Game theory is a branch of mathematics that deals with the study of strategic decision-making. One of the fundamental concepts in game theory is information sets.
An information set is a collection of nodes in a game that share the same information. In this article, we will discuss how to count information sets in game theory.
What are Information Sets?
In game theory, an information set represents a player’s knowledge about the choices made by other players in the game. These choices are not observable to the player and hence are considered uncertain. An information set contains all possible actions that could have led to a particular node in the game.
For example, consider a simple two-player game where each player can choose to play either A or B. If both players choose A, then Player 1 receives a payoff of 4 and Player 2 receives a payoff of 3.
If both players choose B, then Player 1 receives a payoff of 2 and Player 2 receives a payoff of 1. If Player 1 chooses A and Player 2 chooses B, then Player 1 receives a payoff of 0 and Player 2 receives a payoff of 5. If Player 1 chooses B and Player 2 chooses A, then Player 1 receives a payoff of 5 and Player 2 receives a payoff of 0.
In this game, there are two information sets – one for each player. Each player knows his own choice but does not know the choice made by the other player until it is revealed later in the game.
How to Count Information Sets
Counting information sets can be challenging because it requires identifying all possible sequences of actions that lead to each node in the game. One way to do this is by constructing a tree diagram that represents all possible outcomes of the game.
Let’s take an example again – consider a two-player game where each player can choose to play either A or B. If Player 1 chooses B and Player 2 chooses A, then Player 1 receives a payoff of 5 and Player 2 receives a payoff of
0.
We can represent this game using the following tree diagram:
Tree Diagram
- A
- A (4,3)
- B (0,5)
- B
- A (5,0)
- B (2,1)
In this tree diagram, each node represents a possible outcome of the game. The first move is represented by the two branches at the top of the tree – one for each player’s choice. Each subsequent move is represented by additional branches that extend from each node to new nodes.
To count information sets in this game, we need to identify all possible sequences of actions that lead to each node in the tree. For example, the information set for the first player contains two nodes – one for when he/she plays A and one for when he/she plays B.
Counting Information Sets
- Player One:
- Information Set #1: A
- Node 1: (4,3)
- Node 2: (0,5)
- Information Set #2: B
- Node 3: (5,0)
- Node 4: (2,1)
- Player Two:
- Information Set #1: A
- Node 3: (5,0)
- Node 4: (2,1)
- Information Set #2: B
- Node 1: (4,3)
- Node 2: (0,5)
In this example, we have identified two information sets for each player. Each information set contains the nodes that share the same information. For example, Information Set #1 for Player One contains Nodes 1 and 2 since they represent the possible outcomes of Player One’s move when he/she plays A.
In Conclusion
Counting information sets in game theory is an essential concept that helps us understand the strategic decision-making of players in a game. By constructing a tree diagram and identifying all possible sequences of actions that lead to each node in the game, we can count information sets for each player. This enables us to analyze the players’ choices and their impact on the overall outcome of the game.