3.3 Subgame Perfect Equilibrium
2 min read•july 22, 2024
refines for . It ensures players act rationally at every decision point, considering future consequences. This concept is crucial for understanding strategic behavior in multi-stage interactions.
To find subgame perfect equilibrium, use . Start at the game's end and work backwards, determining optimal actions at each node. This process eliminates non- and yields strategies optimal for the entire game and every subgame.
Subgame Perfect Equilibrium
Properties of subgame perfect equilibrium
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- Refinement of Nash equilibrium for sequential games ensures players' strategies are optimal for the entire game and every subgame
- Players act rationally at each decision point, considering the consequences of their actions (e.g., anticipating future moves)
- Assumes players have about the game structure and payoffs (e.g., knowing the and possible outcomes)
- Strategies are sequentially rational, meaning they are optimal given the strategies of other players (e.g., choosing the best response at each stage)
Subgames in extensive form games
- A subgame is a portion of an extensive form game starting at a single decision node and including all subsequent nodes and branches
- To identify a subgame:
- Find a node where the player has perfect information about previous moves
- Include all nodes and branches following from that node
- must have a single starting point and contain all subsequent decision nodes and (e.g., a branch of the game tree)
- are subgames that do not include the entire game tree (e.g., a smaller portion of the game)
Equilibrium in sequential-move games
- To find the subgame perfect equilibrium, use backward induction:
- Start at the end of the game tree and work backwards
- At each decision node, determine the for the player, assuming all subsequent players will also act optimally
- Eliminate branches corresponding to suboptimal actions
- The resulting path from the root to a terminal node represents the subgame perfect equilibrium
- In subgame perfect equilibrium, players' strategies are optimal at every decision point, considering the strategies of other players (e.g., making the best choice at each stage based on anticipated future moves)
Subgame perfect vs other equilibria
- Nash equilibrium (NE):
- Applies to both simultaneous and sequential games
- Strategies are optimal given other players' strategies, but not necessarily optimal for every subgame (e.g., may include non-credible threats)
- (WPBE):
- For games with incomplete information (e.g., auctions with unknown valuations)
- Requires players to update beliefs according to Bayes' rule and play optimally given those beliefs
- :
- Refinement of subgame perfect equilibrium for games with imperfect information (e.g., poker)
- Requires players to have consistent beliefs about off-equilibrium path events and play optimally given those beliefs
Key Terms to Review (16)
Backward induction: Backward induction is a method used in game theory to analyze decision-making processes by reasoning backwards from the end of a problem to determine optimal strategies. This technique involves considering the final outcomes of a game and working back through each player's possible choices to identify the best decisions at each step. It is crucial for understanding how players can make rational choices in extensive form games, evaluate subgame perfect equilibria, and establish credible threats and promises.
Credible Threats: Credible threats refer to potential actions or strategies that a player in a game can take to influence the behavior of other players, which are believable and likely to be executed. These threats can serve as a strategic tool in negotiations, helping to deter unwanted actions from opponents or encourage compliance with desired behaviors. The effectiveness of credible threats relies on the player's ability to commit to the action, making it more likely that others will take them seriously in their decision-making processes.
Extensive Form Games: Extensive form games are a representation of strategic interactions among players that illustrate the sequence of actions and decisions made over time. They use decision trees to depict how players make choices at different points, taking into account the possible moves and outcomes. This format allows for a clearer understanding of the game's dynamics, including information sets, where players may not know what others have chosen at earlier stages.
Game tree: A game tree is a graphical representation of a sequential game, showcasing the possible moves and outcomes at each decision point made by players. It helps visualize the strategies available to each player and illustrates how their choices lead to different payoffs, making it essential for understanding complex strategic interactions. The structure of a game tree allows players to analyze their options and the potential consequences of their decisions in a clear and organized manner.
Nash Equilibrium: Nash Equilibrium is a concept in game theory where players, knowing the strategies of their opponents, choose their optimal strategies resulting in a situation where no player has anything to gain by changing their own strategy unilaterally. This balance occurs when each player's strategy is the best response to the strategies chosen by others, highlighting the interdependence of player decisions and strategic decision-making.
Optimal Action: Optimal action refers to the strategy or decision that yields the best possible outcome for a player in a game, taking into account the choices of other players and the structure of the game itself. This concept is crucial in understanding how players evaluate their options and choose actions that maximize their payoffs. In the context of strategic interactions, optimal actions are determined through processes like backward induction, particularly when analyzing sequential games and ensuring that every action remains a best response in every possible subgame.
Perfect Information: Perfect information refers to a situation in a game where all players have complete knowledge of the game's structure, payoffs, and the actions taken by other players at all times. This condition allows players to make fully informed decisions based on the available data, leading to more predictable outcomes. In contexts like subgame perfect equilibrium and long-term relationships, perfect information plays a crucial role in determining strategies and ensuring that players can anticipate responses from others.
Proper Subgames: Proper subgames are subsets of a game that include a specific player's decision nodes and subsequent outcomes, starting from a given point in the game. These subgames must consist of all the possible actions and payoffs that follow the decision made at that node, ensuring that they are complete in terms of the strategic possibilities available to players. Proper subgames are crucial for analyzing strategies and determining equilibrium points, especially in the context of subgame perfect equilibrium.
Rationality: Rationality refers to the principle that individuals make decisions based on logic and reason, aiming to maximize their utility or payoff in uncertain situations. This concept is foundational in understanding how players behave in strategic settings, as it influences the choices they make when interacting with others. Rationality assumes that players are aware of their preferences and act accordingly to achieve the best possible outcomes based on the information available to them.
Sequential Equilibrium: Sequential equilibrium is a refinement of Nash equilibrium applicable in dynamic games where players make decisions at various points in time. It combines the concepts of Nash equilibrium and perfect Bayesian equilibrium, ensuring that players' strategies are not only optimal given their beliefs but also consistent with those beliefs throughout the game. This means that players should be able to update their beliefs based on the observed actions of others and that the strategies they choose must remain optimal given these updated beliefs.
Sequential games: Sequential games are a type of game in which players make decisions one after another, rather than simultaneously. This structure allows players to observe the actions of others before making their own choices, which can significantly influence the outcome of the game. The analysis of these games often involves constructing extensive form representations to understand the strategic interactions and potential payoffs involved.
Strategy Profile: A strategy profile is a combination of strategies chosen by all players in a game, outlining their respective actions in a given situation. This concept is crucial for understanding how players make decisions and interact with one another, particularly in scenarios involving multiple players or rounds. The strategy profile is foundational to analyzing outcomes like equilibrium points, which are essential in strategic settings and can vary based on whether the game is finite or infinite, or whether it involves pure or mixed strategies.
Subgame Perfect Equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium used in dynamic games, where players make decisions at different stages. It requires that players' strategies constitute a Nash equilibrium in every subgame of the original game, ensuring that players' strategies are optimal even when the game reaches any point in the future. This concept helps analyze decision-making processes in extensive form games and supports the evaluation of credible threats and promises in strategic interactions.
Subgames: Subgames are segments of a game that can be analyzed independently from the larger game structure. These portions are defined by the remaining choices available to players after a certain point in the game, and they maintain their own strategic elements and decision-making processes. Understanding subgames is crucial for analyzing complex games, particularly when determining strategies and outcomes in the context of subgame perfect equilibrium.
Terminal Nodes: Terminal nodes are the endpoints in a decision tree or extensive form game, where no further actions or decisions occur. They represent final outcomes that can lead to payoffs for players involved in the game, and understanding these nodes is crucial for analyzing strategies, outcomes, and the overall structure of the game. They help in visualizing the end results of various strategic paths and play a key role in evaluating the effectiveness of decisions made along the way.
Weak Perfect Bayesian Equilibrium: Weak Perfect Bayesian Equilibrium is a refinement of Bayesian equilibrium that incorporates players' beliefs and strategies in dynamic games with incomplete information. In this concept, players not only make optimal decisions based on their beliefs but also update these beliefs upon observing the actions of other players, ensuring consistency across all possible paths of the game. This equilibrium concept is particularly important in situations where players' strategies depend on both their information and the observed actions of others.