11.4 Sequential games and subgame perfect equilibrium
3 min read•august 16, 2024
Game theory gets real when we look at . These are like chess matches where players take turns, and each move affects the next. Understanding how these games work is key to grasping strategic decision-making in the real world.
takes things up a notch. It's all about making smart choices at every step, not just overall. This concept helps us figure out why some strategies work better than others in complex, multi-step situations.
Sequential Games and Game Trees
Game Tree Structure and Elements
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- Game trees graphically represent sequential games showing order of play, decisions, and outcomes
- represent decision points while show possible actions or choices
- indicate when a player is unaware of previous moves
- typically appear at terminal nodes indicating outcomes based on decision sequences
- Example: Chess with initial move (e4, e5, etc.) branching to subsequent possible moves
Backward Induction Method
- Solves sequential games by reasoning backwards from end to beginning
- Process identifies optimal choices at each decision node starting from final moves
- Used to find perfect equilibria by determining at every decision point
- Example: Solving ultimatum game by first considering responder's optimal choice, then proposer's
Subgame Perfect Equilibrium
Concept and Refinement of Nash Equilibrium
- Subgame perfect equilibrium (SPE) refines for sequential games
- Requires optimal play in every subgame of the original game
- Subgame defined as any part of game tree considered as separate game from single decision node
- SPE eliminates by ensuring strategies optimal at every decision point
- Addresses Nash equilibrium limitation in sequential games where some rely on non-credible strategies
- Example: where SPE differs from some Nash equilibria relying on non-
Finding and Analyzing SPE
- Identify all subgames within original game
- Solve for Nash equilibria in each subgame working backwards to game start
- Number of SPE always less than or equal to Nash equilibria count
- SPE unique in solved by (no payoff ties)
- Example: Solving by first analyzing subgame after potential entry, then working backwards
Credibility in Sequential Games
Assessing Threat and Promise Credibility
- Credible threats/ carried out if conditions occur non-credible ones not in player's best interest
- crucial requiring strategy remain optimal as game progresses
- SPE identifies credible threats/promises by ensuring optimal strategies at all decision points
- illustrates how rational one-shot game threats become non-credible in sequential context
- Example: Analyzing credibility of incumbent firm's threat to engage in predatory pricing against new entrants
Enhancing Credibility in Games
- make non-credible threats/promises credible by limiting future options or altering incentives
- in repeated games enhance credibility by creating long-term follow-through incentives
- Game-theoretic analysis compares short-term deviation benefits with long-term lost credibility consequences
- Example: Firm building excess capacity as commitment device to deter market entry by competitors
Key Terms to Review (21)
Backward induction: Backward induction is a method used in game theory to analyze sequential games by reasoning backward from the end of a game to determine optimal strategies at each previous stage. It helps players anticipate the moves of others, enabling them to make informed decisions based on the expected outcomes of future actions. This technique is particularly useful in dynamic scenarios where players take turns making decisions, as it ensures that all players act rationally and strategically throughout the game.
Branches: In the context of sequential games, branches represent the possible actions or decisions that players can take at different points in the game. Each branch corresponds to a specific choice made by a player, leading to a new state in the game and influencing the subsequent choices available to other players. Understanding branches is crucial for analyzing how decisions unfold over time and the strategies players may adopt based on previous moves.
Centipede Game: The centipede game is a multi-stage game in game theory where two players alternately choose whether to take a larger share of an increasing payoff or pass the opportunity to the other player, leading to a possible larger future payoff. This game illustrates the concepts of strategic decision-making and subgame perfect equilibrium, as players must consider not only immediate payoffs but also the potential outcomes of future moves. It highlights the complexities of rational decision-making in sequential games, where the optimal strategy may not always be immediately apparent.
Chain-store paradox: The chain-store paradox refers to a situation in game theory where a firm operating multiple stores faces a dilemma about whether to engage in aggressive price competition or to maintain higher prices across its stores. This paradox highlights the conflict between short-term profit maximization and long-term strategic considerations, particularly when considering how competitors may react in sequential pricing games.
Commitment devices: Commitment devices are tools or strategies that help individuals stick to their long-term goals by restricting their future choices. These devices are used to counteract tendencies like procrastination and impulsive decision-making, often seen in intertemporal choices. By making certain future actions more difficult or costly, commitment devices help align short-term behavior with long-term objectives, particularly when preferences change over time.
Credible threats: Credible threats are actions or statements made by a player in a strategic interaction that are believable and can influence the behavior of others. In sequential games, where players make decisions one after another, the concept of credible threats is crucial for establishing strategic outcomes. If a threat is credible, it can alter the strategies chosen by other players, leading to different equilibria based on the expectations of future actions.
Entry Deterrence Game: An entry deterrence game is a strategic scenario where an incumbent firm takes actions to prevent potential entrants from entering the market. These actions can include setting prices low or making heavy investments to signal commitment and discourage new competitors. The game highlights the strategic interactions between existing firms and potential entrants, where the incumbent's choices affect the entrant's decisions, reflecting the dynamics of competition in sequential games and the concept of subgame perfect equilibrium.
Game Tree: A game tree is a graphical representation of a sequential game that illustrates the possible moves and outcomes for each player at every stage of the game. It helps in visualizing the decision-making process, showing how players can make choices based on the actions of others and the unfolding events in the game. This structure is vital for analyzing strategies and determining subgame perfect equilibria.
Information Sets: Information sets are used in game theory to describe a situation where a player cannot distinguish between different states of the game at a given point in time. This concept is crucial for understanding how players make decisions in sequential games, particularly when they must choose actions without knowing the prior moves of other players. Information sets help to analyze strategic behavior, as they reflect the knowledge and beliefs players have about the game state when making their choices.
Nash Equilibrium: Nash Equilibrium is a concept in game theory where players in a strategic interaction choose their optimal strategy, given the strategies of others, resulting in no player having an incentive to deviate from their chosen strategy. This concept is crucial in understanding how firms operate in competitive markets, particularly where their decisions are interdependent.
Nodes: In game theory, nodes are points in a decision tree or game tree that represent a player's choice or an outcome of a previous choice. They serve as critical junctions where players make decisions based on the strategies available to them and the potential reactions of their opponents. Understanding nodes is essential for analyzing sequential games, where the timing and order of moves can significantly impact the strategies employed by players.
Non-credible threats: Non-credible threats are statements or commitments made by players in a strategic interaction that lack the power to influence the behavior of other players because they are unlikely to be carried out. These threats are often viewed as empty posturing that do not create real fear of retaliation or consequences, which is crucial in determining strategic decisions in sequential games. The effectiveness of a threat often hinges on its credibility, and when threats are perceived as non-credible, they fail to shape the actions of opponents.
Optimal Strategy: An optimal strategy refers to the best plan of action that a player can adopt in a game, maximizing their payoff given the strategies of other players. In the context of sequential games, it involves making decisions at each stage that anticipate future moves, ensuring that every action taken is the best response to the anticipated actions of opponents. This concept is essential for determining outcomes that are sustainable and can lead to a subgame perfect equilibrium.
Payoffs: Payoffs represent the rewards or outcomes received by players in a game, often expressed in terms of utility or monetary value. In sequential games, these payoffs are crucial for determining the best strategies for each player, as they reflect the consequences of their actions at different stages of the game. Understanding payoffs allows players to anticipate others' responses and make informed decisions to maximize their own outcomes.
Perfect information games: Perfect information games are a type of strategic interaction where all players have complete knowledge about the game's structure, payoffs, and the actions previously taken by all other players. This complete transparency allows players to make fully informed decisions at every stage of the game, leading to predictable outcomes. In such games, strategies are based on the assumption that opponents will act rationally and consider all available information when making their choices.
Promises: In game theory, particularly in the context of sequential games, promises refer to commitments made by players to take specific actions in the future, often influencing subsequent decisions. These promises can shape strategies and outcomes by creating expectations that guide player behavior, making them crucial for achieving cooperative outcomes or establishing credibility in competitive settings.
Reputation effects: Reputation effects refer to the influence that a player's history of actions has on their future interactions in strategic settings, particularly in sequential games. These effects can shape players' strategies as they consider how their past behaviors will impact their opponents' perceptions and decisions in subsequent rounds, creating a dynamic where maintaining a positive reputation can lead to better outcomes over time.
Sequential games: Sequential games are a type of game in game theory where players make decisions one after another, rather than simultaneously. This structure allows for strategic interactions where players can observe the actions of previous players before making their own decisions, leading to a more dynamic understanding of strategies and outcomes. Such games often involve the analysis of how players can optimize their choices based on the responses of others in the sequence.
Subgame: A subgame is a part of a larger game that begins at a particular decision node and includes all subsequent moves available to players. It allows for the analysis of strategic decision-making in a smaller, more manageable context, focusing on how players can respond to various situations that arise during the game.
Subgame Perfect Equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium applicable in dynamic games, where players' strategies form a Nash equilibrium in every subgame of the original game. This concept ensures that players make optimal decisions at every point in the game, anticipating future actions and outcomes. By focusing on sequential rationality, it highlights how players can strategically navigate situations where they have to make decisions at different stages.
Time consistency: Time consistency refers to a situation in which a decision-maker's preferences or strategies do not change over time, ensuring that optimal plans remain optimal as time progresses. This concept is critical in sequential decision-making scenarios, where players' actions can influence future outcomes, highlighting the importance of commitment and credibility in maintaining stable strategies.