I found an article that mentions that reputation does matter. It seems the if more people know what sorts of offers you accept they will provide offers to you at that level. So, if you are known to take low offers in the Ultimatum Game, they will offer that value to you. If you reject low offers, you, of course lose out for that deal, but you bought a reputation of only accepting higher offers.
So, the question is how many times does someone reject an offer, waiting for that big offer, before it is too late?
Reference:
Science 8 September 2000:
Vol. 289. no. 5485, pp. 1773 - 1775
Fairness Versus Reason in the Ultimatum Game
Martin A. Nowak, Karen M. Page, Karl Sigmund
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