Glossary

6.2 Subgame perfect equilibrium

6 min read•july 30, 2024

refines in sequential games, eliminating non-credible threats. It ensures players' strategies are optimal at every decision point, considering the game's dynamic nature and allowing for strategy revisions based on observed actions.

This concept is crucial for understanding how players make decisions in multi-stage games. By applying , we can predict more accurate outcomes and identify strategies that are sustainable throughout the entire game, not just at the beginning.

Subgame Perfect Equilibrium

Definition and Properties

Top images from around the web for Definition and Properties

1 of 1

Top images from around the web for Definition and Properties

1 of 1
  • A subgame perfect equilibrium is a refinement of Nash equilibrium used in of complete and that eliminates non-credible threats
  • In a subgame perfect equilibrium, players' strategies constitute a Nash equilibrium in every subgame of the original game
    • A subgame is a part of the original game that could be considered a separate game in itself
  • Subgame perfection requires that the players' strategies are optimal at any stage of the game, regardless of the history of play
    • This means that players cannot make empty threats or promises that they would not rationally carry out
  • The concept of subgame perfect equilibrium is based on the idea of backward induction, where the game is solved from the end to the beginning
    • Players anticipate future actions and work backwards to determine their optimal strategies
  • In a subgame perfect equilibrium, players cannot make non-credible threats or promises, as their actions must be optimal at every decision point
    • Example: In the ultimatum game, the proposer cannot credibly threaten to offer a very low amount, as the responder would reject such an offer

Applications in Extensive Form Games

  • To find a subgame perfect equilibrium in an extensive form game, start by identifying the proper subgames, which are parts of the game tree that could be considered separate games
    • Subgames must start at a single decision node and include all subsequent nodes and branches
  • Apply backward induction to solve the game, starting from the last decision nodes and working backwards to the root of the game tree
    • At each decision node, the player chooses the action that maximizes their payoff, assuming that all future actions will also be optimal
  • The resulting , which specifies the optimal action at every decision node for each player, constitutes the subgame perfect equilibrium
    • Example: In the centipede game, backward induction leads to the unique subgame perfect equilibrium where players always choose to "take" rather than "pass"
  • In games with multiple subgame perfect equilibria, additional refinements or criteria may be needed to select the most plausible or relevant equilibrium
    • Pareto dominance or risk dominance can be used to narrow down the set of equilibria

Solving Games with Subgame Perfection

Backward Induction Procedure

  • Identify the proper subgames of the extensive form game
    • Each subgame must have a single initial decision node and include all subsequent nodes and branches
  • Start at the last decision nodes of the game tree and determine the optimal action for the player at each node
    • Assume that all future actions will also be optimal
  • Work backwards through the game tree, determining the optimal action at each decision node based on the anticipated future actions
    • At each node, the player chooses the action that maximizes their payoff, given the expected actions of other players
  • The resulting strategy profile, which specifies the optimal action at every decision node for each player, constitutes the subgame perfect equilibrium
    • If there are multiple subgame perfect equilibria, additional refinements may be needed to select the most relevant one

Solving Repeated Games

  • Subgame perfection can also be applied to solve , where players interact over multiple periods
    • In a repeated game, a strategy specifies actions for each player in every possible history of play
  • To find a subgame perfect equilibrium in a repeated game, consider the game as a whole and apply backward induction
    • Start from the last period and work backwards, determining the optimal actions for each player in each possible history of play
  • The resulting strategy profile, which specifies the optimal action for each player in every possible history of play, constitutes the subgame perfect equilibrium of the repeated game
    • Example: In the infinitely repeated prisoner's dilemma, the "grim trigger" strategy (cooperate until the other player defects, then defect forever) can be a subgame perfect equilibrium under certain conditions

Nash Equilibrium vs Sequential Games

Limitations of Nash Equilibrium

  • In sequential games, Nash equilibrium may include non-credible threats or promises, as it does not consider the dynamic nature of the game
    • Players may have an incentive to deviate from their Nash equilibrium strategies once a certain point in the game is reached, making the equilibrium unsustainable
  • Nash equilibrium does not account for the possibility of players revising their strategies based on the observed actions of other players
    • In sequential games, players can update their beliefs and strategies as the game progresses, which Nash equilibrium fails to capture
  • In some cases, Nash equilibrium may lead to suboptimal outcomes or fail to predict the actual behavior of players in sequential games
    • Example: In the centipede game, the Nash equilibrium predicts that players will always choose to "take," but in practice, players often choose to "pass" in early stages of the game
  • Subgame perfect equilibrium addresses these limitations by ensuring that players' strategies are optimal at every decision point, eliminating non-credible threats and promises
    • It takes into account the dynamic nature of sequential games and the ability of players to revise their strategies based on observed actions

Advantages of Subgame Perfection

  • Subgame perfect equilibrium refines Nash equilibrium by eliminating non-credible threats and promises
    • It ensures that players' strategies are optimal at every decision point, regardless of the history of play
  • Subgame perfection takes into account the dynamic nature of sequential games, allowing players to revise their strategies based on observed actions
    • It captures the idea that players can update their beliefs and strategies as the game progresses
  • Subgame perfect equilibrium often provides a more accurate prediction of actual player behavior in sequential games compared to Nash equilibrium
    • It considers the of threats and promises and the incentives of players at each decision point
  • The concept of subgame perfection helps to identify and eliminate unsustainable or unrealistic equilibria in sequential games
    • It focuses on equilibria that are consistent with rational behavior at every stage of the game

Credibility of Threats and Promises

Defining Credibility

  • A threat or promise is credible if it is in the player's best interest to carry it out when called upon to do so
    • Credibility depends on the incentives of the player at the relevant decision point
  • Non-credible threats or promises are those that a player would not rationally carry out if the other player were to take the action that the threat or promise is meant to deter or encourage
    • Example: In the entry deterrence game, the incumbent's threat to fight entry is non-credible if fighting is more costly than accommodating entry
  • In a subgame perfect equilibrium, players cannot make non-credible threats or promises, as their actions must be optimal at every decision point
    • This ensures that the equilibrium is consistent with rational behavior throughout the game

Assessing Credibility

  • To assess the credibility of a threat or promise, consider the player's incentives at the relevant decision node and determine whether carrying out the threat or promise is their optimal action
    • Compare the payoffs of carrying out the threat or promise to the payoffs of not carrying it out, given the other player's expected action
  • If carrying out the threat or promise is not optimal for the player at the relevant decision node, then the threat or promise is non-credible
    • The other player should recognize this and not be influenced by the non-credible threat or promise
  • The presence of non-credible threats or promises in a game can lead to outcomes that differ from the subgame perfect equilibrium, as players may not believe or act upon these threats or promises
    • Example: In the chain store paradox, the incumbent's threat to fight entry is non-credible, leading to a different outcome than the subgame perfect equilibrium prediction
  • Credible threats and promises, on the other hand, can influence the behavior of other players and lead to outcomes consistent with the subgame perfect equilibrium
    • Example: In the ultimatum game, the proposer's threat to offer a low amount is credible, as the responder would accept any positive offer

Key Terms to Review (15)

Backward induction: Backward induction is a method used in game theory to solve extensive form games by reasoning backward from the end of the game to determine optimal strategies at each point. This approach starts from the final decision nodes and works backward to the initial decision, ensuring that every player's strategy is optimal given the future actions of other players. It connects with concepts like subgame perfect equilibrium and is particularly useful in analyzing strategic bargaining situations.
Bargaining model: The bargaining model is a framework used to analyze how parties negotiate and reach agreements over resources, outcomes, or actions, focusing on the strategies and payoffs involved. This model assumes that players have different preferences and can influence the outcome based on their offers and counteroffers, highlighting the importance of strategic interaction. Understanding this model is essential for analyzing situations where cooperation or competition occurs, especially in the context of sequential decisions.
Credibility: Credibility refers to the belief or perception that a player’s threat or promise is believable and can be trusted in strategic situations. It is crucial for players to make credible commitments in order to influence others' decisions, especially in dynamic environments where actions unfold over time. Establishing credibility can lead to more favorable outcomes in games and negotiations, as it affects how other players respond to threats and promises.
Dynamic games: Dynamic games are strategic interactions where players make decisions at various points in time, taking into account the history of the game and the potential future actions of others. These games allow for strategies to evolve over time and incorporate elements like commitment and timing, making them essential for understanding scenarios where players' actions influence one another's payoffs across different stages. The analysis of dynamic games can lead to concepts such as subgame perfect equilibrium and perfect Bayesian equilibrium, both crucial for evaluating players' strategies in these complex settings.
Extensive form games: Extensive form games are a type of game representation that allows for a detailed visualization of players' strategies, decisions, and possible outcomes over time. They are depicted using a tree-like structure, where nodes represent decision points and branches represent the choices available to players. This format is particularly useful in analyzing dynamic interactions, where the order of moves and possible subgames play a crucial role in determining optimal strategies.
Nash equilibrium: Nash equilibrium is a concept in game theory where no player can benefit from changing their strategy while the other players keep theirs unchanged. This situation arises when each player's strategy is optimal given the strategies of all other players, leading to a stable state in strategic interactions.
Normal Form Representation: Normal form representation is a way to describe a game using a matrix or a table format that outlines the players, their strategies, and the resulting payoffs for each combination of strategies. This format makes it easier to analyze strategic interactions, as it clearly lays out all possible actions and outcomes for the players involved, highlighting their preferences and choices. It's particularly useful in identifying equilibrium concepts and understanding the dynamics of strategic decision-making.
Payoff Matrix: A payoff matrix is a table that shows the possible outcomes of a strategic interaction between players in a game, detailing their payoffs based on different strategies they may choose. This matrix is fundamental in understanding how players make decisions, as it encapsulates all possible strategies and their respective outcomes, leading to insights on rational choice and strategic behavior.
Perfect Information: Perfect information is a situation in a game where all players have complete knowledge of the actions, payoffs, and strategies available to them and other players at every point in the game. This level of transparency allows for strategic decision-making based on the full set of available data, leading players to make rational choices that can be predicted based on this information. Perfect information is a critical concept in understanding strategic interactions and analyzing equilibrium states, as it ensures that players can foresee the consequences of their actions in various scenarios.
Rationality: Rationality refers to the principle of making decisions based on logic, consistent preferences, and maximizing utility. It is the foundation of strategic decision-making, where individuals or entities choose actions that lead to the best possible outcomes based on their preferences and available information. Understanding rationality helps explain behaviors in various contexts, such as predicting responses in strategic interactions and assessing the effectiveness of different strategies.
Repeated games: Repeated games are strategic interactions where the same game is played multiple times, allowing players to consider past actions when making current decisions. This setup can lead to different outcomes compared to a one-time game, as players can develop strategies based on previous interactions and can foster cooperation or competition over time. The dynamics of repeated games highlight the importance of strategy, trust, and potential for long-term relationships among players.
Sequential Game: A sequential game is a type of game in game theory where players make decisions one after another, rather than simultaneously. This structure allows players to observe the previous actions of others before making their own choices, which can significantly influence the strategies they adopt. The strategic implications of these games are often analyzed through extensive form representations, such as game trees, and can lead to concepts like subgame perfect equilibrium where players optimize their strategies at every possible decision point.
Stackelberg competition: Stackelberg competition is a strategic game in economic theory where firms make decisions sequentially rather than simultaneously. In this model, one firm, known as the leader, sets its output level first, and the other firm, the follower, makes its decision based on the leader's choice. This dynamic creates a hierarchy in decision-making and can lead to different outcomes compared to simultaneous competition.
Strategy Profile: A strategy profile is a combination of strategies chosen by all players in a game, representing their actions simultaneously. It provides a complete description of how each player will act, allowing the analysis of outcomes based on different strategic choices. Understanding strategy profiles is crucial for identifying equilibria and assessing how players' decisions interact within various frameworks, including payoff matrices and equilibria concepts.
Subgame Perfect Equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium applicable to dynamic games, where players' strategies are optimal not only for the entire game but also for every subgame that could be reached. This concept helps ensure that strategies are credible at every point in the game, thus avoiding non-credible threats and promises that could undermine strategic reasoning.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide