Game theory and real estate negotiations

Here is a puzzle: You meet a real estate agent for a property you are interested in. The house has an asking prize and you haven’t made any offers yet. The realtor mentions casually that she has just had an offer for the house which she has rejected. Would you ask what the offer was? Would the realtor tell you? Is it a fair question to ask? (obviously, the realtor is under no obligation to reveal the truth value of the rejected offer and there is no way for me to verify the answer).

Here is a formalized description of the problem: the Seller adn the Buyer can be each of two types – High or Low.  High Buyers and Sellers prefer High Deal to No Deal no Low Deal, and Low Buyers and Sellers prefer High Deal to Low Deal to No Deal. First, the Seller announces whether she has rejected a Very low or a Moderate offer. If a Moderate offer has been (announced as) rejected, the Buyer can make either a High offer (which all Sellers accept) or No offer which ends the game. If a Very low offer has been (announced as) rejected, the Buyer can make a Low offer, No offer or a High offer (the latter two end the game). If a Low offer has been made, the Seller can either Accept or Reject it. In the case of rejection the Buyer can make a High offer or No offer – both actions end the game. Here is the game tree.

Essentially, by making an announcement that she has rejected a Moderate offer the Seller credibly commits to reject any Low offers. Importantly, Buyers suffer a cost from a rejected offer (which is realistic given the costs of the compulsory technical surveys one has to do before an offer). There is no penalty for a late deal (no time discounting). The game is of two-sided incomplete information – neither the Buyers nor the Sellers know the type of the opponent. So the questions:

1) Should you ask what the rejected offer was?
2) Should the realtor (the Seller) tell you?
3) Would the answer (announcement) of the Seller be informative?
4) Does the Seller do better under this game or a game with no signal (announcement)?
5) Does the Buyer do better under this game or a game with no signal?
6) Is this game Pareto-improving under any circumstances?

My answers are after the fold.

1) Should you ask what the rejected offer was?
No.
2) Should the realtor (the Seller) tell you?
Yes.
3) Would the answer (announcement) of the Seller be informative?
Yes. The Seller reveals its type.
4) Does the Seller do better under this game or a game with no signal (announcement)?
Yes.
5) Does the Buyer do better under this game or a game with no signal?
No.
6) Is this game Pareto-improving under some circumstances?
Yes.

I will post my analysis in a week but here is the reasoning in short.  A High Seller always says that the rejected offer has been Moderate because otherwise it would just waste time. A Low Seller always says that a Very low offer has been rejected – there is no reason to put off potential buyers at this stage if she can still reject Low offers at the next round. Surprisingly, knowing the type of the Seller does not help the Buyer – a High Buyer still cannot commit not to improve on a rejected Low offer. On hearing Moderate, High Buyers make High offers and Low Buyers make no offers. On hearing Very low, all Buyers make a Low offer which only High Buyers improve on if rejected. If the probability that the Buyers is High is above a certain threshold (to be defined in the formal analysis), Low Sellers always Reject, and if it is below that threshold Low Sellers always accept Low offers. Low Sellers have a higher payoff under this game than under a game with no signaling – for High Sellers it doesn’t make a difference. High Buyers do worse under the signaling game if the probability of a High Buyer is above the threshold  – in the remaining cases the signaling game gives the same payoffs as the one without an announcement. So if the probability is below the threshold, the signaling game is Pareto improving as Low Sellers have a higher payoff while High Sellers and all Buyers have the same payoff.

P.S. Needles to say this post should not be taken as a bargaining advice. Don’t blame me if your realtor kicks you out of the door.

4 thoughts on “Game theory and real estate negotiations

  1. Announcing that a lower offer has just been rejected seems to be quite a common strategy of real-estate agents. I’ve recently experienced exactly the same situation, although I would characterize it a bit differently. By showing serious interest in a house advertised at a certain asking price, buyers credibly signal that they are able and possibly willing to buy something in that price range (unless they have a lot of time on their hands, do house viewings for a hobby, and are therefore not potential buyers anyway). This in turn means that the seller has nothing to loose by announcing that an offer moderately lower than the asking price has just been rejected. For buyers who really want the house, the question is not whether to buy it or not, but at exactly what price to buy it. At the same time, the seller has a lot to win if s/he is able, through the announcement, to tilt the range of potential bargaining outcomes in his or her favour. Quite forseeably, the final agreement between the buyer and the seller will be located, after some haggling, somewhere in the middle between the asking price and the first offer made by the buyer. Therefore, by ‘lifting’ the lower end of that range of bargaining outcomes upward, the seller can achieve a higher selling price in the end. Not quite sure whether all that makes sense and is consistent from a game-theoretic point of view, but than again, I would not be surprised if people’s behavior wasn’t quite rational in these types of situations either (I know mine wasn’t…;-)).

  2. c – costs for making an offer (e.g. technical survey of the property)
    D and V are just utilities from a deal set up in such a way so that certain player types prefer a low price deal to no deal, while others prefer no deal to a low price deal. Still, all buyers prefer low price deal to a high price deal, and all sellers prefer high price deal to low price deal.

  3. For my interests in the payoffs: are D and V considered equal and does it matter which letter is assigned to the buyer or seller?

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