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September 21, 2004

Does not Calling Indicate not Liking? Ted and Sheila

Via Alex Tabarrok at MR, I discover Glen Whitman at Agoraphilia

Say Ted would like to talk on the phone every two days , whereas Sheila would like to talk every day . You might think Sheila would call Ted about two-thirds of the time -- but in fact, she will call him every time . If they talk on Monday, Ted plans to call on Wednesday; but then Sheila calls him Tuesday. His clock reset, Ted plans to call on Thursday. And then Sheila calls on Wednesday. Eventually, Sheila decides Ted doesn’t care about her, because he never calls....

Sheila's conclusion is not "rational" in the economic sense, because she ought to have figured this out. If her prior belief is that there are equal probabilities that Ted would want to call her every half-day, every two days (which is in fact the truth) and never, then after the experience described above, she should revise it to put 50-50 probability on Two Days and Never.

Furthermore, if she cares enough, she can experiment and learn. She can purposely refrain from calling. If Ted does not call after Two Days, she can conclude that the truth is he Never wants to call.

...This is a truly useful idea.

I can carry it a little further than Whitman did. Suppose Sheila is rational. She therefore continues to call every day, knowing that there is a 50% probability Ted does like talking, if not as much as she does. In fact, though, let us change the story so Ted's true preference is Never. If Ted thinks that never calling Sheila is going to give her the message that he doesn't like her, he is mistaken. She will continue to put the probability that he doesn't like her at just 50%.

This last story sounds more realistic if we make Sheila's priors on Two Days and Never at 95-5. Remember, these are subjective beliefs of Sheila, so it would not be surprising if she put a high probability on Ted liking her. When Ted never calls, she will continue to hold the 95-5 beliefs. With beliefs that skewed, she will also find no point in experimenting by incurring the cost of not calling and letting two days go by. Ted therefore must bite the bullet and tell her he doesn't like her, or else endure those phone calls indefinitely.

Posted by erasmuse at September 21, 2004 10:14 AM

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