Game theory is a branch of mathematics that deals with the study of decision-making, conflict resolution, and strategy in social situations. In game theory, subgames are a crucial concept that helps to analyze complex games and gain insights into the strategies that players can adopt.

**What Are Subgames?**

A subgame is a smaller game within a larger game. In other words, it is a part of a larger game that is treated as an independent game.

Subgames arise when players make decisions that lead to new decision points or nodes in the game tree. These decision points create new games within the larger game, which are known as subgames.

__Example__

Consider the popular children’s game of tic-tac-toe. Tic-tac-toe has only one possible outcome: either player one wins, player two wins, or it’s a tie.

However, we can create subgames within tic-tac-toe by limiting the number of moves each player can make. For example, we could limit each player to only two moves before passing the turn to their opponent. This creates a new subgame where each player has two turns to try and win.

**How Are Subgames Analyzed?**

Subgames are analyzed using backward induction, which is a common technique in game theory. Backward induction involves working backwards from the final node of the game tree to determine the optimal strategy for each player at every node.

Let’s consider an example where there are two players in a game and four possible outcomes: A1, A2, B1, and B2. Each outcome has its own payoffs for both players:

– If both players choose A1: Player 1 gets 2 points and Player 2 gets 3 points. – If both players choose A2: Player 1 gets 1 point and Player 2 gets 1 point.

– If both players choose B1: Player 1 gets 3 points and Player 2 gets 2 points. – If both players choose B2: Player 1 gets 0 points and Player 2 gets 0 points.

This game has two decision points, which creates four possible subgames. We can analyze each subgame using backward induction to determine the optimal strategy for each player at every node. We start by analyzing the final node of the game tree (i.e., the outcome node) and work our way backwards to the root node (i., the initial decision point).

**Conclusion**

Subgames are an essential concept in game theory that helps to analyze complex games by breaking them down into smaller, independent games. By analyzing subgames using backward induction, we can gain insights into the strategies that players can adopt at every decision point in a game.

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