Microeconomics — Chapter 7 Test: Game Theory — Prisoner's Dilemma and Nash Equilibrium

🔒 The prisoner's dilemma is the most famous example in game theory.
When each player acts rationally (chooses their dominant strategy), the outcome is worse for both than if they had cooperated.

⚖️ Nash equilibrium:
Each player chose the best response to the choices of the others. No one regrets and no one wants to change - even if the overall outcome is not optimal!

🎯 In the prisoner's dilemma, informing is a dominant strategy:
• If the other stays silent - it is preferable to inform (gets out free)
• If the other informs - it is preferable to inform (lighter punishment)
In any case it is preferable to inform!

📊 Nash equilibrium: (Inform, Inform)
Both choose to inform and each receives -5.
The tragedy: If both had stayed silent, they would have received only -1 each!
But individually, each prefers to inform.

🌟 Pareto efficiency:
A situation in which it is impossible to improve someone's situation without making someone else worse off.
In the prisoner's dilemma: (Silence, Silence) is Pareto efficient, but (Inform, Inform) which is the equilibrium - is not efficient!

🔄 Repeated game:
The same game is played over and over. This changes the strategy!
In a one-shot game - there is no point in cooperating.
In a repeated game - there is an incentive to cooperate because there is a "future" to the relationship.

👊 Tit-for-Tat strategy:
1. First round: Cooperate
2. From here on: Do what the opponent did in the previous round
A simple yet very effective strategy in repeated games!

🏭 Cartel = prisoner's dilemma:
• Cooperation: Everyone keeps the price high, good profits
• Defection: Lower the price, steal customers, higher profit
Each firm has a dominant strategy to cheat - therefore cartels are unstable!

🔍 The arrows/lines method:
1. For each column - mark the row best for player 1
2. For each row - mark the column best for player 2
3. A cell with two markings = Nash equilibrium!

📈 Nash ≠ efficiency!
The prisoner's dilemma proves: the equilibrium (Inform, Inform) is inefficient.
Both players could have improved if both had stayed silent - but it is not stable!

🎮 Nash check:
• (Bottom, Right): P1 chooses Bottom because 2>1, P2 chooses Right because 2>1 ✓
• (Top, Left): P1 would prefer Bottom (4>3), P2 would prefer Right (4>3) ✗
Nash equilibrium: (Bottom, Right) - even though (3,3) is better for both!

🤝 Coordination game:
Both players gain when both choose the same thing.
Example: Which side of the road to drive on - the main thing is that everyone is on the same side!
Usually there are several Nash equilibria.

🔢 The number of equilibria varies:
• Prisoner's dilemma - one (Inform, Inform)
• Coordination game - several (e.g. Left-Left and Right-Right)
• Certain games - zero in pure strategies

💰 This is a classic prisoner's dilemma!
"Low" is a dominant strategy for both:
• If the opponent chooses High: 120>100
• If the opponent chooses Low: 50>20
The outcome (50,50) is worse than (100,100), but stable.

🔄 The key: the future!
In a repeated game, if you defect today - the opponent will punish you tomorrow.
A credible threat of punishment creates an incentive to maintain cooperation.
This is why cartels are more stable in industries with ongoing interaction.

⏰ End Game Problem:
In the last round - there is no future, so they will defect.
In the round before last - they know that in the last round they will defect, so they will defect now too.
So actually from the start! (This is a theoretical result - in reality sometimes they still cooperate)

🎯 Best Response:
Given that I know what the opponent is doing - what is best for me?
Nash equilibrium = a situation in which each player plays their Best Response to the choices of the others.

📊 Pure coordination game:
• (A, A): P1 will not change (5>0), P2 will not change (5>0) ✓
• (B, B): P1 will not change (3>0), P2 will not change (3>0) ✓
Two equilibria! (A, A) is preferred for both, but (B, B) is also stable.

🏆 John Nash (1928-2015)
An American mathematician who developed the idea in his doctoral dissertation.
Won the 1994 Nobel Prize in Economics.
His life story was told in the film "A Beautiful Mind".

🎲 Mixed strategy:
Instead of choosing one strategy with certainty, the player randomizes between options.
Example: In penalties, the goalkeeper jumps right 60% and left 40%.
Important when there is no equilibrium in pure strategies.

⚠️ Coordination failure:
Even when everyone is rational, the overall outcome can be bad.
Examples: environmental pollution, arms race, overconsumption of common resources.
This is the justification for government intervention!

🌍 Arms race = prisoner's dilemma:
• Both countries prefer peace (not to spend on weapons)
• But each country fears the other will arm itself
• Therefore both arm themselves - an expensive and inefficient outcome!
More examples: excessive advertising, industrial pollution.

🔐 Solutions to the prisoner's dilemma:
1. Binding contracts: An agreement that can be enforced
2. Repeated game: There is an incentive to maintain reputation
3. External punishment: Laws and fines
4. Social norms: Public pressure

📉 The difference:
Dominant strategy: The best choice no matter what the opponent does
Nash: The best choice given what the opponent does
Every dominant strategy equilibrium is also a Nash, but not vice versa!

🎮 Another prisoner's dilemma!
"Advertise" dominates: 40>30 and 70>60
Both will advertise → (40, 40)
If both had refrained → (60, 60)
The conclusion: Competitive advertising can be wasteful - everyone spends, no one gains an advantage!