🆚Game Theory and Economic Behavior Unit 5 – Sequential Games & Subgame Equilibrium

Key Concepts

• Sequential games involve players making decisions in a specific order, with each player aware of the previous player's actions
• Players have perfect information about the game structure and payoffs, allowing them to make informed decisions
• Extensive form representation uses a game tree to illustrate the sequence of moves, decision points, and payoffs
• Backward induction is a solution concept used to determine the subgame perfect equilibrium by reasoning backwards from the end of the game
• Subgame perfect equilibrium ensures that players' strategies constitute a Nash equilibrium in every subgame of the original game
  • This concept eliminates non-credible threats and ensures rational decision-making at every stage
• Payoff structure determines the incentives and outcomes for each player based on their decisions and the decisions of others
• First-mover advantage refers to the potential benefit gained by the player who makes the first move in a sequential game (Stackelberg competition)

Types of Sequential Games

• Perfect information games where players have complete knowledge of all previous moves and payoffs (Chess, Go)
• Imperfect information games involve some level of uncertainty about previous moves or payoffs (Poker, Bridge)
  • Players may not have full knowledge of the other players' actions or the game's history
• Finite horizon games have a fixed number of moves or a predetermined endpoint (Ultimatum game, Dictator game)
• Infinite horizon games have no predetermined endpoint and can potentially continue indefinitely (Repeated Prisoner's Dilemma)
• Stackelberg games involve a leader-follower dynamic, where one player makes a decision, and the other player responds (Quantity leadership, Price leadership)
• Signaling games involve one player sending a signal to another player, who then takes an action based on the interpretation of the signal (Job market signaling, Advertising)

Game Tree Representation

• Game trees visually represent the structure and sequence of moves in a sequential game
• Nodes represent decision points for players, while edges represent the actions available at each decision point
• Terminal nodes indicate the endpoints of the game and specify the payoffs for each player based on the sequence of actions taken
• Subgames are smaller portions of the game tree that can be analyzed independently using backward induction
  • Subgames must have a single initial node and contain all successive nodes and edges
• Information sets are used in imperfect information games to group together nodes where a player cannot distinguish between different game histories
• Payoffs are typically represented as a tuple or vector at each terminal node, specifying the outcomes for each player (e.g., (2, 1) for a two-player game)

Backward Induction

• Backward induction is an algorithmic approach to solving sequential games and determining the subgame perfect equilibrium
• The process begins at the terminal nodes of the game tree and works backwards to the initial decision point
• At each decision point, the player making the decision chooses the action that maximizes their payoff, assuming that all subsequent players will also make optimal decisions
  • This assumes that players are rational and have perfect information about the game structure and payoffs
• The resulting strategy profile determined by backward induction constitutes the subgame perfect equilibrium of the game
• Backward induction can be applied to finite horizon games with perfect information, ensuring that players make optimal decisions at every stage
• The process helps identify and eliminate non-credible threats, as it assumes that players will make rational decisions in every subgame

Subgame Perfect Equilibrium

• Subgame perfect equilibrium (SPE) is a refinement of the Nash equilibrium concept for sequential games
• In an SPE, players' strategies must constitute a Nash equilibrium in every subgame of the original game
  • This means that no player has an incentive to deviate from their strategy at any decision point, assuming that all other players also play their equilibrium strategies
• SPE eliminates non-credible threats and ensures that players make rational decisions at every stage of the game
• To find the SPE, backward induction is applied to the game tree, starting from the terminal nodes and working backwards to the initial decision point
• In games with multiple Nash equilibria, the SPE helps select the most plausible and rational outcome
  • It eliminates equilibria that rely on non-credible threats or irrational behavior in subgames
• SPE is a stronger solution concept than Nash equilibrium, as it requires stability in every subgame, not just the overall game

Strategies and Outcomes

• Strategies in sequential games specify a complete plan of action for each player, detailing the moves they will make at every decision point
• Pure strategies indicate a specific action to be taken at each decision point, while mixed strategies assign probabilities to different actions
• Strategy profiles are combinations of strategies chosen by each player in a game, determining the outcome and payoffs
• Outcomes are the results of the game based on the strategies employed by the players, typically represented as payoffs or utilities
• Dominant strategies are those that provide a player with the highest payoff regardless of the strategies chosen by other players
  • Games with dominant strategies for all players have a unique Nash equilibrium
• Cooperative outcomes can be achieved when players have the ability to communicate and make binding agreements (Collusion in oligopolies)
• Non-cooperative outcomes arise when players make decisions independently, without the ability to make binding agreements (Price wars, Advertising campaigns)

Real-World Applications

• Sequential games have numerous applications in economics, business, and political science
• Stackelberg competition models market situations where a leader firm makes a decision (quantity or price), and follower firms respond (Cournot duopoly, Bertrand duopoly)
• Bargaining and negotiation scenarios often involve sequential decision-making, with parties making offers and counteroffers (Labor negotiations, International treaties)
• Sequential auctions, such as English auctions and Dutch auctions, can be analyzed using sequential game theory (Art auctions, Government procurement)
• Principal-agent problems, such as contract design and incentive structures, can be modeled as sequential games (Executive compensation, Regulation of monopolies)
• Political campaigns and electoral competition often involve sequential decision-making by candidates and voters (Campaign spending, Voter turnout)

Common Pitfalls and Misconceptions

• Assuming that all players have the same level of rationality or access to information can lead to incorrect predictions
  • Players may have different cognitive abilities, biases, or information sets that affect their decision-making
• Failing to consider the credibility of threats or promises can result in overestimating the likelihood of certain outcomes
  • Non-credible threats should be eliminated using backward induction and subgame perfect equilibrium
• Focusing solely on the overall game and ignoring the stability of subgames can lead to selecting equilibria that are not plausible or rational
  • Subgame perfect equilibrium ensures that players make optimal decisions in every subgame
• Neglecting the potential for communication, cooperation, or collusion among players can overlook important strategic considerations
  • In some cases, players may have incentives to cooperate or make binding agreements that alter the game's outcome
• Overreliance on simplified models or assumptions may not capture the full complexity of real-world situations
  • Factors such as incomplete information, bounded rationality, and psychological biases can influence decision-making in ways that deviate from standard game-theoretic predictions