-type sellers and low-type ones, though the interpretation we intend is that there is but one seller, of unknown type. There is a formal equivalence between buying once from a single unknown seller drawn from a given distribution and buying repeatedly, once from each of the selected known members of that distribution (Gul et al., 1986).

For reference, we first solve the one-period game. In this case, the buyer's offer will be either s, or g, since an intermediate offer still does not attract the high-type sellers, but loses the buyer benefits from the low-type sellers she attracts. The choice between the offers depends on the perceived likelihood ja 1 that the seller is of the low type. Comparing the outcomes of the two choices

> ~ 1 1 ( ~ - - & ) < ~ 1 , ( ~ - - S) -1- (1 - bh ) (b - Y), (1)

we see that the critical cut-off belief occurs at

f f l = ( / ~ - ~ ) / ( / ~ - s_) (2)

so that only for gl < ~1 will the likelihood that the seller is of low type become suf- ficiently small that the buyer will abandon the attempt to extract the maximum rent from the low-type seller (p = a) and concentrate on insuring a sale from the high-type seller (p = 7).

We now turn to the full two-period model. There will be three prices of interest: ~, ~, and g = 83 + (1 - 8)a. The last is the price in the first period that makes a low- type seller indifferent between selling then and waiting a period to receive the high price 7. (Recall that by waiting a period of low-type seller foregoes a service flow of value (1 - 8)s.)

Note that if the second price isp2 = 7, while the first price ispl < ~, then the first price should in fact be 2. Indeed, the high-type sellers are then going to wait until the second period, so the question reduces to whether the low-type sellers should be made to sell in the present or the future. Clearly, if they are to be made to sell in the present, then this should be done atpl = L the lowest price that will have the ef- fect. Finally, one sees that trade in the present is in fact in the buyer's interests; that is, since/2 > a, we obtain

h - ~ = h - ( s f + (1 - 8 )a) > 8 ( ~ - f ) . (3)

Consider any price 7 > p~ > ~. Then the low-type sellers will certainly sell in the first period, since they prefer this to P2 = 7, the best price they can hope for in the second period. The high-type sellers will not, however, sell in the first period, since their reservation price is not met. A rational buyer, anticipating this, will infer that g2 = 0. But since ~2 = 0 < ~1, it follows thatp2 will be 7, in which case the best first- period price is Pl = g and not any of the supposed prices.

Now consider any price ~ > Pl > s. Suppose that this leads to the inference g2 < gl- Then P2 = ~; but as we have seen, this would then require the best price to be

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Pl = S , and not one of the indicated prices. Suppose instead, that the inference is b~2 > ~- The n we obta inp2 = s, and so all low-type sellers trade in the first period. Anticipating this, the rational buyer infers ~t 2 = 0, in contradict ion to the supposed inference g2 > ~. Thus, the only remaining possible posterior is bt2 = g~ for any supposedly best price in the interval. Posteriors are determined by the probabil i ty r(pl) of a low-type seller accepting the first offer, together with the given pr ior ~a~. The fixed posterior then requires the same value r to occur, no matter which price in the interval is selected. However, given acceptances, the price s dominates any of the candidate prices along the interval. Cont inui ty also shows that s must have the same probabi l i ty of acceptance, r.

This probabi l i ty o f acceptance r must be the one that leaves the buyer indifferent between offering~ a n d s in the second period, since we saw that lu2 = ~ was required. Applying Bayes' Rule, we obtain

~ = b - x = (1 - a)p~ + (1 - ~1) (4)

o r

~ ( 5 - ~ ) - ( 5 - ~ ) = ( 5 )

We have shown that the only possible equi l ibr ium first-period prices are s, L and ~. Which price prevails depends on the buyer 's prior, jal. One cutoff belief dividing regimes occurs where the buyer is indifferent between p~ = g and p l = ~; that is, where

~h(& - Y) + (1 - g~)(5 - ~-) = ~(& - ~) + (1 - ja~)8(/~ - ~). (6)

The solution is seen to be Vl = ~. The second cutoff occurs where the buyer is indifferent between PI = ~ and pl = s. Denot ing the solution to this as ~2, we have

p2(h - ~) + (1 - ff2)8(b - Y) = ffzr(h - s) + 592(1 - r ) ( h - s_). (7)

While we omit the expression for ~2, it is not difficult to show that gz > P~- Gather ing our arguments, we have:

theorem There is a unique equi l ibr ium for each pr ior gl (omitting the nongener ic cases g~ = g~ or g~ = g2.) For Pl < ~l, equi l ibr ium specifies Pl ---P2 = g. All sellers accept the first offer. The equi l ibr ium for g~ < ~ < g2 requires that p~ = ~ and Pz -- g. Low-type sellers accept the first offer and high-type sellers the second. The equi l ibr ium for Pz < Pl specifies p~ = P2 = s High-type sellers refuse both offers, while low-type sellers accept one of them, the former one with probabi l i ty r and the latter one with probabi l i ty 1 - r.


5. Efficiency of equilibria

Inefficiency can be in either of two forms in the current model, arising from delay or from disagreement. We have assumed the buyer values the good more than the seller, so efficiency requires the good to be allocated to her. Therefore, any situation ending in no trade is inefficient. This will occur when the buyer plays the tough strategy of offering onlys against a seller who may be of the high type. In addition, though, efficiency requires that the trade occur in the first period. Although it is true that the seller continues to enjoy the services of the house in the period between offers, this is worth only N to him, while the buyer is forgoing these same services, which are worth the greater amount 8b to her. The ascending offer equilibrium is therefore inefficient, due to the possible delay. Only the equilibrium where the buyer acts weakly and offers g from the beginning is ex ante efficient. Note that whatever the equilibrium form, the high-type seller receives no surplus, either because he receives only his reservation price g or because no transaction occurs at all. This no-gain result for high types is, of course, typical of adverse selection problems.

6. Ascending offer equilibria and buyer's remorse

The solution of greatest interest is the one with ascending offers, associated with intermediate priors. As the number of periods of trade is allowed to expand, there will be a whole succession of such ascending offers, occuring over a widened interval of priors (Hart and Tirole, 1987). Whatever the number of periods, the idea is to screen the lower-type sellers from the higher-type ones by offering only lower prices in the early stages. Sellers with a lower reservation price earn more rents from a given offer and so act more impatiently. That is, they suffer a greater loss from waiting for a higher offer and so opt for earlier, lower offers. This ability to price-discriminate works to the buyer's advantage.

Ascending offers yield a separating solution, so that after the fact, the buyer will know whether or not the seller was of low or high type. This may constitute a source of remorse, since if the seller accepts in the first period, the buyer will real- ize that in principle she could have had the house for as little as x < ~ = Pl- There is, however, no way to exploit this ex post information before the fact, and so any remorse is based on illusion.

Note that we have allowed the value of the house to the buyer to be correlated with the value to the seller. The buyer in our solution incorporates this fact into her cut-off beliefs, though not into the prices, which depend only on the seller's valuations. Examining equation (7), we see that if the value of a house to the buyer were uniformly b, this would raise the cut-off belief ~2. In this sense, lower-type sellers' being associated with houses of lower value causes the buyer to play less tough. The buyer is more anxious to concede and pay the higher price, in order that she attract the high-type sellers, to whose houses she attaches greater value.

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A naive buyer expecting to obtain a higher valued house with ex ante probability (1 - gl) will clearly be disappointed if she follows the ascending offer strategy and receives an acceptance on the first bid. This is, of course, an instance of the celebrated winner's curse, arising when a bidder fails to anticipate the information to be infered from the fact that her bid is accepted.

The analysis has preceded without requiring/2 > g. Even taking/2 to be the certain value of a low-type house to the buyer, rather than just the expected value, we still see that a buyer following the weak strategy of always offering ~ may experi- ence an extreme form of regret in case/2 < ~. For if it is a low-type seller that accepts her offer g, she will wind up with a house worth less than what she paid. The case may even arise that/2 < g. Then a buyer making escalating offers will experi- ence regret as soon as her first offer is accepted, since she will know with certainty that this is a low-type house and not worth the price offered. These situations arise when the buyer is fairly certain beforehand that the seller is of high type. She will then clearly offer~ to attract the high-type sellers; the question is merely whether this should be done fight away or whether the lower price ~ should first be offered to flush out the low-type buyers. Though a seller's acceptance ofg means a certain loss, the alternative would have been to offer g right away, and so have lost even more. If, as we have suggested,/2 is treated not as a certain value, but as the expected value to the buyer for that type seller's houses, the potential for such extreme remorse is further increased.

7. Commitment solutions and buyer's remorse

It is interesting to compare the commitment solution in our bargaining problem with the noncommitment solution already developed. It is not hard to show that with commitment one obtains the same solution as in the static one-period problem. Always offer ~ if Pl < ~1 and always offer x if Pl > ~1 (Sobel and Takahashi, 1983). Notice that, unlike the tough noncommitment solution, the tough commitment solution has all low-type parties assenting in the first period. Comparing the two-period commitment and noncommitment solutions for the buyer, the dif- ference in the buyer's strategies arises in the range ~1 • ~11 < ~2, where noncommitment calls for escalating offers. The reason that a noncommited buyer cannot play the apparently superior tough commitment solution is that she cannot guarantee that she will in fact play tough in the second stage. If all low-type sellers were to accept a in the first period, then the buyer would actually offer ~ in the second period to attract the high-type sellers. The low-type sellers know this and so in fact they will not all accept the first period offer. The buyer knowing this will in turn con- clude that the ascending-offer strategies provide greater returns in this belief range. This problem does not arise if commitment is possible, since the buyer can then guarantee that she will pays in the second period, and so low-type sellers will all be willing to trade in the first period.


The inability to commit might be another source of remorse to the buyer. If the seller accepts the first offer and so reveals himself to be of low type, the buyer may come to reprimand herself for not following a commitment strategy, which even in an ex ante sense seems the better strategy. However, without a firm ability to commit, this appearance is illusory, since standing firm is then not a credible strategy.

8. Coase's conjecture and buyer's remorse

One interesting issue that has arisen in the literature on bargaining is the so-called Coase conjecture. The original context was one where a monopolist attempted to price-discriminate by offering a succession of different prices over time. Coase (1972) conjectured that without the ability to commit, the monopolist would cave in and sell everything near the lowest possible price. This conjecture has been formally verified (Gul et al., 1986) as the limiting case when offers come increasingly quickly. In our context, this can be captured by letting the rate of discount 8 go to one, since the degree of discount associated with shorter periods will become pro- gressively less. One sees that ~ converges to s and so the ascending offer case collapses into the weak strategy of offering g right away. If the buyer finds herself under pressure to make offers quickly, as for instance when the seller has other buyers lined up, the buyer's bargaining power dwindles and rents go to the seller. This can again be a source of remorse, though once more, without the ability to delay, the regret is misplaced, since the outcome cannot be otherwise.

9. Conclusion

One of the points of this article has merely been to take advantage of the fact that advances in noncooperative game theory, particularly for problems of incomplete information, have made available the means to discuss issues of bargaining that previously could not be treated in a formal way. We have adapted a type of model available in the literature in order to obtain results that have at least some resemblance to the complex situation that arises in bidding for real estate. Typical bargaining takes the form of a succession of ascending offers ending in acceptance.

In particular, we have treated the problem of buyer's remorse within this framework. Game theoretic equilibria represent an extreme use of the principle of rationality, whereas buyer's remorse must clearly reflect some sort of irrationality, even if it is after the fact. Nonetheless, the revelation of information in the process of bargaining was shown to take on complex and subtle forms that might not have been apparent without the use of such game theoretic reasoning. Within this context, buyer's remorse may be treated as the buyer's ex post response to the information acquired in the trading process itself.

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Acknowledgments

I would like to thank an anonymous referee for his comments. Financial support from California State University, Los Angeles, is acknowledged.

Notes

1. Since these games are posed in the extensive form, they are capable of reflecting details of the actual institutional structure of bargaining transaction. Solutions can be quite sensitive to certain of these details. While this frustrates attempts to reach general results, it does contribute to the understanding of what characteristics of a bargaining procedure are crucial. A normative comparison of different procedures awaits further development in equilibrium choice. For static games in strategic form, the theory of mechanism design has already succeeded in providing optimal procedures. This has led to a deep understanding of auction theory (Milgrom and Weber, 1982; Myerson, 1981), the appropriate technique when a seller faces several buyers simultaneously.

2. See Charterjee and Samuelson (1987), Crampton (1984), Grossman and Perry (1986), and Gul and Sonnenschein (1988).

3. In models with multiple equilibria, buyer's remorse could be considered regret that a buyer finds herself in one equilibrium and not another.

4. Note that while we have emphasized the inability to commit across periods, it is necessary to our analysis that each period's offer be binding in that period. Thus, while a seller often suggests an open- ing price, it usually is nonbinding and, so, of no essential consequence.

5. There is in fact a difficulty in interpreting the analysis if the good does not yield a flow of services. As an example, a low-type seller would not then be indifferent between ~ in the second period and

= ~ + (1 - 8)s in the first period (see section 5 for discussion). While one can say his net gain from the first period sale is g - s, one cannot say his net present value of gain from the second period sale is 8(~ - s), as would be required for indifference. This follows since losing s_ in the second period is not the true opportunity cost of selling then: the opportunity cost is the best alternative foregone, which would entail selling in period one, not period two.

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Bidding in real estate: A game theoretic approach