Hi is there a subgame perfect equilibrium or a Nash equilibr

Hi is there a subgame perfect equilibrium or a Nash equilibrium for the following? If so, how do I find this out??

Consider the two-player extensive-form game represented below. Player 1\'s payoff is written at the top of the vector of payoffs at each terminal history and player 2\'s payoff is written at the bottom. Start Across 4 2 0) Stop Down 1 4 1

Solution

You can find it by backward induction. Across is not optimal strategy for 2 since this will give 0 payoff. B will like

down because given his options this maximises his payoff. He will prefer stop but it is not his discretion. It depends on action of A. A will not prefer stop because it gives him less payoff than across and atleast same payoff of 1 if B chooses Down. So he will definitely choose start.

Since players play only smaller game that consisted of only one part of larger game it is a subgame perfect equilibria.