GT extensive form games

Introduction to Representative Games in Extensive Form

Overview of Game Representation

• The lecture focuses on representative games in extensive form, emphasizing shorter recordings for easier topic navigation.

• A game requires five components: players, actions, knowledge at the time of action, outcomes from actions, and player preferences over outcomes.

Non-Cooperative vs. Cooperative Games

• The focus is on non-cooperative games where decisions are made individually rather than jointly.

• Non-cooperative games allow for individual incentives while acknowledging that cooperation can exist within strategic interactions.

Understanding Game Trees

Structure of Game Trees

• An extensive form or game tree illustrates decision nodes where players make choices; Player 1 moves first at the initial node.

• Player 1's options (alt and in) create branches at the decision node, indicating possible actions.

Information Sets and Perfect Information

• Player 2 observes Player 1's move; this scenario represents perfect information.

• In contrast, if Player 3 does not observe Player 1’s choice (a or b), it creates an information set represented by dotted lines connecting decision nodes.

Nature's Role in Games

Modeling Uncertainty with Nature

• Nature introduces uncertainty into the game through its own moves; it is not a player but affects outcomes based on probabilities.

• For example, there is a probability distribution regarding whether O observes S’s action (p), affecting subsequent decisions.

Assumptions About Knowledge in Games

• Common knowledge assumptions include that all players know the structure of the game and payoffs involved.

• This common knowledge extends beyond public knowledge; each player understands what others know about the game's structure.

Strategies in Game Theory

Definition of Strategy

• A strategy is defined as a complete contingent plan outlining how a player will act based on different potential scenarios throughout the game.

Understanding Strategies in Game Theory

Defining Strategy in Games

• A strategy for a player in a game is defined as a complete contingent plan, which can be challenging to grasp for some students. The instructor notes that objections arise every semester regarding this definition.

• In the centipede game example, Player One has two choices: 'i' or 'o'. Choosing 'o' ends the game, while choosing 'i' allows Player Two to make their choice.

• A strategy must specify actions at both the initial and subsequent decision nodes. It serves as instructions on how to play the game, akin to guidelines provided by leaders or managers.

Importance of Complete Contingent Plans

• The significance of having a complete contingent plan lies in its ability to account for unforeseen circumstances during gameplay. This ensures players are prepared for various scenarios.

• The selected strategy by Player One can influence Player Two's decisions even off the equilibrium path. A pure strategy involves selecting an action with certainty (probability one), without randomization.

Notation and Strategy Profiles

• Player One's set of pure strategies is denoted as S_1 , including combinations like i_a , i_b , and others. The notation helps clarify why certain specifications are necessary when defining strategies.

• Notation also includes generic references to players (e.g., using i ) and distinguishes between individual players and groups (e.g., -i denotes all players except player i ).

Payoffs and Strategy Profiles

• A strategy profile consists of each player's chosen strategies, represented as vectors from player 1 through player n. Each player's payoff depends on this collective strategy profile.

• If nature does not intervene, a given strategy profile defines a unique path through the game's decision tree. If nature does move, it still influences the outcome based on the defined strategies.

Transitioning to Normal Form Representation

• The discussion will transition from extensive form representation of games to normal form representation using matrices, although matrices may not always be applicable depending on specific cases.

• Before concluding this section, students should ensure they can represent strategic situations effectively in both extensive and normal forms.