Two movie fans Ali and Ayse have only one pre-paid ticket to one of the films in Istanbul

Film Festival, which is sold out, and they have to decide who gets the ticket. They both

value seeing the movie equally, and each gets a payoff u > 0 from watching the movie. To

determine who gets the ticket, they agree to play the following game: They simultaneously

pick a number from the interval [0,). Whoever picks the larger number gets the ticket.

If they pick the same number, then Ayse gets the ticket. However, picking a large number

is costly. They both incur a cost equal to the number they pick regardless of receiving the

ticket or not. Find all pure strategy Nash equilibrium profiles, if any.