game theory and business decisions unit 3 study guides

What's This Unit All About?

• Sequential games involve players making decisions in a specific order rather than simultaneously
• Players have knowledge of the actions taken by previous players before making their own decision
• Introduces the concept of subgame perfect equilibrium, a refinement of Nash equilibrium for sequential games
• Focuses on solving sequential games using backward induction to determine the optimal strategies for each player
• Explores real-world applications of sequential games in various fields (business, economics, political science)
• Provides a framework for analyzing strategic interactions where timing and order of moves are crucial factors
• Helps develop skills in strategic thinking, decision-making, and anticipating the actions of others

Key Concepts and Definitions

• Sequential game: a game where players make decisions in a specific order, with each player aware of the actions taken by previous players
• Player: an individual or entity involved in the decision-making process within a game
• Strategy: a complete plan of action that specifies what a player will do in every possible situation throughout the game
• Payoff: the outcome or reward a player receives based on the strategies chosen by all players
• Subgame: a portion of a sequential game that can be treated as a separate game, starting from a specific decision node
• Subgame perfect equilibrium: a refinement of Nash equilibrium where players' strategies constitute a Nash equilibrium in every subgame of the original game
  • Ensures that the strategy profile is optimal and self-enforcing at every decision point
• Backward induction: a solution method for sequential games that involves reasoning backwards from the end of the game to determine the optimal strategies for each player
  • Starts by identifying the optimal decision at the last decision node and then works backwards to the earlier decision nodes

Types of Sequential Games

• Perfect information games: games where all players have complete knowledge of the previous actions taken by other players
  • Examples include chess, tic-tac-toe, and stackelberg competition
• Imperfect information games: games where players may not have complete knowledge of the previous actions taken by other players
  • Can involve simultaneous moves or hidden information
  • Examples include poker, bridge, and auctions with sealed bids
• Finite horizon games: sequential games with a fixed and known number of stages or rounds
  • Backward induction can be easily applied to solve these games
• Infinite horizon games: sequential games that continue indefinitely or have an unknown number of stages
  • May require more advanced solution concepts (subgame perfect equilibrium in repeated games)
• Stackelberg games: sequential games where a leader makes the first move, and a follower responds after observing the leader's action
  • Commonly used to model market competition with a first-mover advantage

Subgame Perfect Equilibrium Explained

• Subgame perfect equilibrium (SPE) is a refinement of Nash equilibrium for sequential games
• Ensures that the strategy profile is a Nash equilibrium in every subgame of the original game
• Eliminates non-credible threats and promises that are not optimal for players to follow through on
• Requires players to make optimal decisions at every decision node, considering the future consequences of their actions
• Can be found using the backward induction method, starting from the end of the game and working backwards
• In an SPE, no player has an incentive to deviate from their strategy at any decision point, assuming that all other players will also play their equilibrium strategies
• Provides a more robust and realistic solution concept for sequential games compared to Nash equilibrium alone

Solving Sequential Games: Step-by-Step

1. Identify the players, their available actions, and the order in which they make decisions
2. Represent the game using a game tree or extensive form representation
   • Decision nodes represent points where players make choices
   • Terminal nodes represent the end of the game and the resulting payoffs for each player
3. Start at the terminal nodes and work backwards through the game tree using backward induction
   • At each decision node, determine the optimal action for the player making the decision, assuming that all future players will also make optimal choices
4. Continue the backward induction process until reaching the initial decision node
   • The optimal action at the initial node, combined with the optimal actions at all subsequent nodes, constitutes the subgame perfect equilibrium strategy profile
5. Verify that the resulting strategy profile is a Nash equilibrium in every subgame of the original game
   • Ensure that no player has an incentive to deviate from their strategy at any decision point
6. Interpret the results and discuss the implications of the subgame perfect equilibrium for the specific game and its real-world applications

Real-World Applications

• Business decisions: sequential games can model market entry, pricing strategies, and investment decisions
  • Example: a duopoly market where firms sequentially choose production quantities (Stackelberg competition)
• Negotiation and bargaining: sequential games can analyze the strategic aspects of negotiations, such as labor disputes or international treaties
  • Example: a union and a firm engaging in a series of offers and counteroffers during a wage negotiation
• Political science: sequential games can model political campaigns, legislative decision-making, and international relations
  • Example: a presidential election where candidates sequentially choose their campaign strategies and respond to each other's actions
• Evolutionary biology: sequential games can study the evolution of cooperation and conflict in animal populations
  • Example: the hawk-dove game, which models animal contests over resources
• Legal proceedings: sequential games can analyze the strategic interactions between plaintiffs, defendants, and judges in legal cases
  • Example: a plea bargaining situation where the prosecutor and defendant sequentially decide whether to accept a plea deal or go to trial

Common Mistakes and How to Avoid Them

• Failing to consider all possible subgames and decision points when applying backward induction
  • Carefully map out the entire game tree and ensure that all decision nodes are accounted for
• Confusing subgame perfect equilibrium with Nash equilibrium
  • Remember that SPE is a refinement of Nash equilibrium specific to sequential games and requires optimality in every subgame
• Incorrectly assuming that players will always follow through on threats or promises
  • SPE eliminates non-credible threats and promises, so only consider strategies that are optimal for players to follow through on
• Neglecting the importance of the order of moves and the information available to players at each decision point
  • Pay close attention to the sequence of actions and the information sets of players when analyzing sequential games
• Misinterpreting the results of the subgame perfect equilibrium
  • Carefully consider the implications of the SPE strategy profile for the specific game and its real-world context
• Attempting to apply backward induction to games with imperfect information or simultaneous moves
  • Backward induction is most suitable for games with perfect information and sequential moves; other solution concepts may be needed for more complex games

Practice Problems and Examples

1. The Ultimatum Game:
   • Player 1 proposes a division of a fixed sum of money (e.g., $10) between themselves and Player 2
   • Player 2 can either accept the proposal, in which case both players receive the proposed amounts, or reject it, resulting in both players receiving nothing
   • Find the subgame perfect equilibrium of this game
2. The Centipede Game:
   • Two players take turns choosing either to "Take" a larger share of an increasing pot of money or to "Pass" and let the other player decide
   • If a player chooses to "Take," the game ends, and the players receive their respective payoffs
   • If both players always choose to "Pass," they both receive a larger payoff at the end of the game
   • Determine the subgame perfect equilibrium and discuss why it differs from the socially optimal outcome
3. The Sequential Prisoner's Dilemma:
   • Two suspects are being interrogated separately by the police
   • The suspects can either "Confess" or "Remain Silent," and their sentences depend on the actions of both players
   • The game is played sequentially, with Suspect 1 making their decision first, followed by Suspect 2
   • Find the subgame perfect equilibrium and compare it to the simultaneous move version of the Prisoner's Dilemma
4. The Stackelberg Duopoly Game:
   • Two firms (a leader and a follower) compete in a market by sequentially choosing their production quantities
   • The leader firm chooses its quantity first, and the follower firm observes this decision before making its own choice
   • The firms face a linear inverse demand function and have constant marginal costs
   • Determine the subgame perfect equilibrium quantities and profits for each firm and compare them to the simultaneous move Cournot duopoly outcome