# Find a subgame perfect equilibrium and a Nash equilibrium

I want to know if my thinking is correct. Look at the following game.

As the game has only one subgame (i.e., the game itself) then the Nash Equilibria will coincide with the subgame perfect equilibria.

In this case, we have two Nash equilibria: {U, u} and {D, d}. By my statement before, the subgame perfect equilibria will be {U, u} and {D, d} too.

Am I right?

• You are correct!
– user17900
Nov 6 '18 at 15:11
• The presentation suggests that it's a sequential game. If so, then "u" is not a strategy, so {U,u} is not a Nash equilibrium. Nov 6 '18 at 20:22

The SPNE (Subgame Perfect Nash Equilibrium) is a refinement of the NE (Nash Equilibrium). So, let's call $$S$$ the set of all SPNE and $$N$$ the set of all NE in a game. Then:
$$S \subseteq N.$$
$$S = N.$$