A Is An Extensive Form Representation Of A Game

The extensive form representation of a game is a detailed and comprehensive way to illustrate the sequential structure of a game, specifying the players, the order of their moves, their available actions at each decision point, the information they possess at those points, and the resulting payoffs for all possible outcomes. Unlike the normal form representation, which focuses solely on strategies and payoffs, the extensive form provides a rich, dynamic view of the game, capturing nuances like timing, information asymmetry, and credibility of threats.

Understanding the Extensive Form

The extensive form represents a game as a tree, where each node represents a decision point, each branch represents an action, and each terminal node represents an outcome with associated payoffs. This visual and structured format allows for a clear understanding of the game's dynamics and the strategic interactions between players. Key components of the extensive form include:

Nodes: Represent decision points in the game.
Branches: Represent actions available at each decision point.
Players: Identify who makes a decision at each node.
Information Sets: Group nodes where a player has the same information.
Payoffs: Represent the outcome for each player at the terminal nodes.

Key Elements of an Extensive Form Game

To fully grasp the extensive form representation, it is essential to understand its core elements. These elements collectively define the game's structure, rules, and potential outcomes.

1. Players

The extensive form clearly identifies the players involved in the game. Each decision node is labeled with the player who has the move at that point. This could be a human player, a computer algorithm, or even nature (representing random events).

2. Order of Moves

The extensive form explicitly shows the sequence in which players make their moves. This is crucial for understanding how actions influence subsequent decisions and the overall flow of the game. The tree structure visually depicts the order, with branches extending from one decision node to the next.

3. Actions

At each decision node, the extensive form specifies the set of actions available to the player whose turn it is to move. Each action corresponds to a branch emanating from that node, leading to either another decision node or a terminal node.

4. Information Sets

Information sets are a critical component for representing situations where players have incomplete information about the history of the game. An information set is a collection of decision nodes that a player cannot distinguish between at the time of making a decision. This means that when a player is at a node within an information set, they do not know which specific node they are at. All nodes within the same information set must have the same set of available actions. This reflects the idea that a player cannot base their choice on information they do not have.

5. Payoffs

At the end of each possible sequence of moves, the extensive form specifies the payoffs that each player receives. Payoffs are typically represented as a tuple of numbers, one for each player, indicating their respective gains or losses from that outcome.

Constructing an Extensive Form Game

Building an extensive form representation involves a systematic process of mapping out the game's structure and rules. Here’s a step-by-step approach:

Identify the Players: Determine all the participants involved in the game.
Define the Starting Point: Establish the initial node, representing the beginning of the game.
Map Decision Nodes: For each player, identify the situations where they have to make a decision. Represent these as nodes in the tree.
Draw Branches for Actions: From each decision node, draw a branch for each possible action the player can take. Label each branch with the corresponding action.
Specify Information Sets: Group nodes into information sets if a player cannot distinguish between them. Ensure all nodes in an information set have the same available actions.
Assign Payoffs: At each terminal node (where the game ends), assign payoffs to each player, representing the outcome of that particular sequence of actions.

Example: The Ultimatum Game

A classic example to illustrate the extensive form is the Ultimatum Game. In this game, Player 1 is given a sum of money (e.g., $10) and proposes how to divide it with Player 2. Player 2 can either accept the offer, in which case the money is divided as proposed, or reject the offer, in which case both players receive nothing.

Players: Player 1 and Player 2
Order of Moves: Player 1 makes an offer, then Player 2 responds.
Actions: Player 1 chooses an amount to offer to Player 2 (from $0 to $10). Player 2 can either accept or reject the offer.
Information Sets: Player 2 knows the offer made by Player 1 when deciding whether to accept or reject.
Payoffs: If Player 2 accepts, the payoffs are the agreed-upon division. If Player 2 rejects, both players get $0.

The extensive form representation would start with a node for Player 1, who has multiple branches representing different offers (e.g., offering $0, $1, $2, ..., $10). Each of these branches leads to a node for Player 2, who then has two branches: accept or reject. The terminal nodes would show the corresponding payoffs for each player based on the actions taken.

Advantages of Using Extensive Form

The extensive form representation offers several advantages over other game representation methods, particularly in capturing the dynamic and sequential nature of strategic interactions.

1. Capturing Sequentiality

Unlike the normal form, the extensive form explicitly represents the order in which players make their moves. This is essential for analyzing games where timing and sequence are critical factors.

2. Modeling Imperfect Information

The use of information sets allows for the representation of situations where players have incomplete or imperfect information about the game's history. This is crucial for modeling real-world scenarios where players often make decisions without full knowledge of past actions.

3. Analyzing Credibility of Threats

The extensive form is particularly useful for analyzing the credibility of threats and promises. By tracing the consequences of different actions through the game tree, one can assess whether a player would rationally carry out a threat or fulfill a promise.

4. Supporting Backward Induction

The extensive form facilitates the use of backward induction, a solution concept that involves reasoning backward from the end of the game to determine the optimal strategy at each decision node. This allows for the identification of subgame perfect equilibria, which are strategy profiles that are Nash equilibria in every subgame of the original game.

Limitations of Extensive Form

While the extensive form offers many advantages, it also has some limitations that should be considered.

1. Complexity

For complex games with many players, actions, and stages, the extensive form can become extremely large and unwieldy. The tree structure can quickly become difficult to manage and analyze.

2. Representation of Simultaneous Moves

The extensive form is primarily designed for sequential games. Representing simultaneous moves can be challenging and may require the introduction of artificial sequentiality or the use of mixed strategies.

3. Focus on Specific Details

The extensive form's emphasis on the detailed structure of the game can sometimes obscure the broader strategic issues. It may be difficult to see the forest for the trees, so to speak.

Applications of Extensive Form

The extensive form representation is widely used in various fields to analyze strategic interactions.

1. Economics

In economics, the extensive form is used to model bargaining games, auctions, and dynamic pricing strategies. It helps economists understand how firms compete, negotiate, and make decisions over time.

2. Political Science

In political science, the extensive form is used to analyze voting behavior, legislative processes, and international relations. It can model how politicians make strategic decisions, how interest groups exert influence, and how countries negotiate treaties.

3. Computer Science

In computer science, the extensive form is used in the design of artificial intelligence agents and algorithms for playing games. It provides a framework for representing the game's rules and the agent's decision-making process.

4. Biology

In biology, the extensive form is used to model evolutionary games, such as predator-prey interactions and cooperation among individuals. It helps biologists understand how different strategies evolve and persist over time.

Extensive Form vs. Normal Form

The extensive form and normal form are two fundamental ways of representing games in game theory. Each has its strengths and weaknesses, making them suitable for different types of analysis.

Feature	Extensive Form	Normal Form
Representation	Tree structure	Matrix
Sequentiality	Explicitly represented	Implicitly represented
Information	Modeled using information sets	Not explicitly modeled
Complexity	Can become complex for large games	Can become complex for many strategies
Solution Concepts	Backward induction, subgame perfect equilibrium	Nash equilibrium
Use Cases	Dynamic games, games with imperfect information	Static games, games with complete information

The normal form represents a game as a matrix, where each row represents a strategy for one player, each column represents a strategy for another player, and the cells contain the payoffs for each player given the corresponding strategy profile. The normal form is useful for analyzing games where players make decisions simultaneously, or where the order of moves is not important.

Backward Induction and Subgame Perfect Equilibrium

One of the most powerful solution concepts associated with the extensive form is backward induction. This involves starting at the end of the game tree and reasoning backward to determine the optimal strategy at each decision node.

Backward Induction: A process of reasoning backward from the end of the game to determine the optimal action at each decision node.
Subgame Perfect Equilibrium: A strategy profile that is a Nash equilibrium in every subgame of the original game.

A subgame is a part of the game that starts at a decision node and includes all subsequent nodes and branches. A subgame perfect equilibrium is a refinement of the Nash equilibrium concept that eliminates non-credible threats by requiring that strategies be optimal in every subgame.

Examples in Detail

To further illustrate the concept of extensive form representation, let's delve into more detailed examples.

1. The Centipede Game

The Centipede Game is a sequential game where two players alternately get a chance to take a larger share of a growing pot. The game starts with a small amount of money in the pot. Each player, in turn, can either take the larger share (and end the game) or pass the pot to the other player, which increases the total amount. The game continues for a fixed number of rounds, and if neither player takes the pot, they split it equally at the end.

Players: Player 1 and Player 2
Order of Moves: Players alternate turns.
Actions: Each player can either "Take" or "Pass."
Payoffs: Depend on when the game ends and who takes the pot.

The extensive form representation of the Centipede Game would show a series of decision nodes for each player, with branches representing the "Take" and "Pass" actions. The payoffs at the terminal nodes would reflect the division of the pot based on the sequence of actions. Backward induction predicts that both players will "Take" at every opportunity, leading to a suboptimal outcome where the game ends immediately.

2. Entry Deterrence Game

The Entry Deterrence Game models a situation where an incumbent firm faces the threat of entry by a potential competitor. The incumbent can choose to either accommodate the entry or fight it by lowering prices.

Players: Incumbent Firm and Entrant Firm
Order of Moves: Entrant decides whether to enter, then the Incumbent decides whether to fight or accommodate.
Actions: Entrant can "Enter" or "Stay Out." Incumbent can "Fight" or "Accommodate."
Payoffs: Depend on the entry decision and the incumbent's response.

The extensive form representation would show a decision node for the Entrant, with branches for "Enter" and "Stay Out." If the Entrant chooses "Enter," the game moves to a decision node for the Incumbent, who can choose to "Fight" or "Accommodate." The payoffs at the terminal nodes would reflect the profits for each firm based on the sequence of actions.

Conclusion

The extensive form representation of a game provides a powerful tool for analyzing strategic interactions in a dynamic and sequential context. By explicitly modeling the order of moves, information sets, and payoffs, the extensive form allows for a detailed understanding of the game's structure and the strategic considerations of the players involved. While it has limitations in terms of complexity and representation of simultaneous moves, the extensive form is invaluable for analyzing games where timing, information asymmetry, and credibility of threats are crucial factors. Its applications span various fields, including economics, political science, computer science, and biology, making it an essential concept for anyone studying game theory and strategic decision-making.