This paper introduces games with a saddle function. A saddle function is a real valued function on the set of action profiles such that, for one player, minimizing the function implies choosing her best-response, and, for the other players, maximizing it implies choosing their best-responses. We provide a new sufficient condition for robustness to incomplete information of sets of equilibria in a sense of Kajii and Morris (1997, Econometrica), Morris and Ui (2005, J. of Econ. Theory) for games with a saddle function. Our result unifies and generalizes sufficient conditions for zerosum and best-response potential games.
We study a home rental market where hosts compete in prices and decide which guests to accept. Such screening results in inefficiently many rejections and can reduce welfare. In a unique symmetric equilibrium, hosts randomize over prices so that it may be unprofitable for them to accept some efficient trades. Moreover, every equilibrium is inefficient when there are few hosts. We further show that screening increases welfare when there are enough hosts or when the potential loss of surplus from inefficient trades is high. We discuss the implications of our findings to online platforms such as Airbnb and HomeAway.
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