Game theory - Classic Games and Concepts

Well-Known Game-Theory Examples Game theory provides a framework for analyzing strategic situations where the outcomes depend on the choices of multiple decision-makers. Several classic games have become standard examples in economics and strategic analysis because they illustrate fundamental concepts like Nash equilibrium, dominant strategies, and coordination problems. This section explores the most important ones you need to know. The Prisoner's Dilemma CRITICALCOVEREDONEXAM The Prisoner's Dilemma is perhaps the most famous game in game theory. It demonstrates a profound tension: individual incentives can lead to outcomes that are worse for everyone involved. The Setup Imagine two suspects arrested for a crime. The prosecutor offers each suspect a deal: if you confess (defect), you receive a reduced sentence. If you stay silent (cooperate), you receive a longer sentence—unless the other suspect confesses, in which case you receive the longest sentence possible. The payoff structure looks roughly like this: if both cooperate, both receive moderate sentences. If both defect, both receive lengthy sentences. If one defects while the other cooperates, the defector receives a short sentence while the cooperator receives the longest sentence. Why It's Called a Dilemma The puzzle emerges when you analyze each suspect's incentives. Defection is a dominant strategy—it's the best response regardless of what the other player does. If your partner cooperates, you're better off defecting. If your partner defects, you're still better off defecting (a shorter sentence is better than a longer one). Yet when both players follow their dominant strategy and defect, they both end up worse off than if they had both cooperated. Mutual cooperation yields higher payoffs for both players, but it's unstable because each player has an incentive to cheat. The unique Nash equilibrium is mutual defection, even though mutual cooperation is Pareto superior (no one could be made better off without making someone worse off). Why This Matters The Prisoner's Dilemma appears throughout economics and life: firms setting prices (they'd all benefit from collusion, but each has incentive to undercut), countries negotiating arms control (all would benefit from disarmament, but each fears defection), and even everyday cooperation problems like public goods provision. Battle of the Sexes CRITICALCOVEREDONEXAM The Battle of the Sexes illustrates a different type of strategic problem: coordination with conflicting preferences. The Setup Two players (traditionally a couple) want to spend the evening together but prefer different activities. One prefers going to a football game; the other prefers going to the opera. Crucially, being together is more important than which activity they choose—but each would prefer their favored activity. This creates a very different payoff structure than the Prisoner's Dilemma. Both players benefit from coordinating, but they disagree about how to coordinate. Multiple Equilibria This game has two pure-strategy Nash equilibria: one where both attend the football game, and one where both attend the opera. In each equilibrium, one player gets their preferred outcome, but both players are satisfied because they're together. The game also has a mixed-strategy Nash equilibrium where each player randomizes between their two options. This reflects the fundamental problem: absent some coordination mechanism, players face uncertainty about what the other will do. Key Insight Unlike the Prisoner's Dilemma where the Nash equilibrium is inefficient, the pure-strategy equilibria in Battle of the Sexes are efficient. The problem isn't that players end up in a bad outcome—it's that they face coordination risk. Without communication or an established convention, they might fail to coordinate and end up at different activities, which is worse for both. The Ultimatum Game CRITICALCOVEREDONEXAM The Ultimatum Game is a classic behavioral economics experiment that reveals how actual human behavior often deviates from simple self-interest predictions. The Setup Two players split a sum of money (say, $10). The proposer suggests a division: how much goes to them, how much to the responder. The responder then decides: accept the offer (both players get what was proposed) or reject it (both players get nothing). The Rational Prediction Standard economic theory predicts that the proposer should offer the responder almost nothing—say, $0.01. The responder should accept because something is better than nothing. Yet empirically, this prediction fails dramatically. Actual Behavior In practice, proposers typically offer 40-50% of the money, and responders frequently reject unfair offers (like 10-20% of the total), even though rejection leaves them with zero. This suggests people care not just about absolute payoffs but also about fairness and are willing to punish unfair behavior even at a cost to themselves. Why It Matters The Ultimatum Game demonstrates that self-interest alone doesn't fully explain human behavior. Preferences include fairness concerns, reciprocity, and the desire to punish norm-violations. This has profound implications for economic institutions and policy design. The Trust Game CRITICALCOVEREDONEXAM The Trust Game measures the willingness to trust others and the reciprocal tendency to reward that trust. The Setup An investor receives an initial endowment (say, $10) and can transfer any amount to a trustee. Whatever is transferred gets tripled by an experimenter. The trustee then decides how much to return to the investor. The trustee keeps the remainder. The Rational Prediction If everyone acts purely out of self-interest, the investor should transfer nothing. The trustee has no incentive to return anything since the investor cannot punish them afterward. Actual Behavior Investors typically transfer substantial amounts, and trustees frequently return money even though they aren't obligated to. Moreover, trustees tend to return more when investors transfer more, showing reciprocal behavior. This suggests trust (the investor's willingness to be vulnerable) actually encourages reciprocal behavior rather than opportunism. Key Insight The Trust Game reveals that trusting behavior can be self-fulfilling to some degree. When people trust others, others often respond with reciprocity, validating that trust. This helps explain why markets and cooperative endeavors can sustain themselves. Cournot Competition CRITICALCOVEREDONEXAM Cournot competition models how firms compete when they choose production quantities simultaneously. This is fundamental to understanding oligopolistic markets. The Setup Multiple firms produce a homogeneous product (one that customers view as identical). Each firm simultaneously chooses how much to produce. The market price is determined by total supply—firms can't control price, only their output. Each firm wants to maximize profit given what the others produce. A firm's best response is the quantity that maximizes its profit given the quantities chosen by competitors. The Nash Equilibrium A Cournot equilibrium occurs where each firm's quantity is a best response to every other firm's quantity. In other words, at equilibrium, no firm wants to change its output given what others are producing. The diagram shows this with best response curves. Each firm's curve shows optimal quantity given the competitor's output. The intersection point is the Nash equilibrium, where both firms are simultaneously optimizing. Key Results For two firms with identical constant marginal costs: The Cournot equilibrium quantity and price lie between the perfectly competitive outcome (price = marginal cost) and the monopoly outcome Each firm produces the same quantity at equilibrium Equilibrium price exceeds marginal cost, giving firms positive profit As more firms enter (with the same costs), the equilibrium approaches perfect competition Economic Significance Cournot competition shows that even without explicit collusion, oligopolistic firms don't engage in pure price wars. Instead, they choose quantities that balance their desire for profit with the reality that others are also optimizing. This model has been influential in antitrust economics and industrial organization. Bertrand Competition CRITICALCOVEREDONEXAM Bertrand competition models how firms compete through price-setting rather than quantity-setting. It provides a striking contrast to Cournot outcomes. The Setup Multiple firms simultaneously set prices for a homogeneous product. Customers buy from the cheapest firm (or split between firms with the same lowest price). The Surprising Result When all firms have the same constant marginal cost, the Nash equilibrium price equals marginal cost. This is the same outcome as perfect competition, even with only two firms! Why This Happens Suppose firms are charging a price above marginal cost. Any firm could undercut slightly and capture the entire market. So each firm has incentive to lower its price. This competitive pressure continues until price falls to marginal cost. Once price equals marginal cost, no firm can profitably undercut further (it would lose money on each sale). This is the unique Nash equilibrium. The Bertrand Paradox The fact that duopoly (two firms) yields the same competitive outcome as perfect competition seems paradoxical—it's called the Bertrand Paradox. However, it disappears when you relax assumptions: with differentiated products, firms can sustain markups; with capacity constraints, firms can't always meet all demand at the lowest price. Cournot vs. Bertrand: Which is Realistic? The contrast between Cournot and Bertrand competition highlights that how firms compete matters. In markets where firms set quantities first (or capacity constraints matter), Cournot outcomes are more realistic. In markets with price flexibility and undifferentiated products, Bertrand pressure applies. Real markets often exhibit features of both models. Stag Hunt CRITICALCOVEREDONEXAM The Stag Hunt is another coordination game, but one that emphasizes the tension between efficiency and security. The Setup Two hunters can hunt a stag (a large deer) together or each hunt a hare alone. Hunting a stag is more efficient—it yields a high payoff if both cooperate. But if your partner abandons you to hunt a hare instead, you get nothing (you can't hunt a stag alone). Hunting a hare alone gives a modest but certain payoff, regardless of what the other hunter does. The Equilibria This game has two pure-strategy Nash equilibria: Both hunt stag: More efficient, but risky (vulnerable to the other person abandoning you) Both hunt hare: Less efficient, but safe (you get your payoff regardless) There's also a mixed-strategy equilibrium. The Key Tension Unlike the Prisoner's Dilemma, mutual cooperation is a Nash equilibrium here. The problem isn't that cooperation is irrational—it's that it's risky. You must trust your partner, and trust can be violated. This illuminates real-world cooperation problems. Many beneficial cooperative activities share this structure: they work well if everyone participates but fail if some defect. Examples include participation in democracy, workplace teamwork, and market transactions. Distinguished from Other Games The Stag Hunt differs crucially from the Prisoner's Dilemma: there's no dominant strategy to defect. Instead, the choice depends on your beliefs about what the other player will do. This makes it a game of trust and common knowledge. <extrainfo> Related Concepts for Further Reading Several advanced concepts build on these foundational games but are less likely to be the primary focus of exam questions. The Core is the set of allocations where no group of players can improve their situation by forming a smaller group and negotiating independently. It represents allocations that are stable against all possible deviations. Stackelberg equilibrium describes a sequential game where a leader moves first and other players respond. Unlike simultaneous-move games, the leader can often achieve better outcomes by moving first and committing to a choice. Common knowledge means that all parties know something, and moreover, they know that everyone knows it, and everyone knows that everyone knows it, and so on. Many game-theoretic results require common knowledge of the game structure itself—if players aren't certain about payoffs or rules, equilibrium predictions may not hold. Mechanism design reverses the usual game-theory question: instead of analyzing a given game, it asks how to design a game or institution to achieve desired outcomes. This is crucial for auction design, voting systems, and market regulation. </extrainfo>