(Based on “MA Microeconomics” textbook and UGC NET syllabus)
1. Introduction
Game Theory is a mathematical framework that analyzes strategic interactions among rational decision-makers (players), where the outcome of one’s decision depends on the choices of others.
It was first formalized by John von Neumann and Oskar Morgenstern in their classic book Theory of Games and Economic Behavior (1944).
In Microeconomics, Game Theory is particularly useful in studying oligopolistic markets, where few firms make interdependent decisions regarding price, output, and advertising.
2. Classification of Games
| Basis | Types | Explanation |
|---|---|---|
| Number of Players | Two-player, n-player | Duopoly, Oligopoly, etc. |
| Nature of Payoffs | Zero-sum, Non-zero-sum | In zero-sum, one’s gain = another’s loss. |
| Nature of Cooperation | Cooperative, Non-cooperative | Cooperative involves binding agreements; non-cooperative involves independent strategies. |
| Timing of Moves | Simultaneous, Sequential | Firms act together or one after another. |
| Information Availability | Complete, Incomplete | Players may or may not know each other’s payoffs. |
3. Non-Cooperative Games: Meaning and Features
A non-cooperative game is one where players make decisions independently, without collaboration or binding agreements.
Each player selects a strategy that maximizes their own payoff, given their beliefs about others’ choices.
Features
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Independent decision-making
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Strategic interdependence
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Use of payoff matrices
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Focus on Nash Equilibrium
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May involve dominant or mixed strategies
4. Basic Concepts of Non-Cooperative Games
A. Players and Strategies
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Players: The decision-makers (e.g., firms in oligopoly).
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Strategies: Plans of action available to each player (e.g., “Raise Price” or “Cut Price”).
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Payoff: The reward or outcome for each combination of strategies.
A payoff matrix shows all possible outcomes.
B. Payoff Matrix (Example)
| Firm B ↓ / Firm A → | High Price | Low Price |
|---|---|---|
| High Price | (10, 10) | (2, 15) |
| Low Price | (15, 2) | (5, 5) |
Each cell shows the profits (A, B) from their chosen strategies.
5. Dominant Strategy
A dominant strategy is one that provides a higher payoff to a player, regardless of what others do.
Example:
If Firm A earns higher profit by always choosing “Low Price,” then “Low Price” is its dominant strategy.
If both firms have dominant strategies, the resulting outcome is called the Dominant Strategy Equilibrium.
6. Nash Equilibrium
Introduced by John Nash (1950), the Nash Equilibrium occurs when no player can improve their payoff by unilaterally changing their strategy, given the other’s choice.
In the above payoff matrix:
-
(Low Price, Low Price) = (5, 5)
Neither A nor B gains by changing strategy → Nash Equilibrium.
7. The Prisoner’s Dilemma Model
One of the most famous examples of a non-cooperative game, used to demonstrate strategic interdependence and conflict between individual and collective rationality.
The Setup:
Two prisoners (Ranga and Billa) are arrested.
They can either Confess or Deny the crime.
| Billa Confess | Billa Deny | |
|---|---|---|
| Ranga Confess | (5 yrs, 5 yrs) | (0 yrs, 10 yrs) |
| Ranga Deny | (10 yrs, 0 yrs) | (2 yrs, 2 yrs) |
Analysis:
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Confession gives each prisoner a dominant strategy.
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Both confess → each gets 5 years, though mutual denial (2,2) was better.
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This is a Nash Equilibrium but Pareto inefficient.
Economic Application:
In oligopoly, firms face similar situations:
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If both cut prices → lower profits.
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If both cooperate (keep prices high) → higher profits.
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But mutual distrust prevents cooperation.
8. Application of Non-Cooperative Games in Oligopoly
Case Example: Advertising Game
Two firms (Sony and Suzuki) must decide whether to increase advertising or not.
| Suzuki ↑ | Suzuki ↓ | |
|---|---|---|
| Sony ↑ | (20, 20) | (30, 10) |
| Sony ↓ | (10, 30) | (25, 25) |
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Both increasing ads (20,20) is Nash equilibrium.
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Even though mutual restraint (25,25) would be better, competition pressures drive firms toward less optimal outcomes.
This illustrates strategic rivalry and inefficiency of non-cooperative outcomes.
9. Mixed Strategy Equilibrium
Sometimes, no pure strategy equilibrium exists.
A mixed strategy involves players randomizing among available actions with specific probabilities.
Example: In sports (e.g., penalty kicks), goalkeepers and players mix strategies unpredictably.
Nash proved that every finite game has at least one equilibrium (pure or mixed).
10. Zero-Sum vs Non-Zero-Sum Games
| Type | Description | Example |
|---|---|---|
| Zero-Sum Game | One player’s gain = another’s loss | Poker, war games |
| Non-Zero-Sum Game | Both players may gain or lose together | Oligopoly, trade negotiations |
Non-cooperative games are often non-zero-sum, as mutual cooperation or defection affects both players’ outcomes.
11. Repeated and Sequential Games
| Type | Description | Example |
|---|---|---|
| Repeated Game | Players interact repeatedly over time → reputation and punishment possible | Firms maintaining cartel pricing |
| Sequential Game | One player moves first, others follow | Stackelberg model |
Repeated games can sustain cooperation through threat of retaliation, unlike one-shot games.
12. Equilibrium in Non-Cooperative Games
| Concept | Definition | Relevance |
|---|---|---|
| Dominant Strategy Equilibrium | Both choose dominant strategies | Always stable but may be inefficient |
| Nash Equilibrium | No incentive to deviate unilaterally | Common in duopoly |
| Pareto Optimality | No one can be better off without making another worse off | Often violated in non-cooperative settings |
13. Real-World Examples
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Oligopoly Pricing: Firms deciding whether to collude or compete.
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Trade Policy: Countries deciding whether to impose tariffs.
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Arms Race: Nations choosing between arming or disarming.
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Advertising: Firms allocating budget between ads and price cuts.
14. Criticisms of Non-Cooperative Game Theory
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Assumes perfect rationality.
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Ignores emotions and bounded rationality.
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Difficult to predict outcomes in multi-player, dynamic settings.
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Relies heavily on payoff quantification.
15. Key Models and Theorists
| Theorist | Contribution |
|---|---|
| John von Neumann & Oskar Morgenstern | Founders of Game Theory |
| John Nash | Concept of Nash Equilibrium |
| Martin Shubik | Applied Game Theory to Oligopoly |
| Tucker | Formalized the Prisoner’s Dilemma |
16. Mathematical Representation
For a 2-player game with strategies :
At Nash Equilibrium:
for all
🔹 17. Summary
| Concept | Key Points |
|---|---|
| Game Theory | Analyzes strategic decision-making |
| Non-Cooperative Games | Independent strategies without binding agreements |
| Dominant Strategy | Always best regardless of others |
| Nash Equilibrium | No incentive to deviate individually |
| Prisoner’s Dilemma | Explains failure of cooperation |
| Mixed Strategies | Randomization in strategy choice |
| Applications | Oligopoly, advertising, trade, politics |
🔹 18. UGC NET Key Focus Areas
| Topic | Importance | Common Questions |
|---|---|---|
| Nash Equilibrium | ⭐⭐⭐⭐ | Definition, calculation |
| Dominant Strategy | ⭐⭐⭐ | Identification in payoff matrices |
| Prisoner’s Dilemma | ⭐⭐⭐⭐ | Application in oligopoly |
| Mixed Strategy | ⭐⭐ | Concept and example |
| Zero-sum vs Non-zero-sum | ⭐⭐ | Distinction |
| Repeated Games | ⭐⭐ | Collusion and punishment models |
19. Key Equations
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Expected Payoff (Mixed Strategy):
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Nash Condition:
No unilateral improvement possible. -
Dominance Rule:
Eliminate dominated strategies iteratively to simplify analysis.