Suppose you and a friend of the opposite sex are recruited for an experiment. You are brought into separate rooms and told that you will be asked some questions and, unless you give consent, all of your answers will be kept secret.
First you are asked whether you would like to hook up with your friend. Then you are asked whether you believe your friend would like to hook up with you. These are just setup questions. Now come the important ones. Assuming your friend would like to hook up with you, would you like to know that? Assuming your friend is not interested, would you like to know that? And would you like your friend to know that you know?
Assuming your friend is interested, would you like your friend to know whether you are interested? Assuming your friend is not interested, same question. And the higher-order question as well.
These questions are eliciting your preferences over you and your friend’s beliefs about (beliefs about…) you and your friend’s preferences. This is one context where the value of information is not just instrumental (i.e. it helps you make better decisions) but truly intrinsic. For example I would guess that for most people, if they are interested and they know that the other is not that they would strictly prefer that the other not know that they are interested. Because that would be embarrassing.
And I bet that if you are not interested and you know that the other is interested you would not like the other to know that you know that she is interested. Because that would be awkward.
Notice in fact that there is often a strict preference for less information. And that’s what makes the design of a matching mechanism complicated. Because in order to find matches (i.e. discover and reveal mutual interest) you must commit to reveal the good news. In other words, if you and your friend both inform the experimenters that you are interested and that you want the other to know that, then in order to capitalize on the opportunity the information must be revealed.
But any mechanism which reveals the good news unavoidably reveals some bad news precisely when the good news is not forthcoming. If you are interested and you want to know when she is interested and you expect that whenever she is indeed interested you will get your wish, then when you don’t get your wish you find out that she is not interested.
Fortunately though there is a way to minimize the embarrassment. The following simple mechanism does pretty well. Both friends tell the mediator whether they are interested. If, and only if, both are interested the mediator informs both that there is a mutual interest. Now when you get the bad news you know that she has learned nothing about your interest. So you are not embarrassed.
However it doesn’t completely get rid of the awkwardness. When she is not interested she knows that *if* you are interested you have learned that she is not interested. Now she doesn’t know that this state of affairs has occurred for sure. She thinks it has occurred if and only if you are interested so she thinks it has occurred with some moderate probability. So it is moderately awkward. And indeed you know that she is not interested and therefore feels moderately awkward.
The theoretical questions are these: under what specification of preferences over higher-order beliefs over preferences is the above mechanism optimal? Is there some natural specification of those preferences in which some other mechanism does better?
Update: Ran Spiegler points me to this related paper.