Subgame perfect equilibrium

[2] A common method for determining subgame perfect equilibria in the case of a finite game is backward induction.

A subgame perfect equilibrium necessarily satisfies the one-shot deviation principle.

Determining the subgame perfect equilibrium by using backward induction is shown below in Figure 1.

Based on the provided information, (UA, X), (DA, Y), and (DB, Y) are all Nash equilibria for the entire game.

The second normal-form game is the normal form representation of the subgame starting from Player 1's second node with actions A and B.

Because of this, all games prior to the last subgame will also play the Nash equilibrium to maximize their single-period payoffs.

[5] Reinhard Selten proved that any game which can be broken into "sub-games" containing a sub-set of all the available choices in the main game will have a subgame perfect Nash Equilibrium strategy (possibly as a mixed strategy giving non-deterministic sub-game decisions).

The subgame-perfect Nash equilibrium is normally deduced by "backward induction" from the various ultimate outcomes of the game, eliminating branches which would involve any player making a move that is not credible (because it is not optimal) from that node.

One game in which the backward induction solution is well known is tic-tac-toe, but in theory even Go has such an optimum strategy for all players.

The problem of the relationship between subgame perfection and backward induction was settled by Kaminski (2019), who proved that a generalized procedure of backward induction produces all subgame perfect equilibria in games that may have infinite length, infinite actions as each information set, and imperfect information if a condition of final support is satisfied.

Figure 1
Figure 2
Subgame 1 is solved and (3,4) replaces all of Subgame 1 and player one will choose U -> (3,4)Solution for Subgame 1
Solution of Subgame Perfect Equilibrium
One game in which the backward induction solution is well known is tic-tac-toe