3.Types of Games in Game Theory

Understanding the different types of games in game theory is essential as they define how players interact, make decisions, and strategize. We can broadly categorize games into simultaneous games and sequential games.

Simultaneous Games

Definition
In simultaneous games, players make their decisions without knowing the choices of the other players. This lack of information can lead to strategic decision-making based on expectations about what others will do.

Examples

  1. Prisoner’s Dilemma
    The classic example of a simultaneous game, the Prisoner’s Dilemma, illustrates how two rational individuals may not cooperate, even if it appears that it is in their best interest to do so.

    Scenario: Two criminals are arrested and interrogated separately. Each has two choices: to cooperate with the other (remain silent) or to betray the other (confess). The consequences are as follows:

    Payoff Matrix:

    • Outcomes:

      • If both prisoners cooperate and remain silent, they each serve 3 years (3, 3).

      • If one betrays while the other cooperates, the betrayer goes free (0 years) while the cooperator serves 5 years (0, 5).

      • If both betray, they each serve 1 year (1, 1).

    • Analysis: The dilemma arises because betrayal is the dominant strategy for both players. No matter what the other does, each player minimizes their own time by betraying, leading to the paradoxical outcome of both serving more time than if they had cooperated.

  2. Battle of the Sexes
    This game represents a scenario where two players have different preferences for activities but want to coordinate with each other.

    Scenario: A couple wishes to go out together but has different preferences: one prefers the football game while the other prefers the opera.

    Payoff Matrix:

    • Outcomes:

      • If both choose football, the football-lover receives a payoff of 2, while the opera-lover gets 1 (2, 1).

      • If both choose opera, they get payoffs of 1 and 2, respectively (1, 2).

      • If they choose different activities, they receive no payoff (0, 0).

    • Analysis: Players must balance their preferences with the desire to be together. The Nash Equilibria in this game are the two coordinated outcomes: both going to the football game or both going to the opera. If they do not coordinate, they both end up with zero payoffs, highlighting the importance of cooperation.

Key Characteristics of Simultaneous Games

  • No Prior Knowledge: Players make decisions without knowledge of others’ actions.

  • Strategic Anticipation: Players must anticipate the possible choices of their opponents, often leading to mixed strategies.

  • Nash Equilibrium: A state where players’ strategies are optimal given the strategies of others.

Solution Methods

  • Nash Equilibrium Analysis: Identify stable outcomes where players have no incentive to change their strategy unilaterally.

  • Best Response Strategies: Players calculate the best responses to each possible action of their opponents, guiding their own strategic choices.

Sequential Games

Definition
In sequential games, players make decisions one after another, allowing them to consider previous actions when formulating their strategies. This structure can lead to different strategic dynamics compared to simultaneous games.

Examples

  1. Stackelberg Competition
    In this economic model, one firm acts as a leader and sets its output level first, while other firms (followers) observe this decision and react accordingly.

    Scenario:

    • Firm A (the leader) decides on its production quantity first.

    • Firm B (the follower) observes Firm A's choice and then decides its own production.

    Payoff Structure:

    • Firm A anticipates how Firm B will respond to its output. If Firm A produces too much, it risks lowering the price, impacting both firms' profits.

    • The leader’s output decision directly affects the follower's response, making this a strategic interaction based on the timing of decisions.

    Example Output Levels:

    • If Firm A produces 100 units and Firm B responds by producing 80 units, both firms must calculate their respective payoffs based on market prices and costs.

    • The leader's advantage lies in its ability to set the stage for the follower's reaction.

  2. Chess
    Chess is a classic example of a sequential game where players alternate moves, with each move influencing the game's progression.

    Key Concepts:

    • Turn-Based Play: Players take turns to make moves, allowing them to respond to the opponent’s previous actions.

    • Backward Induction: Players consider future potential positions based on current moves, helping them to plan strategies several moves ahead.

    • Subgame Perfection: A refinement of Nash Equilibrium, ensuring that strategies remain optimal at every possible stage of the game, even if the game reaches different paths. For instance, a player’s threat to make a certain move must be credible and rational at each stage of play.

    Analysis:

    • In chess, a player might analyze several moves ahead (e.g., considering their own and the opponent’s possible responses). If a player has a winning strategy, they must ensure their moves lead to favorable positions in all potential subgames.

Key Characteristics of Sequential Games

  • Order of Moves: Decisions are made sequentially, allowing earlier choices to inform later ones.

  • Strategic Depth: Players can plan based on prior actions, introducing more complexity into their decision-making processes.

Solution Methods

  • Backward Induction: Used to determine optimal strategies by analyzing the game from the end back to the beginning.

  • Subgame Perfect Equilibrium: Ensures strategies are rational and credible at every stage of the game.

Conclusion

Understanding the distinctions between simultaneous and sequential games is crucial for analyzing strategic interactions. Each type presents unique challenges and requires different approaches to strategy formulation.

Questions for Reflection

  1. In the Prisoner’s Dilemma, what drives players to betray rather than cooperate?

    • Players are often driven by the fear of being worse off if they cooperate while the other betrays. The rational choice leads to both players betraying, even at a higher collective cost.

  2. How does coordination affect outcomes in the Battle of the Sexes?

    • Successful coordination allows players to maximize their payoffs. If they fail to coordinate, they end up with zero payoffs, highlighting the importance of communication and shared strategy.

  3. What strategic advantages does the leader have in Stackelberg competition?

    • The leader sets the output level first, allowing them to anticipate and influence the follower’s decisions. This can lead to higher profits compared to a simultaneous game where both firms choose outputs without knowledge of the other's decision.

  4. How does backward induction impact strategy in chess?

    • Backward induction allows players to assess potential future positions, leading to strategic planning that anticipates several moves ahead, thus increasing their chances of winning the game.

Next Lesson