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Subgame Perfect Equilibrium

Subgame perfect equilibrium (SPE) is a concept in game theory that describes a situation where two or more subgames of a game are perfectly matched, meaning that each player has an optimal solution to the game. In other words, if players can solve all their subgames simultaneously, then they have achieved a state of “perfect equilibrium.”

In a perfect equilibrium, there is no conflict between two players because they can solve all their subgames simultaneously, and neither player can guarantee that their opponent will not be able to do so. This leads to an optimal solution for both players, as the game has reached a stable state where each player’s objective function (i.e., maximizing their payoff) is minimized or maximized.

SPE is often referred to as a “perfect equilibrium” because it represents a situation where one player can guarantee that their opponent will not be able to do so, even if they have an optimal solution for both players. This is in contrast to “perfect competition,” where two players can achieve perfect equilibrium by solving all their subgames simultaneously.

SPE has several important properties that make it desirable:

  1. Optimality: SPE implies that each player’s objective function must be maximized, meaning that they should have a good chance of achieving optimal solutions for both players.
  2. Stability: The stability of SPE is essential because it ensures that neither player can guarantee that their opponent will not be able to do so. This is because the game has reached a stable state where each player’s objective function must be minimized or maximized, and no player can guarantee that their opponent will not be able to do so.
  3. Consistency: SPE implies that the game has been consistent over time, meaning that it has maintained its optimal equilibrium despite changes in the opponents’ strategies or circumstances.
  4. Flexibility: SPE is also important because it allows for flexibility in response to changing circumstances, such as a player’s opponent’s strategy being different from their own.

SPE has been observed in various games, including:

  1. Game of Life: A game where two players can solve all their subgames simultaneously, but only one player can guarantee that their opponent will not be able to do so.
  2. Chess: A game where two players can solve all their subgames simultaneously, but only one player can guarantee that their opponent will not be able to do so.
  3. Tic Tac Toe: A game where two players can solve all their subgames simultaneously, but only one player can guarantee that their opponent will not be able to do so.
  4. Poker: A game where two players can solve all their subgames simultaneously, but only one player can guarantee that their opponent will not be able to do so.

In summary, SPE is a desirable property of games because it ensures that each player’s objective function must be maximized or maximizes themselves, and that the game has been consistent over time despite changes in the opponents’ strategies or circumstances. This property is essential for maintaining stability, consistency, and flexibility in response to changing circumstances.

See also

Real Business Cycle Theory

Slutsky Equation

Identification in Structural Models

Walrasian General Equilibrium

Screening and Signaling Equilibria