Explain the concept of extensive form games and their representation using game trees.

Extensive form games are a type of strategic interaction model in game theory that captures the sequential nature of decision-making. These games are represented using game trees, which visually depict the players' choices and the possible outcomes at each stage of the game.

In an extensive form game, players make decisions in a specific order, taking into account the actions of previous players. The game starts with a set of players, a set of possible actions for each player, and a set of possible outcomes. Each player has perfect information about the actions taken by previous players and the outcomes that have already occurred.

To represent an extensive form game using a game tree, we start with a root node that represents the initial decision point. From the root node, branches extend to represent the possible actions of the first player. Each branch represents a different action, and the number of branches corresponds to the number of possible actions for that player.

At the end of each branch, we reach a new decision point for the next player. This is represented by additional nodes connected to the previous nodes by branches. The process continues until all players have made their decisions and we reach the final outcome nodes.

The outcome nodes represent the possible outcomes of the game, which can be either terminal or non-terminal. Terminal nodes indicate the end of the game and the associated payoffs for each player. Non-terminal nodes represent intermediate stages of the game where players have not yet made all their decisions.

To determine the payoffs at the terminal nodes, we assign values to each outcome for each player. These values can represent utility, monetary rewards, or any other measure of success. The payoffs reflect the preferences of the players and can be represented as a vector of numbers.

By analyzing the game tree, players can determine their optimal strategies at each decision point, taking into account the actions of other players and the potential outcomes. This analysis involves backward induction, where players reason backward from the terminal nodes to determine the best actions at each stage of the game.

Overall, extensive form games and their representation using game trees allow us to analyze strategic interactions in a sequential manner, capturing the dynamics of decision-making and providing insights into optimal strategies and potential outcomes.