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Game Theory & Mathematics

Explore fascinating concepts in game theory, strategic thinking, and mathematical reasoning.

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Prisoner's Dilemma
Dec 6, 2025
Mathexus Team
Game Theory

The Prisoner's Dilemma: Why Cooperation is Hard

The Prisoner's Dilemma is one of the most famous examples in game theory. Two prisoners are separately interrogated. If both stay silent, they each serve 1 year. If one betrays while the other stays silent, the betrayer goes free while the other serves 3 years. If both betray, they each serve 2 years.

The paradox? Rational self-interest leads both to betray, resulting in a worse outcome (2 years each) than if they had cooperated (1 year each). This demonstrates how individual rationality can lead to collective irrationality.

Key Insight:

The Nash equilibrium (both betray) is not Pareto optimal. This applies to real-world scenarios like arms races, environmental protection, and business competition.

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Nash Equilibrium
Dec 5, 2025
Mathexus Team
Strategic Thinking

Nash Equilibrium: When No Player Wants to Change

Named after mathematician John Nash, a Nash equilibrium occurs when each player's strategy is optimal given the strategies of all other players. No player can benefit by changing their strategy while others keep theirs unchanged.

Consider the classic example: In traffic, if everyone drives on the right (or left), no individual benefits from switching. This stable state is a Nash equilibrium. The concept appears everywhere from economics to biology.

Mathematical Beauty:

Nash proved that every finite game has at least one equilibrium (possibly in mixed strategies). This theorem earned him the Nobel Prize in Economics in 1994.

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Zero-Sum Games
Dec 4, 2025
Mathexus Team
Competitive Strategy

Zero-Sum Games: One Player's Gain is Another's Loss

In zero-sum games, the total benefit to all players sums to zero. Every advantage gained by one player is exactly balanced by losses to other players. Chess, poker, and most competitive sports are classic examples.

The minimax theorem, proved by John von Neumann in 1928, states that in two-player zero-sum games, there exists a value V such that one player can guarantee to win at least V while the other can guarantee to lose no more than V.

Real-World Applications:

While pure zero-sum games are rare in real life, the concept helps analyze competitive scenarios in business, military strategy, and negotiations where interests directly oppose each other.

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