Game Theory

A mathematical theory developed by John von Neumann, John Nash, and others, studying strategic interactions among rational decision-makers. Nash Equilibrium is the core concept of game theory.

๐Ÿ“– Formal Introduction

Game Theory is a mathematical theory that studies how rational decision-makers make optimal decisions in strategic interaction situations. Founded by John von Neumann and Oskar Morgenstern in 1944, John Nash made groundbreaking contributions in the 1950s.

Core Concepts:

  • Nash Equilibrium: Each player's strategy is the best response to others' strategies. No player can benefit by unilaterally changing their strategy.
  • Dominant Strategy: A strategy that always yields the best outcome regardless of the opponent's choice.
  • Pareto Optimality: It is impossible to improve one party's situation without harming another.
  • Zero-sum vs Non-zero-sum Games: In zero-sum games, one's gain equals another's loss; in non-zero-sum games, both can win or lose.
  • Complete vs Incomplete Information Games: The degree to which players know the game structure and payoffs.

Game theory is widely applied in economics (market competition, auction design), political science (voting, international relations), biology (evolutionary games), computer science (algorithm design, AI), and other fields. It provides a rigorous analytical framework for understanding competition, cooperation, negotiation, conflict, and other phenomena.

๐Ÿ’ฌ Plain Language Introduction

Game theory studies the problem of "you know that I know that you know" โ€” when your decisions affect others and others' decisions affect you, how do you make the smartest choice?

๐Ÿ” Classic Example: Prisoner's Dilemma

Two criminals are interrogated separately:
โ€ข Both silent โ†’ 1 year each (insufficient evidence)
โ€ข One confesses, one silent โ†’ confessor goes free, silent gets 10 years
โ€ข Both confess โ†’ 5 years each

Rational Analysis: No matter what the other does, confessing is always better (if silent, I go free; if they confess, I get 5 years instead of 10). The result: both confess and get 5 years each โ€” even though both staying silent would give only 1 year each! This is individual rationality leading to collective irrationality.

๐ŸŒŸ Game Theory in Real Life

  • Price Wars: Two supermarkets want to cut prices to attract customers, but if both cut prices, profits decrease โ€” better to keep prices stable
  • Common Resources: Everyone wants to use more common resources (e.g., overfishing), but if everyone does this, resources deplete and everyone loses
  • Exam Cheating: If no one cheats, it's fair competition; but if others cheat and you don't, you're at a disadvantage
  • Traffic Congestion: Everyone takes the fastest route, but then everyone ends up on the same road, making it slower

Key Insight: In interactive decision-making, you can't just consider your own best choice โ€” you must predict what others will think and do, then respond optimally. Sometimes individual optimality โ‰  collective optimality, which requires institutional design, trust-building, or repeated games to resolve.

Choose Classic Game Scenarios

Prisoner's Dilemma

Two prisoners are interrogated separately. They can choose to cooperate (stay silent) or defect (confess).

Player 2: Cooperate
Player 2: Defect
Player 1: Cooperate
-1
-1
-3
0
Player 1: Defect
0
-3
-2
-2
โญ Nash Equilibrium
โ— Player 1 Payoff
โ— Player 2 Payoff

๐Ÿ“Š Game Analysis

Interactive Simulator

๐Ÿ”‘ Key Concepts Explained

๐ŸŽฏ Nash Equilibrium

At Nash Equilibrium, no player can benefit by unilaterally changing their strategy. This is the stable state of the game.

Example: In Prisoner's Dilemma, (Defect, Defect) is Nash Equilibrium โ€” not the best outcome, but no one wants to change unilaterally.

๐Ÿค Pareto Optimality

At Pareto Optimal state, it's impossible to improve one party's situation without harming another.

Example: In Prisoner's Dilemma, (Cooperate, Cooperate) is Pareto Optimal โ€” any change would make at least one party worse off. But it's NOT a Nash Equilibrium!

โš–๏ธ Dominant Strategy

A dominant strategy always yields the best payoff regardless of the opponent's choice.

Example: In Prisoner's Dilemma, "Defect" is the dominant strategy โ€” always better regardless of the opponent's choice.

๐Ÿ’ก How to Understand Payoff Matrix?

Payoff Matrix is the core tool of game theory, clearly showing payoffs for different strategy combinations:

  • Rows represent Player 1's strategies, columns represent Player 2's strategies
  • Each cell has two numbers: red is Player 1's payoff, blue is Player 2's payoff
  • How to find Nash Equilibrium: For each cell, check if any player can gain by changing strategy alone. If neither can, it's Nash Equilibrium
  • Green highlighted cells indicate Nash Equilibrium points โ€” the stable outcomes

๐Ÿ’ก Tip: Click the different game scenario buttons below to observe the payoff matrices and Nash Equilibrium positions of different games!