5.2 Subgame perfect equilibrium concept
3 min read•august 7, 2024
takes to the next level in . It ensures players' strategies are optimal at every decision point, not just overall. This concept is crucial for understanding and time-consistent strategies.
SPE eliminates and promises, refining Nash equilibria to more realistic outcomes. It's applied in various scenarios, from chess to market entry games, helping predict rational behavior in complex strategic interactions.
Subgame Perfect Equilibrium Fundamentals
Defining Subgame Perfect Equilibrium
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- perfect equilibrium (SPE) represents a Nash equilibrium of every subgame in an
- SPE requires players' strategies to constitute a Nash equilibrium in every subgame, ensuring no player has an incentive to deviate at any decision point
- Guarantees that the equilibrium strategies remain optimal at every stage of the game, considering all possible future actions
Subgames and Nash Equilibrium
- Subgame defined as a subset of a game that begins at a single decision node and includes all subsequent nodes and branches
- must contain all the successors of the initial node, with the initial node consisting of a single decision point
- Nash equilibrium occurs when each player's strategy is a to the strategies of the other players, resulting in no unilateral incentive to deviate
Credibility and Consistency
Credibility in Strategic Interactions
- refers to the believability of a player's threat or promise in a strategic interaction
- Credible threats or promises must be in the player's best interest to carry out if called upon to do so
- Lack of credibility can undermine the effectiveness of strategic commitments and influence the behavior of other players
Time Consistency and Subgame Perfection
- ensures that a player's optimal strategy at the start of the game remains optimal at every subsequent decision point
- Subgame perfect equilibrium satisfies time consistency by requiring Nash equilibrium in every subgame
- Time-inconsistent strategies may involve empty threats or promises that are not credible, leading to suboptimal outcomes
Refinement of Nash Equilibrium
- Subgame perfect equilibrium serves as a of Nash equilibrium, eliminating Nash equilibria that involve incredible threats or promises
- Refinement narrows down the set of equilibria by imposing additional constraints on players' strategies
- SPE selects Nash equilibria that are robust to and eliminates equilibria based on non-credible strategies
Applications and Examples
Zermelo's Theorem in Chess
- states that chess has a unique subgame perfect equilibrium outcome, either a forced win for white, a forced win for black, or a forced draw
- Applies backward induction to the extensive form representation of chess, determining the optimal move at each decision point
- Demonstrates the existence of a deterministic outcome in finite two-player games with perfect information, such as chess
Selten's Chain-Store Paradox
- illustrates the concept of credibility and subgame perfection in a market entry game
- Considers a monopolist (incumbent) facing potential entry by a competitor in multiple markets
- The incumbent's threat to fight entry in every market is not credible, as it is not a subgame perfect equilibrium
- Backward induction reveals that the incumbent's optimal strategy is to accommodate entry in the final market, undermining the credibility of the threat
Key Terms to Review (22)
Backward induction: Backward induction is a method used in game theory to determine optimal strategies by analyzing a game from the end to the beginning. It involves looking at the last possible moves of players and determining their best responses, then moving sequentially backward through the game tree to deduce the optimal actions of earlier moves. This technique is particularly relevant in analyzing strategic interactions in sequential games and helps in identifying subgame perfect equilibria.
Bargaining Models: Bargaining models are frameworks used to analyze strategic interactions where parties negotiate over the allocation of resources or outcomes. These models help to understand how parties can reach agreements or settlements, factoring in their preferences, alternatives, and information available during negotiations. The insights gained from bargaining models are crucial for determining equitable outcomes and strategies in various economic contexts, including decision-making processes and competitive market scenarios.
Best Response: Best response is the strategy that produces the most favorable outcome for a player, given the strategies chosen by other players. It reflects how rational players will choose strategies that maximize their payoffs, taking into account the decisions of others, which connects to concepts like dominant strategies and Nash equilibrium, where each player's best response leads to stable outcomes in strategic interactions.
Credibility: Credibility refers to the belief that a player's threats or promises in a game are trustworthy and likely to be carried out. In strategic interactions, players must consider the credibility of their commitments, as it can heavily influence others' responses and lead to equilibrium outcomes. A credible commitment is essential for sustaining cooperation in repeated games or ensuring that threats are effective in deterring undesirable actions by opponents.
Credible Threats: Credible threats refer to commitments made by one party that are believable and can influence the behavior of another party in strategic interactions. They play a crucial role in ensuring that strategies are carried out effectively, particularly in competitive situations where one player's actions can impact the outcomes for others. The essence of credible threats is that they must be perceived as real and enforceable by the party making them, thereby affecting the decisions of other players in the game.
Dynamic Games: Dynamic games are a class of games where players make decisions at different points in time, taking into account the actions and reactions of other players over the course of the game. These games are characterized by the sequential nature of decision-making and often utilize extensive form representation, allowing for strategies that evolve as the game progresses. The concepts of equilibrium, beliefs, and information updates become crucial in understanding how players interact within these dynamic frameworks.
Extensive form game: An extensive form game is a representation of a strategic situation that allows players to make decisions at various points in time, depicted through a tree-like structure that illustrates the sequence of moves, choices, and potential outcomes. This format helps analyze strategies in situations where timing and order of moves matter, connecting key concepts like backward induction, sequential rationality, and subgame perfect equilibrium, while also illustrating credible threats and promises as well as the iterative elimination of dominated strategies.
Incredible threats: Incredible threats are commitments made by players in a game that they have no intention or ability to follow through on, often because doing so would not be in their best interest. In the context of strategic interactions, these threats can distort the expected outcomes, as rational players recognize that such threats lack credibility and are unlikely to influence the actions of others.
John Nash: John Nash was an influential mathematician and economist best known for his groundbreaking work in game theory, particularly the concept of Nash equilibrium. His theories have fundamentally shaped our understanding of strategic interactions among rational decision-makers, making them essential for analyzing competitive behaviors in various fields, including economics, political science, and biology.
Nash Equilibrium: Nash Equilibrium is a concept in game theory where no player can benefit by unilaterally changing their strategy if the strategies of the other players remain unchanged. This means that each player's strategy is optimal given the strategies of all other players, resulting in a stable outcome where players have no incentive to deviate from their chosen strategies.
Payoff Matrix: A payoff matrix is a table that shows the payoffs or outcomes for each player in a game, given all possible combinations of strategies chosen by the players. It visually represents the choices available to players and their potential results, making it essential for analyzing strategic interactions in various types of games.
Proper Subgame: A proper subgame is a part of a larger game that satisfies specific conditions allowing it to be analyzed as a standalone game. For a subgame to be considered proper, it must start at a decision node and include all future decisions from that node, ensuring players' strategies remain consistent throughout the game. This concept is crucial for identifying subgame perfect equilibria, where players’ strategies are optimal not just overall but also within each proper subgame.
Refinement: Refinement is a concept in game theory that involves improving the predictive power of equilibria by eliminating those that are considered unreasonable or unrealistic in certain contexts. This process helps to identify more precise equilibria, such as subgame perfect equilibria, which require that players' strategies are optimal not just for the entire game, but also for every possible subgame. Refinement enhances our understanding of players' strategic behavior and decision-making processes, making it easier to predict outcomes in dynamic situations.
Reinhard Selten: Reinhard Selten was a prominent German economist and game theorist, known for his foundational contributions to the understanding of strategic behavior in games. His work particularly emphasized the importance of subgame perfect equilibrium, which refines Nash equilibrium by ensuring that players make optimal decisions at every possible point in the game, thereby enhancing the predictability of outcomes in dynamic settings.
Selten's Chain-Store Paradox: Selten's Chain-Store Paradox highlights a situation in game theory where a chain store, facing potential entry by competitors, may be deterred from entering the market despite having the capacity to do so. The paradox arises because, intuitively, one might expect that a firm can always deter entry through aggressive pricing or other strategies; however, the rationality of players can lead to unexpected outcomes where entry occurs despite the firm's capabilities to deter it.
Sequential games: Sequential games are a type of game in game theory where players make decisions one after another, rather than simultaneously. In these games, the order of moves matters and players can take into account the previous actions of others when deciding their next move. This structure allows for strategies that can exploit the timing of decisions, leading to various outcomes based on the knowledge and expectations players have about each other's actions.
Stackelberg competition: Stackelberg competition is a strategic game in economics where firms compete on quantity rather than price, with one firm acting as a leader and the other as a follower. This model highlights the importance of sequential decision-making, where the leader firm sets its output level first, allowing the follower to optimize its response based on the leader's choice. Understanding this dynamic is crucial for analyzing market behavior, especially in industrial organization and bargaining situations, where firms' strategies significantly affect market outcomes.
Strategy Profile: A strategy profile is a complete description of the strategies chosen by each player in a game, detailing what action each player will take in every possible situation they may encounter. This concept is crucial as it helps analyze the overall outcome of strategic interactions and assists in determining the equilibrium points within various game formats. Understanding strategy profiles is essential for evaluating decision-making processes, especially when converting between different game representations or analyzing equilibria in games with incomplete information.
Subgame: A subgame is a portion of a larger game that consists of a decision tree starting from a particular node, where the strategies and payoffs are analyzed as if the subgame were an independent game. It allows for examining the strategic interactions and outcomes from a specific point in the larger game's structure. Subgames are crucial for understanding the concept of subgame perfect equilibrium, where strategies must be optimal at every point of the game, not just overall.
Subgame Perfect Equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium applicable to dynamic games where players make decisions at various stages. It requires that players' strategies constitute a Nash equilibrium in every subgame of the original game, ensuring that the strategies are credible and optimal, even if the game is played out from any point along the decision path.
Time Consistency: Time consistency refers to a situation where a decision-maker's preferences or strategies remain the same over time, even as circumstances change. This concept is crucial in game theory and economics, particularly when analyzing the strategic interactions of players in dynamic settings. Time consistency is essential for ensuring that decisions made at one point in time align with future decisions, maintaining credibility and trust among players.
Zermelo's Theorem: Zermelo's Theorem is a foundational result in game theory that asserts every finite extensive game with perfect information has a solution in the form of a strategy for each player. This theorem is significant as it establishes the concept of backward induction, which helps in determining optimal strategies by analyzing the game from its final stages back to the initial decision points.