Alex and Barbara will simultaneously choose to go to the beach or to go cycling without talking to one another. Alex prefers beach and Barbara prefers cycling, but they prefer to be with one another. If both choose beach, Barbara's payoff is 2 and Alex's payoff is 3. If both choose cycling, Barbara's payoff is 3 and Alex's payoff is 2.

If they make different choices, the person who chose cycling (the cyclist) has an option to find the other player. If the cyclist chooses not to find the other player, both get a payoff of 0. If the cyclist does choose to find the other person, the other player can now choose where they will both go cycling or they will both go to the beach. In that case they each get the same payoff as outlined above, except that the cyclist now has his/her payoff reduced by 1 (because the cyclist spends some effort to find the other person).

For example, suppose Barbara chose beach and Alex chose cycling in the first stage of the game. Now if Alex chooses to find Barbara and Barbara decides they will go to the beach together, Barbara's payoff will be 2 and Alex's payoff will be 3-1 = 2.

(a) Draw the game tree.
(b) Find the subgame perfect Nash equilibrium.