  We consider an auction market in which market makers fill the order book
during a given time period while some other investors send market orders. We
define the clearing price of the auction as the price maximizing the exchanged
volume at the clearing time according to the supply and demand of each market
participants. Then we derive in a semi-explicit form the error made between
this clearing price and the efficient price as a function of the auction
duration. We study the impact of the behavior of market takers on this error.
To do so we consider the case of naive market takers and that of rational
market takers playing a Nash equilibrium to minimize their transaction costs.
We compute the optimal duration of the auctions for 77 stocks traded on
Euronext and compare the quality of price formation process under this optimal
value to the case of a continuous limit order book. Continuous limit order
books are found to be usually sub-optimal. However, in term of our metric, they
only moderately impair the quality of price formation process. Order of
magnitude of optimal auction durations is from 2 to 10 minutes.
