Sequential Games

Sequential Games are a type of game in game theory where players make decisions one after another, rather than simultaneously. Each player, when making a decision, is fully aware of the actions taken by the players who moved before them. This sequential nature allows players to observe earlier moves and adjust their strategies accordingly. Sequential Games are often represented using extensive form, typically with a game tree that visually displays the order of moves, possible actions at each decision point, and the resulting payoffs.

Extensive Form Representation

Game Trees

In Sequential Games, the extensive form is a crucial tool for representation. A game tree consists of nodes, branches, and terminal nodes:

• Nodes represent decision points for players.
• Branches represent the possible actions a player can take at each decision node.
• Terminal nodes indicate the end of the game, with associated payoffs for each player.

The tree structure explicitly shows the order of moves, the players making decisions, and the available actions at every stage. This clarity allows analysis of strategies that depend on the history of moves.

Information Sets

In some sequential games, players may not have perfect information about earlier moves, leading to the concept of information sets. An information set groups decision nodes that a player cannot distinguish between at the time of their move. In games with perfect information, each information set contains exactly one node, but in games with imperfect information, multiple nodes may belong to the same set, reflecting uncertainty.

Solution Concepts in Sequential Games

Backward Induction

Backward Induction is a method used to solve finite sequential games with perfect information. It involves analyzing the game tree from the end (terminal nodes) back to the beginning:

1. At the terminal nodes, the payoffs are known.
2. At each decision node, the player selects the action that maximizes their payoff, assuming rational play in subsequent moves.
3. This process continues backward until the initial node, determining the optimal strategy for every player at every decision point.

Backward Induction identifies which strategies are credible and optimal, considering the entire sequence of future actions.

Subgame Perfect Equilibrium (SPE)

Subgame Perfect Equilibrium refines the Nash equilibrium concept for sequential games. An SPE is a strategy profile that constitutes a Nash equilibrium in every subgame of the original game. A subgame is any part of the game that can be considered a game on its own, starting from a particular node and including all its successors.

SPE eliminates non-credible threats or promises because strategies must be optimal not only for the whole game but also for every subgame. The backward induction outcome is an example of an SPE in games with perfect information.

Strategic Interaction and Commitment

Impact of Sequential Moves

The order of moves strongly influences strategic behavior. The first mover can shape the game by committing to a strategy that influences the second player's response. This commitment can create strategic advantages or disadvantages depending on the context.

Credible Threats and Promises

In Sequential Games, players may attempt to influence opponents' actions by making threats or promises. However, only those that are credible—meaning they are optimal responses in the subgames where they are relevant—affect equilibrium strategies. Non-credible threats are disregarded in the analysis.

Applications of Sequential Games

Sequential Games are widely used in economics and managerial decision-making where timing and order of moves matter. Examples include:

• Investment decisions: Firms deciding whether to enter a market knowing competitors’ prior actions.
• Pricing strategies: Firms setting prices sequentially to capture market share.
• Bargaining: Offers and counteroffers exchanged in a sequence.
• Negotiations: When parties alternately propose terms.

This framework allows modeling real-world strategic interactions involving timing, commitment, and observation.

Summary of Key Features

Sequential Games provide a rich analytical framework to study strategic interactions where the timing and order of decisions are crucial, enabling the identification of optimal and credible strategies throughout the game.