Game Theory – Non-Cooperative Games

(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

Independent decision-making
Strategic interdependence
Use of payoff matrices
Focus on Nash Equilibrium
May involve dominant or mixed strategies

4. Basic Concepts of Non-Cooperative Games

A. Players and Strategies

Players: The decision-makers (e.g., firms in oligopoly).
Strategies: Plans of action available to each player (e.g., “Raise Price” or “Cut Price”).
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:

Confession gives each prisoner a dominant strategy.
Both confess → each gets 5 years, though mutual denial (2,2) was better.
This is a Nash Equilibrium but Pareto inefficient.

Economic Application:

In oligopoly, firms face similar situations:

If both cut prices → lower profits.
If both cooperate (keep prices high) → higher profits.
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)

Both increasing ads (20,20) is Nash equilibrium.
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

Oligopoly Pricing: Firms deciding whether to collude or compete.
Trade Policy: Countries deciding whether to impose tariffs.
Arms Race: Nations choosing between arming or disarming.
Advertising: Firms allocating budget between ads and price cuts.

14. Criticisms of Non-Cooperative Game Theory

Assumes perfect rationality.
Ignores emotions and bounded rationality.
Difficult to predict outcomes in multi-player, dynamic settings.
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 $S_{A}, S_{B}$ :

$P_{A} = f_{A} (S_{A}, S_{B})$ $P_{B} = f_{B} (S_{A}, S_{B})$

At Nash Equilibrium:

$f_{A} (S_{A}^{*}, S_{B}^{*}) \geq f_{A} (S_{A}, S_{B}^{*})$ $f_{B} (S_{A}^{*}, S_{B}^{*}) \geq f_{B} (S_{A}^{*}, S_{B})$

for all $S_{A}, S_{B}$

🔹 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

Expected Payoff (Mixed Strategy):

$E (U) = \sum p_{i} \times u_{i}$
Nash Condition:
No unilateral improvement possible.
Dominance Rule:
Eliminate dominated strategies iteratively to simplify analysis.

Tag: Game Theory – Non-Cooperative Games

UGC NET Economics Unit 1 – Game Theory – Non-Cooperative Games