Question

What is a sequential game?

Short answer

A sequential game is a strategic game where players make their decisions one after another, implying that some players have more information than others because they make decisions later in the sequence. Sequential games often represented with a game tree, and a practical example is a game of chess.

Step-by-step solution

Definition of a sequential game

A sequential game is a type of game in strategic form games where players make their decisions one after another. This means the player who makes a decision later in the sequence has more information than those making decisions earlier in the sequence.

Comparison to simultaneous games

In contrast to simultaneous games where every player makes a decision without being aware of their opponents' choices, a player in a sequential game has an opportunity to respond to the actions of their opponent.

Representation of Sequential Games

Sequential games are often represented through a game tree (also known as an extensive form). Each vertex represents a player’s decision, and each edge is labeled by a decision made by a player at that point. The payoffs are displayed at the leaf nodes (the end of the game tree).

Example of Sequential Games

One common example of a sequential game is chess. Players take turns to move their pieces, and each player makes decisions based on the observation of the previous moves from another player.

Key concepts

Game Theory

Game theory is an area of study focused on mathematical models of conflict and cooperation between intelligent decision-makers. It is an important tool used to understand the strategies that people or organizations employ when interacting with one another. In game theory:

• A 'game' is any situation in which players make strategic decisions that have an impact on each other.
• Players are participants in the game who make decisions within the scenario.
• Strategies are the actions or decisions available to players.
• Payoffs are the outcomes of these strategies, typically represented as rewards or penalties.

By modeling complex interactions, game theory helps predict the likelihood of different outcomes based on the strategic choices of different players. It's used in various fields such as economics, political science, biology, and computer science. Understanding game theory is invaluable for anyone looking to grasp the dynamics of strategic situations.

Extensive Form

The extensive form is a way of representing games in game theory that highlights the sequential nature of decisions. Unlike the strategic (or normal) form, which might jointly list all possible strategies, the extensive form uses a tree structure to display how the game unfolds over time.

• Decision Nodes: Represent the points at which players must make decisions.
• Branches: Indicate the choices available at each decision node.
• Leaf Nodes: Show the final outcomes or payoffs of the game after all decisions are made.

In the extensive form, you can visualize the order of play, as well as observe how earlier actions influence later choices. For instance, in a sequential game like chess, each move and counter-move is mapped out in its structure. This representation helps players assess the potential consequences of their decisions more clearly.

Strategic Decision-Making

Strategic decision-making involves choosing among different strategies to achieve the best outcome. In a sequential game, strategic decision-making is crucial because each player has the opportunity to adapt their strategy based on the actions of others. This includes:

• Anticipating Opponent Moves: Players must think ahead and consider how opponents might react to their choices.
• Utilizing Information: In sequential games, players can use information from previous moves to adjust their strategies in real-time.
• Optimal Strategies: These are strategies designed to maximize a player’s payoff, taking into account the possible responses from other players.

Strategic decision-making doesn't just apply to games like chess, but also to real-world scenarios such as business negotiations and competitive sports. By understanding how choices impact outcomes, individuals and organizations can make more informed and effective decisions.