🎱Game Theory Unit 6 – Sequential Games and Backward Induction

Key Concepts

• Sequential games involve players making decisions in a specific order, with each player aware of the previous players' actions
• Game trees visually represent the structure and possible outcomes of sequential games
• Backward induction is a solution concept used to determine the subgame perfect equilibrium in sequential games
• Nash equilibrium in sequential games occurs when each player's strategy is a best response to the strategies of the other players
• Subgame perfect equilibrium refines Nash equilibrium by ensuring that the equilibrium strategies are optimal at every decision point (node) in the game tree
  • This means that the strategies must be optimal not only for the entire game but also for every subgame within the game tree
• Credible and non-credible threats play a crucial role in sequential games and can influence the outcomes
• Information sets represent the knowledge available to players at different decision points in the game

Game Tree Representation

• Game trees are diagrams that illustrate the structure and possible outcomes of sequential games
• Nodes in the game tree represent decision points for players, while edges represent the available actions or choices at each node
• The initial node (root) represents the starting point of the game, and terminal nodes represent the possible outcomes or payoffs
• Information sets are used to group together nodes where a player has the same information and available actions
  • Dashed lines or ovals are often used to denote information sets in game trees
• Payoffs are typically listed at the terminal nodes, showing the outcomes for each player based on the sequence of actions taken
• Game trees help visualize the strategic interactions and decision-making processes in sequential games
• Example: In the classic "Ultimatum Game," the game tree would have two levels, with the proposer making an offer at the initial node and the responder accepting or rejecting the offer at the second level

Backward Induction Process

• Backward induction is a solution concept used to solve sequential games and determine the subgame perfect equilibrium
• The process involves starting at the terminal nodes of the game tree and working backward to the initial node
• At each decision node, the player making the decision chooses the action that maximizes their payoff, assuming that all subsequent players will also make optimal decisions
  • This is based on the principle of rationality, where players are assumed to make decisions that maximize their own payoffs
• The backward induction process eliminates non-credible threats and identifies the optimal strategies for each player at every decision point
• By working backward through the game tree, the subgame perfect equilibrium is determined, which represents the optimal strategies for all players in the game
• Backward induction ensures that the equilibrium strategies are not only optimal for the entire game but also for every subgame within the game tree
• Example: In a two-stage sequential game, the backward induction process would start by analyzing the second player's optimal decision at each possible outcome of the first player's action, and then determine the first player's optimal action based on the anticipated responses

Nash Equilibrium in Sequential Games

• Nash equilibrium in sequential games occurs when each player's strategy is a best response to the strategies of the other players
• In a Nash equilibrium, no player has an incentive to unilaterally deviate from their chosen strategy, given the strategies of the other players
• Nash equilibrium in sequential games takes into account the sequential nature of decision-making and the ability of players to observe and respond to previous actions
• Subgame perfect equilibrium is a refinement of Nash equilibrium that ensures the equilibrium strategies are optimal at every decision point in the game tree
  • This means that the strategies must be optimal not only for the entire game but also for every subgame within the game tree
• Nash equilibrium in sequential games can be determined using the backward induction process, which identifies the optimal strategies for each player at every decision point
• In some cases, there may be multiple Nash equilibria in a sequential game, and additional solution concepts (such as subgame perfect equilibrium) may be needed to select among them
• Example: In the "Stackelberg Competition" game, the Nash equilibrium involves the leader firm choosing the optimal quantity based on the anticipated best response of the follower firm

Subgame Perfect Equilibrium

• Subgame perfect equilibrium (SPE) is a refinement of Nash equilibrium that ensures the equilibrium strategies are optimal at every decision point (node) in the game tree
• SPE requires that the strategies chosen by players must be optimal not only for the entire game but also for every subgame within the game tree
  • A subgame is a portion of the game tree that starts at a particular decision node and includes all the subsequent nodes and branches
• SPE eliminates non-credible threats and ensures that players' strategies are rational and optimal at every stage of the game
• The backward induction process is used to determine the subgame perfect equilibrium in sequential games
• In an SPE, players' strategies must be optimal given the strategies of the other players, and this optimality must hold for every subgame, not just the overall game
• SPE is a stronger solution concept than Nash equilibrium, as it requires optimality at every decision point and eliminates non-credible threats
• Example: In the "Centipede Game," the subgame perfect equilibrium involves players choosing to "take" at every decision node, even though this leads to a lower overall payoff compared to the cooperative outcome

Real-World Applications

• Sequential games and backward induction have numerous real-world applications in economics, business, politics, and other fields
• Stackelberg competition: In markets with a leader-follower dynamic, firms make sequential decisions based on the anticipated reactions of their competitors
  • The leader firm chooses its strategy first, taking into account the expected best response of the follower firm
• Bargaining and negotiations: Sequential bargaining models, such as the Ultimatum Game and the Alternating Offers Game, analyze how parties make offers and counteroffers in negotiations
  • Backward induction helps determine the equilibrium outcomes and strategies in these bargaining situations
• Political decision-making: Sequential games can model political processes, such as legislative bargaining or international negotiations, where actors make decisions in a specific order
• Reputation and credibility: Sequential games can analyze how players build or maintain reputation and credibility over time through their actions and commitments
• Dynamic pricing and revenue management: Firms may use sequential decision-making to optimize prices and allocate resources based on the anticipated behavior of customers and competitors
• Investment and entry decisions: Sequential games can model how firms make investment or market entry decisions based on the expected reactions of incumbents or rivals

Common Mistakes and Pitfalls

• Failing to consider the sequential nature of decision-making and assuming that players make decisions simultaneously
• Ignoring the credibility of threats and assuming that all threats will be carried out, even if they are not optimal for the threatening player
• Not properly identifying the subgames within the game tree and applying backward induction incorrectly
• Overlooking the importance of information sets and assuming that players have perfect information at all decision points
• Confusing Nash equilibrium with subgame perfect equilibrium and not recognizing the need for optimality at every subgame
• Misinterpreting the payoffs or outcomes in the game tree, leading to incorrect analysis and conclusions
• Not considering the possibility of multiple equilibria and focusing only on a single solution
• Overrelying on backward induction without considering other factors, such as bounded rationality, incomplete information, or behavioral biases

Practice Problems and Examples

• Ultimatum Game: Analyze the subgame perfect equilibrium and discuss how fairness considerations may influence players' behavior
• Stackelberg Competition: Determine the optimal strategies for the leader and follower firms and calculate the equilibrium quantities and profits
• Centipede Game: Discuss the subgame perfect equilibrium and explore the reasons why players may deviate from the equilibrium in practice
• Alternating Offers Bargaining: Find the subgame perfect equilibrium and analyze how time preferences and discount factors affect the bargaining outcome
• Sequential Prisoner's Dilemma: Examine how the ability to observe and respond to the other player's action affects the equilibrium strategies and outcomes
• Reputation Game: Analyze how a player can build and maintain a reputation for being tough or cooperative through their actions in a repeated sequential game
• Sequential Auction: Determine the optimal bidding strategies for players in a sequential auction and discuss how information revelation affects the equilibrium outcome
• Sequential Investment Game: Explore how the threat of entry or retaliation affects the investment decisions of incumbent and entrant firms in a sequential setting