Why does the Prisoner's Dilemma produce a bad outcome for both players?

Each player has a dominant strategy to betray (confess), regardless of what the other does. If the other stays silent, betraying yields freedom vs 1 year. If the other betrays, betraying yields 3 years vs 5 years. Betrayal always beats silence — so both betray, getting 3 years each, even though mutual silence would yield only 1 year each. This is the tragedy: rational individual choices produce a collectively worse outcome.

What is Tit-for-Tat and why is it so effective?

Tit-for-Tat (TFT) starts with cooperation, then mirrors whatever the opponent did last round. It is nice (never betrays first), retaliatory (punishes betrayal immediately), forgiving (returns to cooperation after the opponent does), and clear (easy to understand). In Robert Axelrod's famous computer tournaments, TFT consistently outperformed more complex strategies across hundreds of repeated games.

What is the difference between zero-sum and non-zero-sum games?

In a zero-sum game, one player's gain exactly equals the other's loss — the total payoff is always zero (chess, poker, market share competition). In a non-zero-sum game, cooperation can create value so both players can gain or both can lose (Prisoner's Dilemma, arms races, climate agreements). Most real strategic situations are non-zero-sum, which is why game theory is more nuanced than pure competition.

How is game theory applied in business and economics?

Game theory is used across many domains: auction design (Google AdWords, spectrum auctions use Vickrey/sealed-bid mechanisms), oligopoly pricing (Cournot and Bertrand models explain why airlines and oil companies coordinate implicitly), labor negotiations (Nash Bargaining Solution), international trade policy (tariff games between countries), and platform competition (two-sided market entry timing).

What is the difference between dominant strategy and Nash Equilibrium?

A dominant strategy is best for a player regardless of what opponents do. A Nash Equilibrium is a stable outcome where no player wants to deviate given others' strategies. Every dominant strategy equilibrium is a Nash Equilibrium, but not all Nash Equilibria are dominant strategy equilibria. In some games (like coordination games), Nash Equilibria exist without either player having a dominant strategy.

Game Theory — The Math Behind Strategic Choice

Q: What is a Nash Equilibrium?

A Nash Equilibrium is a set of strategies where no player can improve their outcome by changing only their own strategy, assuming other players keep theirs unchanged. It is the stable resting point of a game — not necessarily the best collective outcome, but the outcome no single player wants to deviate from.

Game theory is the mathematical study of strategic decision-making between rational agents. Developed by John von Neumann and Oskar Morgenstern (1944) and extended by John Nash (1950), it provides a rigorous framework for analyzing any situation where the outcome depends on the choices of multiple players — from business competition to international diplomacy to evolutionary biology.

Nash Equilibrium

No Unilateral Gain

Stable strategy profile — no player benefits from deviating alone

Prisoner's Dilemma

Both Betray

Rational individuals produce collectively worse outcome

Zero-Sum Games

Win = Loss

Chess, poker — one player's gain is exactly the other's loss

Tit-for-Tat

Best in Repeats

Start cooperative, mirror last move — wins tournament play

📐 The Core Framework: Nash Equilibrium

John Nash proved that every finite game has at least one equilibrium — a stable point no player wants to deviate from.

Nash Equilibrium — Formal Definition

s_i^* \in \arg\max_{s_i} \; u_i(s_i, s_{-i}^*) \quad \forall i

s*ᵢ Player i's equilibrium strategy — their best response given others' strategies

s*₋ᵢ All other players' equilibrium strategies (held fixed)

uᵢ Player i's utility (payoff) function — what they are trying to maximize

argmax The strategy that maximizes utility — equilibrium requires this for every player simultaneously

출처: Nash, J. (1950) — Nobel Prize in Economics 1994

🔢 The Prisoner’s Dilemma — Payoff Matrix

The most famous game in game theory. Two suspects face identical choices with no communication.

Prisoner's Dilemma payoff matrix — (Player A years, Player B years)
Player A ↓ / Player B →	B Cooperates (Stay Silent)	B Betrays (Confess)
A Cooperates (Stay Silent)	(1 year, 1 year) ← Mutual optimum	(5 years, 0 years) ← A's worst outcome
A Betrays (Confess)	(0 years, 5 years) ← B's worst outcome	(3 years, 3 years) ← Nash Equilibrium

Betrayal is a dominant strategy — but both players suffer for it

From A’s perspective: if B cooperates, betraying is better (0 vs 1 year). If B betrays, betraying is still better (3 vs 5 years). So A always betrays — and B reasons identically. The Nash Equilibrium (Betray, Betray) gives 3 years each, even though (Cooperate, Cooperate) gives only 1 year each. Rational individual logic leads to collective irrationality.

📊 Classic Games in Game Theory

Canonical game theory scenarios — structure, equilibrium, and real-world analog
Game	Structure	Nash Equilibrium	Key Insight	Real-World Analog
Prisoner's Dilemma	Non-zero-sum, simultaneous	(Betray, Betray)	Dominant strategy produces socially inferior outcome	Price wars, arms races, carbon emissions
Battle of the Sexes	Non-zero-sum, coordination	Two equilibria: (Opera, Opera) or (Football, Football)	Multiple equilibria — communication and commitment matter	Industry standards (VHS vs Betamax, USB-C)
Stag Hunt	Non-zero-sum, coordination	(Stag, Stag) or (Hare, Hare)	Payoff-dominant equilibrium exists but requires trust	Team projects, international treaties
Chicken (Hawk-Dove)	Non-zero-sum, anti-coordination	One swerves, one continues — mixed strategy	Commitment and credibility determine outcome	Labor strikes, nuclear standoffs
Matching Pennies	Zero-sum, simultaneous	Mixed strategy: each plays 50/50	No pure strategy equilibrium — randomization is optimal	Rock-paper-scissors, penalty kicks in soccer
Ultimatum Game	Non-zero-sum, sequential	Theoretically: offer minimum, accept anything	Actual behavior: offers near 50/50 — fairness matters	Wage negotiations, VC term sheets

🔄 Strategies in Repeated Games

When the same game is played repeatedly, cooperation becomes sustainable through reputation and retaliation.

Strategies in iterated Prisoner's Dilemma — Axelrod tournaments (1980, 1984)
Strategy	First Move	Subsequent Moves	Tournament Performance	Key Property
Always Defect	Betray	Always betray	Wins single encounters; loses repeated play	Exploitative — destroys long-term value
Always Cooperate	Cooperate	Always cooperate	Exploited by defectors; loses overall	Naive — cannot punish betrayal
Tit-for-Tat (TFT)	Cooperate	Mirror opponent's last move	Won both Axelrod tournaments	Nice, retaliatory, forgiving, clear
Tit-for-Tat with forgiveness	Cooperate	Mirror last move; occasionally cooperate despite betrayal	Robust against noise/mistakes	Breaks mutual defection spirals
Grim Trigger	Cooperate	Cooperate until first betrayal, then always defect forever	Strong deterrent; fragile to errors	Maximum punishment — lacks forgiveness
Pavlov (Win-Stay, Lose-Shift)	Cooperate	Repeat if outcome was good; switch if bad	Outperforms TFT in noisy environments	Self-correcting — avoids permanent defection spirals

Tit-for-Tat won without ever being the highest single-round scorer

TFT never beats any opponent in a single game — at best it ties. Yet it won the tournament by consistently achieving near-optimal mutual cooperation. The lesson: in repeated interaction, being cooperative and predictable outperforms being exploitative. This explains why reputation, trust, and reciprocity are so economically valuable.

🌍 Real-World Applications of Game Theory

Game theory in practice — domains, models, and outcomes
Domain	Game Type	Key Model	Application
Auction Design	Non-zero-sum	Vickrey (second-price) auction	Google AdWords, spectrum auctions — bidders reveal true valuations; no advantage to overbidding
Oligopoly Competition	Non-zero-sum	Cournot (quantity) / Bertrand (price)	Airlines, oil companies — tacit coordination without explicit agreement
International Trade	Repeated Prisoner's Dilemma	Tariff retaliation game	WTO as commitment device — countries cooperate because repeat play makes defection costly
Labor Negotiations	Bargaining game	Nash Bargaining Solution	Split of surplus proportional to each side's outside options (BATNA)
Platform Competition	Coordination game	Two-sided market entry	Which platform becomes standard (iOS vs Android, Slack vs Teams) depends on early coordination
Climate Policy	Global commons game	Tragedy of the Commons	Each country has incentive to free-ride on others' emissions reductions — classic multi-player Prisoner's Dilemma
Sports Strategy	Zero-sum	Mixed strategy equilibrium	Penalty kick direction, serve placement in tennis — optimal play requires randomization to be unpredictable

🔄 How to Analyze a Strategic Situation

Step-by-step game theory analysis framework

Identify Players, Actions, and Payoffs

Who are the decision-makers? What can each player do? What does each player receive for each combination of actions? Build the payoff matrix.

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Check for Dominant Strategies

Does any player have a strategy that is best regardless of what others do? Eliminate dominated strategies — rational players never use them. This simplifies the analysis.

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Find Nash Equilibria

For each cell in the payoff matrix, check: can either player improve their payoff by switching? If no player can improve, it is a Nash Equilibrium. Multiple equilibria may exist.

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Consider the Game Type

Is this one-shot or repeated? Simultaneous or sequential? Zero-sum or cooperative? Repeated play enables cooperation. Sequential play creates first-mover or follower advantages.

⚖️ Cooperative vs Non-Cooperative Game Theory

Two branches of game theory — individual strategy vs coalition formation

구분	Non-Cooperative Game Theory	Cooperative Game Theory
Focus	Individual player strategies and self-interest	Coalition formation and fair division of collective gains
Key solution concept	Nash Equilibrium — stable individual strategies	Shapley Value — fair distribution of coalition surplus
Communication	Binding agreements not enforceable	Binding contracts and side payments allowed
Examples	Oligopoly competition, auctions, international trade	Joint ventures, labor unions, vote trading in legislatures
Limitation	May reach inefficient equilibria (Prisoner's Dilemma)	Requires enforceable contracts; ignores strategic manipulation

Frequently Asked Questions

What is a Nash Equilibrium and why does it matter?

A Nash Equilibrium is a stable configuration where no player can improve their outcome by unilaterally changing their strategy. It matters because it predicts where rational strategic interaction will settle. Nash proved that every finite game has at least one equilibrium (possibly in mixed strategies). However, equilibria are not always efficient — the Prisoner’s Dilemma has a Nash Equilibrium that is worse for everyone than the cooperative outcome.

Why doesn’t rational self-interest always produce the best collective outcome?

The Prisoner’s Dilemma demonstrates this precisely: each player follows the individually rational dominant strategy (betray), yet both end up worse than if they had cooperated. This conflict between individual and collective rationality underlies many social problems: carbon emissions, overfishing, price wars, arms races. Solutions include binding agreements, reputation mechanisms, regulation, or repeated interaction (where future cooperation becomes valuable enough to outweigh short-term defection gains).

How does the number of repetitions affect cooperation?

In a finitely repeated game with a known end point, backward induction logic unravels cooperation: both players know they will defect on the last round, so they defect on the second-to-last, and so on. In infinitely repeated games (or games with uncertain endpoints), the “folk theorem” shows that cooperation can be sustained if players value future payoffs enough — if the discount factor is high enough, the threat of future punishment makes cooperation individually rational.

What is the difference between pure and mixed strategies?

A pure strategy is a deterministic choice (always cooperate, always betray). A mixed strategy assigns probabilities to different actions (e.g., betray with 60% probability). In zero-sum games like matching pennies, no pure strategy equilibrium exists — the optimal solution requires randomization so opponents cannot predict and exploit your pattern. Mixed strategy equilibria always exist when pure strategy equilibria do not.

How is game theory used in auction design?

Auction mechanism design is a direct application of game theory. The Vickrey (second-price sealed-bid) auction is strategy-proof: bidding your true value is a dominant strategy regardless of others’ bids. Google AdWords uses a generalized second-price auction. Spectrum auctions (for mobile phone bandwidth) use simultaneous ascending auctions designed by game theorists to allocate resources efficiently and prevent collusion.

What is the Shapley Value?

The Shapley Value (Lloyd Shapley, Nobel 2012) is a fair allocation method for cooperative games. It assigns each player their average marginal contribution across all possible coalition orderings. It is used in cost allocation (how much each partner contributes to a joint project), voting power analysis, and machine learning (SHAP values for feature importance — a direct application of the Shapley Value concept).