Is there an example of zero-sum game that has a Nash equilibrium which is not subgame perfect? As a refinement of  Nash equilibrium, it is known that not all Nash equilibria are subgame perfect. But it seems to me in zero-sum games of perfect information, Nash equilibrium coincides with subgame perfect equilibrium. Is it true for all zero-sum games?
 A: It is true that the set of Nash equilibrium and subgame-perfect equilibrium outcomes (distributions over terminal nodes) are equivalent for zero-sum games. However, the sets of equilibrium strategies need not be equivalent. Under subgame-perfect equilibrium, any off-equilibrium play is permitted. 
Consider an extensive game $\Gamma$ and some subgame-pefect equilibrium strategies in which some subgame is reached with probability $0$. A Nash equilibrium strategy which induces the same outcome can be achieved by modifying some player's strategy to be non-credible in the subgame which is never reached along the equilibrium path.
Here is an example. Let player 1 move first. If 1 moves left, the game ends and payoffs are (5,-5). If player 1 moves right, player 2 moves. If player 2 moves left, payoffs are (3,-3) and if player moves right, payoffs are (2,-2). 
Moving right is strictly dominated by left for player 1. The only NE and Subgame outcome is (5,-5) with probability 1. However, while (Left, Right) is the only subgame perfect strategy profile, (Left, Right) and (Left, Left) are Nash equilibrium strategies.  
