Components of a Formal Model in Game Theory

students, welcome to your first step into game theory 🎯. In this lesson, you will learn how to turn a real-world strategic situation into a formal model. That matters because game theory is not just about “who wins” or “who guesses right.” It is about understanding how people or groups make choices when their outcomes depend on each other.

By the end of this lesson, you should be able to:

Identify the players in a strategic situation.
List the feasible actions for each player.
Describe payoffs and information in a model.

Imagine two students deciding whether to study together or separately, two companies deciding whether to lower prices, or two drivers choosing routes to avoid traffic. In each case, the outcome depends on what others do. Game theory helps us describe that situation clearly and precisely. 🤝

What Is a Formal Model?

A formal model is a simplified description of a strategic situation using clear mathematical or logical parts. Instead of focusing on every tiny detail of the real world, we identify the essential ingredients that affect decisions and outcomes.

A game in its most basic form includes four core components:

Players
Actions
Payoffs
Information

These four pieces help us answer questions like: Who is making decisions? What can they do? What happens if they choose one action instead of another? And what do they know when they decide?

For example, suppose two stores are deciding whether to run a sale this weekend. The stores are the players. Each store can choose between actions such as “run a sale” or “do not run a sale.” The payoff for each store might be profit. Information tells us whether each store knows the other store’s choice before deciding. That one detail can completely change the situation.

A formal model is useful because it removes confusion. Real life has noise, emotions, and extra details. A model focuses only on the parts that matter for strategic behavior. Think of it like a map 🗺️: a map is not the whole world, but it helps you find your way.

Players: Who Is Making the Decisions?

The first step in building a formal model is identifying the players. Players are the decision-makers in the game. They can be individuals, groups, companies, countries, teams, or even computer programs.

students, a player is not always a person. For example:

In a business competition, the players may be two firms.
In a political setting, the players may be two candidates.
In a sports setting, the players may be two teams.
In a classroom setting, the players may be two students deciding how much effort to put into a project.

To identify players correctly, ask: Who has a choice that affects the outcome?

Suppose two restaurants are competing for customers. If each restaurant chooses its menu prices independently, then each restaurant is a player. If a landlord and a tenant are negotiating a lease, both are players because each has choices that affect the final agreement.

Sometimes the players are easy to see, but sometimes they are hidden inside a larger system. For instance, if a company makes a decision, the true player may be the company’s management team rather than the company name itself. In formal models, we treat the relevant decision-maker as the player.

A common mistake is including people who do not actually make decisions in the model. If someone is affected by the outcome but does not choose anything, they may matter for the story, but they are not a player unless the model gives them a decision.

Actions: What Can Each Player Do?

Once the players are identified, the next step is to list their feasible actions. A feasible action is a choice that a player is allowed to make within the rules of the situation.

The set of actions for a player is sometimes called that player’s strategy set or action set. In simple models, the actions may be easy to list. For example:

A driver can choose Route A or Route B.
A firm can choose to advertise or not advertise.
A student can choose to study hard or study lightly.
A company can choose a high price or a low price.

In a formal model, actions must be clearly defined. “Do better” is not an action because it is vague. “Choose price $p$” is more precise. “Try harder” is not enough unless the model explains exactly what that means.

Actions can be:

Discrete, meaning there are separate options, like $\{\text{High},\ \text{Low}\}$.
Continuous, meaning the choice can take many values, like any price in a range $[0,100]$.

For example, if two firms choose prices, each firm may have a feasible action set like $A_i = [0,100]$. That means firm $i$ can choose any price between $0$ and $100$. If two students choose whether to attend a study session, then each student’s action set may be $A_i = \{\text{Attend},\ \text{Skip}\}$.

The actions must also be feasible in the real-world setting. A student cannot choose an impossible option like “get a perfect score without taking the test” if that is not actually available in the model. A good model only includes choices that the player can realistically make.

Payoffs: What Do Players Care About?

Payoffs describe how much a player values each outcome. They represent the results of the game from each player’s point of view. Payoffs do not have to mean money only. They can represent profit, satisfaction, grades, time saved, risk avoided, or any other measure of value.

For example:

A business may care about profit.
A student may care about grades and free time.
A country may care about security, trade, or influence.
A sports team may care about winning.

In formal models, payoffs are often written as numbers. Higher numbers mean better outcomes for the player. If player $i$ gets payoff $u_i$, then the model can compare different outcomes by comparing values of $u_i$.

Here is a simple example. Suppose two students choose whether to study together. Their payoffs might depend on whether each studies and how much benefit they get from cooperation. One possible payoff table could show that both do best when they both study, because they learn more and stay accountable. Another possibility is that one student benefits more than the other. The exact numbers depend on the situation, but the key idea is that payoffs rank outcomes.

Payoffs are important because players usually choose actions that they believe improve their payoffs. Game theory studies how each player’s best choice depends on what the others do.

A payoff can be:

Positive or negative
The same across players or different across players
Based on one goal or several combined goals

For example, if a student chooses to study an extra hour, the payoff may increase because the grade is higher, but it may also decrease because leisure time is lower. A model may combine those effects into a single payoff number.

Information: What Does Each Player Know?

Information is the last key ingredient of a formal model. It describes what each player knows when making a decision.

This matters because the same action can mean different things depending on what a player knows. If you know your classmate will not study, you may study more. If you do not know their choice, you may act differently. In game theory, knowledge can change behavior a lot. 🧠

Information can include:

What actions are available
What the other players have already chosen
What the payoffs are
What random events have occurred

In some games, everyone knows everything relevant. That is called complete information in many basic models. In other games, players do not know important facts, such as the other player’s costs, type, or previous move. That is incomplete or imperfect information, depending on the setting.

For example, imagine two companies deciding whether to invest in a new product. If each company knows the other company’s budget and plan, information is more complete. If neither company knows the other’s budget, then each must make decisions under uncertainty.

Information can also be about timing. If one player moves first and the other moves after observing that move, the second player has more information at the time of decision. If both players choose at the same time, neither sees the other’s action before deciding.

This difference is crucial. A game where both players choose simultaneously is not the same as a game where one player acts after observing the other. The same players and same actions can produce very different outcomes depending on information.

Putting the Pieces Together

To build a formal model, students, you ask four questions in order:

Who are the players?
What can each player do?
What are the payoffs for each outcome?
What does each player know when deciding?

Let’s use a simple real-world example: two roommates deciding whether to clean their apartment.

Players: Roommate 1 and Roommate 2.
Actions: Each can choose $\{\text{Clean},\ \text{Not Clean}\}$.
Payoffs: Both prefer a clean apartment, but each dislikes doing all the work alone.
Information: They may choose at the same time, or one may see the other’s action first.

This small example already shows why the four components matter. If both know the other will clean, each may be tempted to avoid the work. If one acts first, the timing changes. If the cost of cleaning is high, the payoff changes. The game is not just about the choices; it is about the structure around the choices.

When these components are written clearly, the model becomes easier to analyze. Later in the course, you will use these ingredients to study equilibrium, incentives, cooperation, conflict, and bargaining.

Conclusion

A formal model is the foundation of game theory. It organizes a strategic situation into four essential parts: players, actions, payoffs, and information. Players are the decision-makers. Actions are the feasible choices they can make. Payoffs show what each player values. Information tells us what each player knows at the moment of choice.

students, once you can identify these four components, you can begin to translate everyday situations into the language of game theory. That is the first major skill in this course. From here, you will be ready to analyze how people choose when their decisions affect one another.

Study Notes

Players are the decision-makers in a strategic situation.
Players can be individuals, firms, teams, governments, or other decision-making units.
Actions are the feasible choices available to each player.
Action sets must be clearly defined and realistic.
Payoffs measure how each player ranks outcomes.
Payoffs can represent money, grades, time, satisfaction, or other values.
Information describes what each player knows when choosing.
Timing and knowledge can change strategic behavior.
A formal model usually includes players, actions, payoffs, and information.
The purpose of a model is to simplify a real situation while keeping the important strategic features.