Extensive form game: Meaning, Criticisms & Real-World Uses

What Is Extensive Form Game?

In game theory, an extensive form game is a detailed representation of a sequential game, illustrating the sequence of players' possible moves, their choices at each decision-making point, the information available to them, and the payoffs for all possible outcomes. This type of game, a core concept within game theory, is typically visualized as a game tree, where nodes represent decision points and branches represent possible actions. It provides a comprehensive framework for analyzing strategic interactions where the timing and order of moves significantly influence the game's progression and results.

History and Origin

Modern game theory, which forms the bedrock for understanding extensive form games, gained significant prominence with the publication of Theory of Games and Economic Behavior in 1944 by mathematician John von Neumann and economist Oskar Morgenstern. This seminal work laid the groundwork for analyzing strategic interactions.¹⁰ The formal definition and representation of extensive form games as we largely know them today were further developed and extended by Harold W. Kuhn in 1953, building upon von Neumann's earlier work from 1928. This evolution was driven by the need to formalize sequential decision-making, especially in scenarios where timing and information played critical roles, transforming game theory into an indispensable analytical tool across various disciplines.

Key Takeaways

An extensive form game visually represents sequential decision-making through a tree diagram.
It explicitly details the order of moves, player choices, available information, and final outcomes (payoffs).
This representation is crucial for analyzing strategic interactions in dynamic environments.
The primary solution concept for extensive form games with perfect information is the subgame perfect equilibrium, often found using backward induction.
Extensive form games provide insights into how strategies evolve over time based on observed actions.

Interpreting the Extensive Form Game

An extensive form game is interpreted by tracing the possible paths of play from the initial node to the various terminal nodes, each representing a final outcome with associated payoffs for all players. The structure allows for the analysis of conditional strategies, where a player's optimal choice at any given point depends on the preceding actions of other players.

The primary analytical method for solving extensive form games, particularly those with perfect information, is backward induction. This process involves starting from the end of the game (the terminal nodes) and working backward to determine the optimal strategy at each decision point. By anticipating the rational choices of subsequent players, each player can make their own optimal decision at every stage. This method helps to identify a Nash equilibrium that is also a subgame perfect equilibrium, meaning that the chosen strategies remain optimal in every subgame, not just the game as a whole.

Hypothetical Example

Consider a simple entry game between an incumbent firm (Player 1) and a potential new entrant (Player 2).

Player 1 (Entrant) decides whether to enter the market (E) or stay out (O).
If Player 1 chooses to stay out (O), the game ends. Player 1 gets a payoff of (0), and Player 2 gets a payoff of (10) (representing the incumbent's continued monopoly profit).
If Player 1 chooses to enter (E), Player 2 (Incumbent) then observes this move and decides whether to fight (F) or accommodate (A) the entry.
If Player 2 chooses to fight (F), both players incur losses. For instance, Player 1 gets (-5), Player 2 gets (-5).
If Player 2 chooses to accommodate (A), both firms share the market. Player 1 gets (3), Player 2 gets (3).

To analyze this extensive form game using backward induction:

Step 1: Look at the last decision. If Player 1 enters (E), Player 2 faces a choice between Fight (F) with payoff (-5) and Accommodate (A) with payoff (3). A rational Player 2 will choose to accommodate, as 3 > -5.
Step 2: Consider Player 1's initial decision. Knowing that Player 2 will accommodate if Player 1 enters, Player 1 compares the payoff of staying out (0) with the payoff of entering and being accommodated (3). A rational Player 1 will choose to enter, as 3 > 0.

Therefore, the predicted outcome of this extensive form game is that Player 1 enters, and Player 2 accommodates, resulting in payoffs of (3, 3) for (Player 1, Player 2). The structure of this decision-making process is visually represented by a game tree with specific payoffs at each terminal node.

Practical Applications

Extensive form games have broad practical applications across various financial and economic domains due to their ability to model sequential decision-making. In corporate finance, they are used to analyze strategic interactions such as capital structure decisions, dividend policies, and mergers and acquisitions.⁹,⁸ For instance, a firm's decision to issue new debt or equity can be modeled as a move in an extensive form game, with subsequent moves by investors or competitors.

In investment strategies, extensive form games can help anticipate market participants' actions, such as how competitors might react to a new product launch or a pricing change.⁷ They are also applied in understanding situations involving information asymmetry, like signaling in financial markets or the behavior of informed traders.⁶,⁵ For example, the strategic moves in a tender offer or a negotiation for a private equity deal can be mapped out and analyzed using this framework. The framework provides insights for optimizing returns and managing risk management effectively.⁴

Limitations and Criticisms

While extensive form games provide a powerful analytical framework, they come with certain limitations and criticisms. A key assumption underpinning their analysis, particularly through backward induction, is that of perfect rationality among all players. This assumes that players will always make choices that maximize their own expected payoffs and that this rationality is common knowledge among all participants.³, In reality, human behavior can deviate from perfect rationality due to cognitive biases, emotions, or incomplete information, leading to outcomes that differ from game-theoretic predictions.²,

Another limitation arises in games with incomplete information, where players might not know all aspects of the game, such as other players' true payoffs or types. While mechanisms like the Harsanyi transformation can convert such games into extensive form games with imperfect information, they add complexity. Additionally, as the number of decision nodes or players increases, the game tree can become exceedingly large and complex, making analysis computationally intensive and challenging to visualize or solve in practice. The procedure is best suited for games with perfect information.

Extensive Form Game vs. Normal Form Game

The extensive form game and the normal form game are two fundamental ways to represent a game in game theory, each offering a different perspective on strategic interactions.

An extensive form game explicitly shows the sequence of moves, the information available to players at each decision point, and the full history of the game. It is typically represented as a game tree, which visually maps out every possible path the game can take from start to finish. This representation is particularly useful for sequential games where the timing and order of decisions are crucial, as it allows for the analysis of conditional strategies and the application of methods like backward induction to find a subgame perfect equilibrium.,

In contr¹ast, a normal form game (also known as strategic form) condenses the game into a matrix or table that lists all possible pure strategies for each player and the resulting payoffs for every combination of strategies. It does not explicitly show the sequence of moves or the flow of information during the game. Instead, it focuses on simultaneous decisions or a summary of potential outcomes given chosen strategies, making it more suitable for analyzing simultaneous-move games or for identifying Nash equilibrium without regard to the specific timing of actions.

FAQs

What is the main difference between an extensive form game and a strategic form game?

The main difference lies in their representation and emphasis: an extensive form game uses a tree to show the sequence of moves and information, ideal for sequential games, while a strategic (or normal) form game uses a payoff matrix to show all possible strategies and outcomes, usually for simultaneous moves.

How is an extensive form game solved?

Extensive form games are typically solved using backward induction. This method involves starting from the end of the game and working backward, determining the optimal choice for the last player, then the second-to-last, and so on, until the first player's optimal strategy is identified. This process helps find a subgame perfect equilibrium.

What is a game tree in the context of extensive form games?

A game tree is the graphical representation of an extensive form game. It consists of nodes that represent decision points for players (or chance events), and branches that represent the actions taken from those decision points, leading to other nodes or final terminal nodes with associated payoffs.

Can extensive form games have imperfect information?

Yes, extensive form games can model imperfect information, meaning that players may not know all previous moves or the exact state of the game when making a decision. This is typically represented using "information sets," which group together decision nodes that a player cannot distinguish between. Such games are more complex to analyze than those with perfect information.