Resolving a family conflict

I was recently able to help family friends, a father and daughter, with a little family conflict using a bit of microeconomics. The problem was this. The daughter, let’s call her Marianne (not her real name) needed dental work. Her Austrian dentist was fully prepared to fix Marianne’s dental problem for a fee in the neighborhood of € 1000. Marianne’s father, let’s call him Franz (not his real name), tends to go to a dentist in a neighboring country and is very happy with his service there. He ascertained that his dentist would charge something in the neighborhood of € 100 for the same dental work. Marianne is a 20 year old student and still relies on her father to pay things such as dental bills for her. When I met them recently they were arguing over which dentist she should go to. In what follows I will explain their positions, and how a little bit of microeconomics helped with the resolution of this conflict, why it worked, and when it would not necessarily work.

This is roughly the sequence of events: Marianne asked Franz to give her the € 1000 so she could go to her dentist to solve her dental problem. Franz then pointed out the alternative and suggested she go there, which he would then be happy to pay for. Marianne then said she was shocked to hear how little her father cared about her (dental) welfare as he seemed to refuse her the by far superior quality treatment with her dentist. Franz then countered that this is nonsense. His dentist is just as good as hers. The discussion became quite heated and the positions hardened along these lines, Franz insisting that both dentists are equally good, Marianne insisting that her dentist is far superior.

Knowing both fairly well, and after hearing about their problem separately from each party, I suggested the following. Franz should offer that, if Marianne goes to his dentist, he would pay Marianne’s dental bill and pay Marianne an additional € 100 for her own use. Receiving this new offer, Marianne’s opinion of the quality of her father’s dentist’s work changed dramatically. Almost immediately she accepted this new offer happily. I think we can say that an efficient decision was reached in the end.

Why and how did it work? What we here have is a binary decision. Marianne has to go to one of the two dentists for treatment (apparently going to other dentists, or not doing anything, was already ruled out by both as infeasible or not really better than the two given options). Given that, we can arbitrarily call “Marianne going to her dentist” as the default option and “Marianne going to her father’s dentist” the alternative option. Both “players” or “agents”, as they are referred to in the game theoretic or mechanism design literature, have a net value (expressed in Euros) for the alternative over the default option. We probably do not quite know their true values but this does not prevent us from giving them a name, let Marianne’s value be VM and Franz’s value be VF. I would guess that Franz’s value for Marianne going to his dentist instead of her dentist is somewhere in the area of VF = € 900, the difference in dentists’ fees. It could be that VF > € 900 if Franz thinks that his dentist actually provides better quality service than Marianne’s dentist. It could also be that VF < € 900 if Franz thinks, as Marianne seems to do, that Marianne’s dentist provides better quality service than Franz’s dentist (and / or if Franz disagrees with Marianne’s assessment of the two dentists’ qualities but cares about Marianne’s opinion).  Marianne’s net value for her going to her dentist instead of Franz’s dentist is probably such that VM < € 0 (meaning she prefers to go to her dentist as long as she does not pay anything in either case, or pays the same amount in either case). That the resolution I proposed was accepted by Marianne, actually indicates that her net value also satisfied VM > € -100. Of course this was not necessarily clear before Marianne accepted this deal and ex-ante (as the game theorist likes to say), that is before any negotiations took place, her net value could well have been less than € -100.

So why did it work? Well I had a hunch that Franz’s value was around € 900 (actually somewhat slightly less I thought) and Marianne’s value between € -100 and € 0. I thus knew (well I was pretty sure about) two things: One, that Marianne going to Franz’s dentist is the unique efficient decision, providing a total net benefit from somewhere between €800 and €900 (as compared to Marianne going to her dentist). And two, that the offer I suggested would be acceptable to both. But why was it acceptable to both? As far as I could see Marianne and Franz did not have clearly defined “property rights” (or here better “right to choose”), meaning it was not clear who, in the absence of an agreement, would eventually be allowed to decide which option would be taken. Thus, my proposal was tantamount to giving Marianne the “right to choose” and then letting bargaining in the form of Franz making a take-it-or-leave-it offer (giving him a lot of bargaining power) take place. This is an example of what is often referred to as Coase’s theorem, although I do not know of a universally accepted formal version of such a theorem (Coase certainly did not have such a theorem in his famous 1960 paper for which this supposed theorem is given his name). But the idea is, and some version of it could be formally proven in the present case, that it does not matter whether we give the ultimate “right to choose” to Marianne or Franz, in either case Marianne will eventually go to Franz’s dentist if we assume that the information I had about their net value was also commonly known between Franz and Marianne and that bargaining is done by means of a take-it-or-leave it offer by Franz. Of course it does matter to the two agents who the “right to choose” is given to. If Franz has the ultimate “right to choose” he will simply tell Marianne to go to his dentist, without any side payments. He might say that he would be willing to let her go to her dentist if Marianne contributes something of the order of €900 to the mayment of the dental bill in that case. This is an offer Marianne would decline, however.

So in either case we get an efficient (i.e. welfare-maximizing) decision. When would this not work? We could have a problem if the two agents (and I, the mechanism designer, as well) are much less sure about the two net values. That is, if the uncertainty about these two values includes the possibility (I mean the probability) that the efficient thing to do could be either that Marianne go to her dentist or that Marianne go to Franz’s dentist. Then we run into the Myerson-Satterthwaite theorem, which states that no mechanism – that is no bargaining protocol – can lead to the two parties always reaching an efficient agreement. But perhaps I should explain this in a future post.

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