Partial perfect balance
The subgame perfect equilibrium is a concept of mathematical game theory developed by Reinhard Selten for games in extensive form . It represents a refinement of the Nash equilibrium ; i.e. every subgame-perfect equilibrium is also a Nash equilibrium. A Nash equilibrium is subgame perfect if it induces a Nash equilibrium in every subgame of G.
Each subgame-perfect equilibrium also represents the strategy profile for a sequential equilibrium .
Subgame
The subgame-perfect balance is based on the concept of the subgame , which in turn was specially developed for this concept. A sub-game is a game that begins in a single decision node in the game tree and contains all nodes that follow that node. In addition, the sub-game must not separate any subsequent information areas. Thus, every extensive game contains at least one sub-game, namely itself. Real sub-games, i.e. further sub-games in addition to the game itself, are not guaranteed, however, even with complex games.
Examples
Example I does not contain a real sub-game, whereas Example II contains two: One beginning in node B and one beginning in node C.
example 1
Example I has a Nash equilibrium in which the players each use a mixed strategy . Since it only has itself as a subgame, the balance is trivial subgame perfect.
Example 2
Example II has the equilibria (1 / (1/1)); (1 / (1/2)) and (2 / (2/2)). However, (2 / (2/2)) and (1 / (1/1)) are not part-game perfect. First about equilibrium: (2 / (2/2)) because when B is reached, no equilibrium is or would be induced for the remaining part of the game. The strategy (2/2) to advertise (before the game) would give player 2 a higher payout if this announcement were believed by player 1, the associated threat (to react to decision 1 of player 1 with decision 2) is however implausible. Similarly, strategy (1 / (1/1)) can also be excluded as a subgame-perfect balance.
"Credible" in this theory is an announcement by a player i to respond to a possible situation s with an action a if and only if a given s maximizes the utility of i. Since in example II player 2 according to strategy (2/2) in situation B would not respond with the benefit-maximizing move 1 but with move 2, the announcement of this action would be "move 2 given B" and the associated strategy (2/2 ) "implausible".
Common belief is a statement p for agents i and j if and only if: i and j believe that p (is true) and i and j believe that i and j believe that p, and i and j believe that i and j believe that p, etc.
If player 1 believes that the rationality of player 1 and player 2 is common belief and believes that the acceptance of the concept of subgame perfection among rational players is common belief and believes that all possible payouts according to situation B in example II (for both players) common belief, then: in example II, the announcement of strategy (2/2) and in particular the threat of move 2 of player 2 given situation B (move 1 of player 1) is in the literal sense implausible for this player 1. In addition, player 1 then also believes that this lack of credibility is common belief among rational players.
The only credible statement from player 2 is to use strategy (1/2) (the strategy that maximizes your profit). This means that in this example (1 / (1/2)) is a subgame-perfect equilibrium.
The point of this equilibrium concept is that the behavior of rational players must not be based on implausible announcements. Example: A monopolist threatens a potential competitor entering the market with ruining it by an aggressive price war when it enters the market. However, this price war would also damage the monopolists more than if they accepted a reduction in profits. Then the threat of the monopoly is implausible. A potential competitor should then not be impressed by the announcement that in the case of the sub-game "market entry" the strategy "aggressive price war" will be played, since this is not an equally weighted (Nash) strategy. As all players know this, their behavior cannot be based on unbalanced announcements in any sub-game. Subgame perfection thus excludes possibly implausible Nash equilibria.
However: Assume that rational gamblers would never make announcements that they believe to be common beliefs. If now, in example II, player 2 (before the game) should make the announcement to reply to move 1 (of player 1) with move 2, then this announcement would be implausible - but then player 1 would have problems, this announcement to be interpreted, provided that he had previously considered player 2 to be rational.