Question

How is each Nash equilibrium a sub game perfect equilibrium? I need this explained further please!

Answer 1

Nash equilibrium is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that if the players played any smaller game that consisted of only one part of the larger game and their behavior represents a Nash equilibrium of that smaller game, then their behavior is a subgame perfect equilibrium of the larger game. Every finite extensive game has a subgame perfect equilibrium.

A common method for determining subgame perfect equilibria in the case of a finite game is backward induction. Here one first considers the last actions of the game and determines which actions the final mover should take in each possible circumstance to maximize his/her utility. One then supposes that the last actor will do these actions, and considers the second to last actions, again choosing those that maximize that actor's utility. This process continues until one reaches the first move of the game. The strategies which remain are the set of all subgame perfect equilibria for finite-horizon extensive games of perfect information. However, backward induction cannot be applied to games of imperfect or incomplete information because this entails cutting through non-singleton information sets.

A subgame perfect equilibria necessarily satisfy the One-Shot deviation principle.

The set of subgame perfect equilibria for a given game is always a subset of the set of Nash equilibria for that game. In some cases, the sets can be identical.

The Ultimatum game provides an intuitive example of a game with fewer subgame perfect equilibria than Nash equilibria.

A Nash equilibrium is said to be subgame perfect if an only if it is a Nash equilibrium in every subgame of the game. The third subgame is the game itself. Note that, in each subgame, the equilibrium computed via backward induction remains to be an equilibrium of the subgame.