How Do Functions Help in Game Theory in Economics?

Game theory is a way to understand how people make choices when they are trying to win or do better than others. In economics, functions play a key role in looking at these choices. They help us see how different actions impact others.

1. Utility Functions

One important way we use functions in game theory is through something called utility functions.

A utility function gives a number to show how much someone likes a certain outcome. For example, if we have a utility function written as $U(x)$, where $x$ is the amount of something someone uses, different amounts change how much happiness or satisfaction they feel.

• Example: If $U(x) = x^2$, then if someone consumes 3 units, they get a satisfaction of $U(3) = 3^2 = 9$. But if they consume 4 units, their satisfaction goes up to $U(4) = 4^2 = 16$.

2. Payoff Functions

In games, payoff functions show what rewards players get based on their choices and the choices made by their opponents.

For example, in a game with two players, the payoff could be written as $P(a, b)$. Here, $a$ is the choice of Player 1, and $b$ is the choice of Player 2.

• Example Use: In a competition where two businesses set prices, their payoff could be how much money they earn based on their prices. If Player 1 sets a price of $p_1$ and Player 2 sets a price of $p_2$, their profits can be represented with $P(p_1, p_2) = (D(p_1, p_2) - C) \cdot p_1$, where $D$ is how many people want to buy and $C$ is the cost of making the product.

3. Best Response Functions

Best response functions show how players change their strategies based on what others do. If we call the best response for Player 1 to Player 2’s choice $b$ as $BR_1(b)$, this helps us see what Player 1 should do to get the best outcome, depending on what Player 2 does.

• Interesting Fact: Research has found that a situation called Nash Equilibrium, where everyone’s best responses match up, happens in about 75% of the games tested in labs.

4. Linear and Non-linear Functions in Game Theory

Functions in game theory can be either linear or non-linear.

Linear functions show a straight-line pattern, meaning they give equal results each time, while non-linear functions can show more complex patterns like decreasing returns or changing benefits. Knowing these types helps economists create better models to explain how businesses compete.

• Example in Use: The Cournot competition model often uses linear functions to suggest that companies will change how much they produce to make the most profit.

In short, functions are really important in game theory. They help us understand preferences, rewards, and strategies in economics, which leads to better predictions about how competition works.