3.2b Subgame Perfect Nash Equilibrium

Subgame Perfect Nash Equilibrium

A strategy profile $s^{\ast }$ constitutes a subgame perfect Nash equilibrium if for every subgame $G_E^d$ with root note $d$ in $G_E$, $s^{\ast }$ constitutes a Nash equilibrium within $G_E^d$.

Equivalently, a strategy profile $s^{\ast }$ constitutes an SPNE if $\forall s_i\in s^{\ast }$, every $s_i$ is sequentially rational at the root node.

The path of play in a subgame perfect Nash equilibrium is known as the equilibrium path.

Note!

The SPNE can never involve mixing in extensive form games of perfect information, since every player’s action is observable.

Intuition

What is the difference between NE and SPNE? Well:

• Nash equilibrium requires rational behavior (best response) only on the path of play.
• SPNE requires rational behavior ON AND OFF the path of play.
  • Done to add credibility to threats; if a strategy is not rational, then threatening with it will be irrational.

How do we find SPNEs? Using backward induction. Start at the last non-terminal node. Check the best response of the player at that node. Do so for all nodes on that level. Go back one level, and do the same for the other player. More on this in the future!

Properties

(Existence) At least one SPNE exists in any finite extensive form game.

(Uniqueness) If no players are indifferent between terminal nodes, there is a unique SPNE.

Examples

Strategic Voting