Here is a puzzle: You meet a real estate agent for a property you are interested in. The\u00a0house has an asking prize\u00a0and you haven’t made\u00a0any offers yet. The realtor mentions casually that she has just had an offer for\u00a0the house which she has rejected. Would you ask what the offer was? Would the realtor tell you? Is it a fair question to ask?\u00a0(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).<\/p>\n
Here is a\u00a0formalized description of the problem: the Seller adn the Buyer can be\u00a0each of two types – High or Low.\u00a0 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\u00a0Moderate 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\u00a0a 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.<\/a><\/p>\n Essentially, by making an announcement that she has rejected a Moderate offer the Seller credibly commits to reject any Low offers.\u00a0Importantly, Buyers suffer a cost from a rejected\u00a0offer (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:<\/p>\n 1) Should\u00a0you ask what the rejected offer was? My answers are after the fold. 1) Should\u00a0you ask what the rejected offer was? I will post my analysis in a week but here is the reasoning in short.\u00a0 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\u00a0Sellers always Reject, and if it is below that threshold Low\u00a0Sellers 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\u2019t make a difference. High Buyers do worse under the signaling game if the probability of a High Buyer is above the threshold\u00a0 – 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>\n 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.<\/p>\n","protected":false},"excerpt":{"rendered":" Here is a puzzle: You meet a real estate agent for a property you are interested in. The\u00a0house has an asking prize\u00a0and you haven’t made\u00a0any offers yet. The realtor mentions casually that she has just had an offer for\u00a0the house which she has rejected. Would you ask what the offer was? Would the realtor tell you? Is it a fair question to ask?\u00a0(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\u00a0formalized description of the problem: the Seller adn the Buyer can be\u00a0each of two types – High or Low.\u00a0 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\u00a0Moderate 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\u00a0a 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<\/p>\n
\n2) Should the realtor (the Seller) tell you?
\n3)\u00a0Would the answer (announcement) of the Seller\u00a0be informative?
\n4) Does the Seller do better under this game or a game with no signal (announcement)?
\n5) Does the Buyer do better under this game or a game with no signal?
\n6) Is this game Pareto-improving under any circumstances?<\/p>\n
\n<\/p>\n
\nNo.<\/strong>
\n2) Should the realtor (the Seller) tell you?
\nYes.<\/strong>
\n3)\u00a0Would the answer (announcement) of the Seller\u00a0be informative?
\nYes. The Seller reveals its type.<\/strong>
\n4) Does the Seller do better under this game or a game with no signal (announcement)?
\nYes.<\/strong>
\n5) Does the Buyer do better under this game or a game with no signal?
\nNo.<\/strong>
\n6) Is this game Pareto-improving under\u00a0some circumstances?
\nYes.<\/strong><\/p>\n