When most people hear the word “game,” they think of fun activities like chess, poker, or video games. But in mathematics, a “game” has a much broader and more precise meaning. Game theory studies strategic situations where the outcome depends not just on your choices, but on the choices of others too.

What Makes Something a “Game”?

In game theory, a game is any situation where:

  1. Multiple decision-makers (called “players”) interact
  2. Each player has choices (called “strategies”) they can make
  3. Each combination of choices leads to an outcome with specific rewards or costs (called “payoffs”)
  4. Players care about these outcomes and try to make rational decisions

The beauty of this definition is that it applies to far more than board games. Business competition, political negotiations, evolutionary biology, and even social dilemmas are all “games” in the mathematical sense.

The Three Essential Components

Every game has three fundamental elements:

%%{init: {'theme':'dark', 'themeVariables': {'primaryTextColor':'#fff','secondaryTextColor':'#fff','tertiaryTextColor':'#fff','textColor':'#fff','nodeTextColor':'#fff'}}}%% graph TD A[Mathematical Game] --> B[Players] A --> C[Strategies] A --> D[Payoffs] B --> B1["Who makes decisions?"] C --> C1["What choices are available?"] D --> D1["What are the outcomes?"] style A fill:#4c6ef5 style B fill:#51cf66 style C fill:#ff6b6b style D fill:#ffd43b

1. Players

Players are the decision-makers in the game. They can be:

  • Individuals (you vs. your friend in chess)
  • Companies (Apple vs. Samsung in smartphone pricing)
  • Countries (trade negotiations)
  • Even genes (in evolutionary game theory)

The key is that each player has preferences and tries to achieve the best outcome for themselves.

2. Strategies

A strategy is a complete plan of action—everything a player might do in every possible situation. For example:

  • In rock-paper-scissors: your strategy could be “always play rock” or “randomly choose”
  • In business: “price high” vs. “price low”
  • In negotiations: “cooperate” vs. “defect”

3. Payoffs

Payoffs represent how much each player values each possible outcome. They’re usually shown as numbers (higher is better), though they don’t always represent money—they could represent utility, satisfaction, or evolutionary fitness.

A Simple Example: The Coffee Shop Game

Let’s see how these components work together in a real-world scenario.

Scenario: Two coffee shops on the same street must each decide whether to stay open late (until 10 PM) or close early (at 6 PM).

%%{init: {'theme':'dark', 'themeVariables': {'primaryTextColor':'#fff','secondaryTextColor':'#fff','tertiaryTextColor':'#fff','textColor':'#fff','nodeTextColor':'#fff'}}}%% graph LR A[Coffee Shop A] --> A1{Stay Open Late?} A1 -->|Yes| A2[Open until 10 PM] A1 -->|No| A3[Close at 6 PM] B[Coffee Shop B] --> B1{Stay Open Late?} B1 -->|Yes| B2[Open until 10 PM] B1 -->|No| B3[Close at 6 PM] style A fill:#4c6ef5 style B fill:#51cf66

Let’s break this down:

Players: Coffee Shop A and Coffee Shop B

Strategies:

  • Each shop has two strategies: “Stay Open Late” or “Close Early”

Payoffs:

  • If both stay open late: They split the late-night customers but pay extra staff costs (moderate profit)
  • If one stays open and one closes: The open shop captures all late customers (high profit), closed shop saves on costs (low profit)
  • If both close early: Both save on costs but miss potential customers (moderate profit)

Types of Games

Games come in many varieties. Here are the main classifications:

%%{init: {'theme':'dark', 'themeVariables': {'primaryTextColor':'#fff','secondaryTextColor':'#fff','tertiaryTextColor':'#fff','textColor':'#fff','nodeTextColor':'#fff'}}}%% graph TD A[Types of Games] --> B[Number of Players] A --> C[Sum of Payoffs] A --> D[Timing] A --> E[Information] B --> B1[Two-player] B --> B2[Multi-player] C --> C1["Zero-sum
(My gain = Your loss)"] C --> C2["Non-zero-sum
(We can both win)"] D --> D1["Simultaneous
(Choose at same time)"] D --> D2["Sequential
(Take turns)"] E --> E1["Perfect information
(Everyone knows everything)"] E --> E2["Imperfect information
(Hidden information)"] style A fill:#4c6ef5 style B1 fill:#51cf66 style B2 fill:#51cf66 style C1 fill:#ff6b6b style C2 fill:#51cf66 style D1 fill:#ffd43b style D2 fill:#ffd43b style E1 fill:#ae3ec9 style E2 fill:#ae3ec9

Classification Examples:

Zero-sum games: Poker (your winnings = others’ losses), chess, most sports

Non-zero-sum games: Business negotiations (both sides can benefit), climate agreements, the Prisoner’s Dilemma

Simultaneous games: Rock-paper-scissors, sealed-bid auctions, most business competition

Sequential games: Chess, bargaining, chess, tic-tac-toe

Why This Matters

Understanding what a “game” is in mathematics gives us a powerful framework for analyzing real-world situations:

In Business: Companies use game theory to decide on pricing, when to enter markets, and how to compete with rivals.

In Economics: Economists model auctions, market competition, and trade negotiations as games.

In Computer Science: Algorithm design, network routing, and AI decision-making all use game theory.

In Biology: Evolution itself can be modeled as a game where different strategies (genes) compete for survival.

In Daily Life: Negotiating salaries, deciding whether to cooperate or compete, even choosing routes in traffic—all are games.

The Power of Mathematical Thinking

By formalizing these situations as games, we can:

  • Predict what rational players will do
  • Find optimal strategies
  • Understand why certain outcomes occur (even when they seem suboptimal)
  • Design better systems and incentives

The coffee shop example might seem simple, but the same framework applies to global supply chains, nuclear deterrence, and climate negotiations. That’s the power of mathematical abstraction—one framework, infinite applications.

What’s Next?

Now that you understand what a game is, the next question is: how do we actually analyze them? In the next post, we’ll explore payoff matrices—a simple visual tool that lets you see the entire structure of a two-player game at a glance.

Once you can visualize a game, finding optimal strategies becomes much clearer. And as you’ll discover, some games have surprising properties that challenge our intuitions about rational behavior.

This is the first post in our Game Theory Series. Understanding these foundations will help you see strategic situations everywhere and make better decisions in competitive environments.