All equilibria found with backwards induction on a tree of a perfect information game are Nash equilibria, but in general the reverse is not true:
y
(1)---+---(0, 20) ← Also a Nash equilibrium when (2) announces
| n y he will play y: (0, 20) > (-10, -2)
+---(2)---+---(-10, -2)
| n
+---( +5, -1) ← Solution through backwards induction
(In class, we've called this the "icecream game", (1) is the mom who needs to decide whether to buy his son an ice cream and (2) is the son who needs to decide whether to cry or not about it.)
However, Chess (or Checkers, or Tic Tac Toe) are different from the "icecream game" because the payoffs are either (1, −1) (white win), (−1, 1) (black win) or (0,0) (draw).
Do these games still allow Nash equilibria that can't be found through backward induction?