# Nash equilibrium that is not subgame perfect in an infinitely repeated game?

All of the literature on infinitely repeated games with perfect information uses the term subgame-perfect equilibrium. However, I cannot find any such game with a Nash equilibrium that is not subgame-perfect. Intuitively, NE and SPE mean the same, but is it true? Why do we distinguish them?

## 1 Answer

There is an important difference between NE and SPNE. In the language of the game theory, in SPNE, we must play Nash Equilibrium in any subgame. In NE, the prescribed play does not have to correspond to a Nash of equilibrium in Each subgame. This distinction applies to both infinitely and finitely repeated games. This means that in NE we might support the equilibrium play by non-credible threats while in SPNE all threats have to be credible (i.e., will be implemented if we ever arrive in that subgame).

Let's consider the following example: $$\begin{bmatrix} & L & R \\ T & 10,4 &6,6& \\ B & -5,-1 & -5,-1 \end{bmatrix}$$

The following is a NE:

-> Player 1 plays T as long as player 2 played L in all periods before. Once player 2 deviates, Player 1 plays B forever.

-> Player 2 plays always L.

This is a NE since no player has incentives to deviate as long as (T,L) has been played in all periods before. But this is not SPNE because once Player 2 deviates player 1 will have no incentives to play B as he is always strictly better off by playing T. So the punishment following a deviation is not credible.