Game Theory: Understanding Strategic Decision-Making and Competition
Have you ever wondered why you see McDonald’s and Burger King next to each other on the same street? Or why gas stations cluster around the same intersection? Or why Netflix and Disney+ offer similar content and prices? These are all examples of strategic decisions made by competing players in a social situation. And the science that studies these decisions is called game theory.
What is game theory?
Game theory is a branch of applied mathematics that provides tools for analyzing situations in which parties, called players, make decisions that are interdependent. This means that each player’s decision affects the outcome for themselves and for others. Game theory helps us understand how rational players would behave in such situations, and what kind of outcomes they would expect.
Game theory was originally developed by John von Neumann and Oskar Morgenstern in the 1940s to solve problems in economics. They argued that traditional mathematics, which describes the workings of nature, was not suitable for economics, which involves human interactions and choices. They proposed a new kind of mathematics, which they called game theory, to model the strategic aspects of decision making.
Game theory was further developed in the 1950s by John Nash, who established the mathematical principles of game theory and showed how it can be applied to various fields, such as politics, sociology, biology, and computer science. Game theory has been used to study topics such as bargaining, voting, auctions, war, cooperation, evolution, and artificial intelligence.
How does game theory work?
Game theory uses mathematical models to represent the players, their actions, their preferences, and the outcomes of their interactions. A game can be described by four elements:
The set of players: who are the decision makers involved in the game?
The set of actions: what are the possible choices that each player can make?
The set of payoffs: what are the consequences or rewards that each player receives from each combination of actions?
The set of information: what do the players know or not know about each other’s actions and payoffs?
A game can be classified into different types based on these elements. For example:
A game can be cooperative or non-cooperative, depending on whether the players can communicate and form binding agreements or not.
A game can be simultaneous or sequential, depending on whether the players make their decisions at the same time or one after another.
A game can be zero-sum or non-zero-sum, depending on whether the total payoff for all players is constant or variable.
A solution to a game is a strategy for each player that specifies what action they should take in every possible situation. A solution is optimal if it maximizes the expected payoff for each player, given what they know and expect about the other players’ strategies. A solution is also called an equilibrium if no player has an incentive to deviate from their strategy, given that the other players stick to theirs.
One of the most famous concepts in game theory is the Nash equilibrium, named after John Nash. A Nash equilibrium is a solution in which every player is playing their best response to the other players’ strategies. In other words, no player can improve their payoff by changing their strategy unilaterally. A Nash equilibrium may not exist, may not be unique, or may not be efficient.
Why do companies open stores next to their competition?
One of the applications of game theory is to explain why companies open stores next to their competition. This phenomenon is known as Hotelling’s law, named after Harold Hotelling, who proposed a simple model of spatial competition in 1929.
Imagine two ice cream vendors who sell identical products on a beach. The beach is one mile long and has customers uniformly distributed along it. The customers will buy ice cream from the closest vendor to minimize their walking distance. The vendors can choose where to locate their carts along the beach.
What would be the optimal location for each vendor? You might think that they would want to spread out evenly along the beach to capture half of the market each. However, this is not a Nash equilibrium. Suppose vendor A locates at 1/4 mile from one end of the beach and vendor B locates at 3/4 mile from the same end. Then vendor A can increase their market share by moving slightly closer to vendor B, say at 1/3 mile. This way, vendor A can attract more customers from vendor B’s side without losing any customers from their own side. Vendor B will then have an incentive to move closer to vendor A as well, say at 2/3 mile. This process will continue until both vendors end up at the middle of the beach.
This is a Nash equilibrium because neither vendor can improve their payoff by moving away from the middle. However, this equilibrium is not efficient because it reduces the total welfare of the customers, who have to walk longer distances to buy ice cream. This is an example of a prisoner’s dilemma, a famous game in which two players have a dominant strategy to defect, but both would be better off if they cooperated.
Hotelling’s model can be extended to more than two players and more than one dimension. For example, suppose there are three coffee shops in a city, and they can choose their location and their price. The customers will buy coffee from the closest and cheapest shop. The shops will compete on both location and price to maximize their profit. The outcome of this game will depend on the demand, the cost, and the distance functions of the customers and the shops.
Game theory can help us understand how companies make strategic decisions in different markets and environments. It can also help us design better policies and regulations to improve social welfare and efficiency. Game theory is not only a science of strategy, but also a science of creativity.
