Game theory is a fascinating branch of mathematics that helps us understand strategic decision-making. One of the most fundamental concepts in game theory is equilibrium, which is a central idea that describes how players in a game can maximize their utility.

What is Equilibrium in Game Theory?

In game theory, equilibrium refers to a state of balance in which no player has an incentive to change their behavior. This means that each player’s strategy is optimal given the strategies of all the other players. In other words, no player can improve their position by changing their strategy if all the other players also keep their strategies unchanged.

There are two main types of equilibria in game theory: Nash equilibrium and Pareto equilibrium. Nash equilibrium, named after Nobel laureate John Nash, occurs when each player’s strategy is the best response to the strategies of all the other players. Pareto equilibrium, on the other hand, occurs when there is no alternative set of strategies that could make at least one player better off without making any other player worse off.

Nash Equilibrium

To understand Nash equilibrium better, let’s consider an example. Imagine two companies are competing for market share by setting prices for their products.

If both companies set high prices, they will both earn low profits because customers will switch to cheaper alternatives. If both companies set low prices, they will both earn high profits because they will capture a larger market share. However, if one company sets a low price and the other sets a high price, the former will earn higher profits than the latter because it will attract more customers.

The table below shows the payoff matrix for this scenario:

• Company A
• Low Price
• High Price
• Company B
• Low Price
• 5,5
• 0,10
• High Price
• 10,0
• 1,1

In this matrix, the first number represents Company A’s profit, and the second number represents Company B’s profit. For example, if both companies set low prices (the top-left cell), they will both earn a profit of 5.

To find Nash equilibrium in this scenario, we need to identify each company’s best response to the other company’s strategy. If Company A sets a low price, Company B’s best response is also to set a low price because it will earn a higher profit than setting a high price.

Similarly, if Company A sets a high price, Company B’s best response is again to set a low price because it will earn a higher profit than setting a high price. Therefore, the Nash equilibrium in this scenario is for both companies to set low prices and earn profits of 5 each.

Pareto Equilibrium

Pareto equilibrium is a more stringent condition than Nash equilibrium because it requires that no player can be made better off without making someone else worse off. To understand Pareto equilibrium better, let’s consider another example.

Imagine two people are deciding how to divide \$10 between them. They can either split it equally (giving each person \$5) or one person can take all \$10 while the other gets nothing. The table below shows the payoff matrix for this scenario:

Person B

Split Equally
B gets nothing
• Person A
• Split Equally
• 5,5
• 10,0

In this matrix, the first number represents Person A’s payoff, and the second number represents Person B’s payoff. For example, if they split the money equally (the top-left cell), they will both receive a payoff of 5.

To find Pareto equilibrium in this scenario, we need to identify a strategy that makes both players better off without making anyone worse off. In this case, splitting the money equally is the only Pareto equilibrium because any other strategy would make at least one person worse off.

Conclusion

Equilibrium is a key concept in game theory that helps us understand how strategic decision-making can lead to balance and optimal outcomes. Nash equilibrium and Pareto equilibrium are two common types of equilibria that are used to analyze different types of games. By understanding these concepts, we can gain insights into many real-world situations ranging from business competition to international politics.