Game theory is a fascinating field of study, and one of the most important concepts in this area is the decision tree. Decision trees are graphical representations of the possible outcomes of a decision, and they’re used to model strategic interactions between multiple players. One of the most common symbols used in decision trees is the tree itself, which represents the possible paths that can be taken.

So, how do you draw a tree in game theory? Well, there are several steps involved, and we’ll go through them one by one.

Step 1: Define the Game

The first step in creating a decision tree is to define the game you want to model. This involves identifying the players involved in the game, their objectives, and the actions they can take.

For example, let’s say we want to model a game between two players: Alice and Bob. Alice wants to sell her car to Bob, and Bob wants to buy it at the lowest possible price. Alice can choose to set a high or low price for her car, and Bob can choose to accept or reject her offer.

Step 2: Identify Possible Outcomes

Once you’ve defined your game, you need to identify all possible outcomes. In our example game, there are four possible outcomes:

1) Alice sets a high price and Bob accepts

2) Alice sets a high price and Bob rejects

3) Alice sets a low price and Bob accepts

4) Alice sets a low price and Bob rejects

Step 3: Create Nodes for Each Decision Point

Next, create nodes for each decision point in your game. A node represents a point at which players must make a decision. In our example game, there are two decision points: when Alice decides on her asking price for the car, and when Bob decides whether or not to accept her offer.

Step 4: Draw Branches Between Nodes

With your nodes in place, draw branches between them to represent the possible outcomes of each decision. In our example game, we’ll draw two branches from Alice’s node (one for a high price and one for a low price), and two branches from Bob’s node (one for accepting and one for rejecting).

Step 5: Assign Payoffs

Finally, assign payoffs to each outcome. A payoff represents the benefit or loss that each player experiences as a result of an outcome. In our example game, we could assign payoffs as follows:

1) Alice sets a high price and Bob accepts: Alice gets a high price for her car, Bob gets the car at a higher price than he wanted. 2) Alice sets a high price and Bob rejects: Alice doesn’t sell her car, Bob doesn’t buy the car.

3) Alice sets a low price and Bob accepts: Alice gets a lower price for her car, Bob gets the car at a lower price than he wanted. 4) Alice sets a low price and Bob rejects: Alice doesn’t sell her car, Bob doesn’t buy the car.

With payoffs assigned, your decision tree is complete! It should look something like this:

## Decision Tree Example:

- Alice decides on asking price
- High Price
- Bob Accepts
- Alice gets high price for car, Bob pays more than he wanted
- Payoff: (3, -3)
- Bob Rejects
- Alice doesn’t sell her car, Bob doesn’t buy the car
- Payoff: (0, 0)
- Low Price
- Bob Accepts
- Alice gets lower price for car, Bob pays less than he wanted
- Payoff: (-1, 1)
- Bob Rejects
- Alice doesn’t sell her car, Bob doesn’t buy the car
- Payoff: (0, 0)

In conclusion, drawing a tree in game theory involves defining your game, identifying possible outcomes, creating nodes for each decision point, drawing branches between nodes to represent possible outcomes, and assigning payoffs. With these steps in mind, you can create a visual representation of your game that helps you analyze strategic interactions between players.