Dominant Strategy Examples

Why This Matters

Game theory questions test whether you understand strategic decision-making: how rational actors choose when their outcomes depend on others' choices. You're not just being tested on definitions. You need to recognize when a dominant strategy exists, when it doesn't, and why following individual incentives sometimes leads to collectively terrible outcomes. These concepts connect directly to market failures, oligopoly behavior, public goods provision, and externalities.

The examples below demonstrate core principles you'll encounter on both multiple-choice and FRQ sections: the tension between Nash equilibrium and Pareto efficiency, why cooperation breaks down even among rational actors, and how payoff structures determine strategic behavior. Don't just memorize game names. Know what concept each game illustrates and be ready to identify dominant strategies (or their absence) from a payoff matrix.

---

Games Where Betrayal Is the Dominant Strategy

These games share a crucial feature: each player's best response is the same regardless of what the other player does. That's what makes it a dominant strategy. The result? Both players rationally choose strategies that leave everyone worse off than if they'd cooperated.

Prisoner's Dilemma

This is the single most important game in the course. Two suspects are interrogated separately. Each can confess (betray) or stay silent (cooperate). The payoff structure makes confessing strictly better no matter what your partner does.

• Betrayal dominates for both players. If your partner stays silent, confessing gets you a lighter sentence. If your partner confesses, confessing still gets you a lighter sentence than staying silent.
• The Nash equilibrium (both confess) is Pareto inefficient. Both players end up worse off than they would under mutual cooperation (both staying silent). This is the core tension of the game.
• Explains why oligopoly collusion breaks down. Two firms would both profit more by keeping prices high, but each firm has an individual incentive to undercut the other. Without an enforcement mechanism, the cooperative outcome unravels.

Traveler's Dilemma

Two travelers must independently claim a value for a lost item (say, between $2 and $100). The airline pays both the lower claim, but gives a bonus to whoever claimed less and a penalty to whoever claimed more.

• Undercutting is always rational. Claiming one dollar less than your opponent earns you the bonus, so there's always an incentive to go lower.
• Iterative reasoning leads to minimum claims. You think your opponent will undercut, so you undercut first, and they reason the same way. This chain of logic drives both claims down to the minimum ($2).
• Demonstrates how competition destroys surplus. Both players receive far less than they would if they'd simply both claimed $100.

---

Games Without Pure Dominant Strategies

Not every strategic situation has a clear "always do this" answer. These games require players to randomize their choices or adapt based on expectations about opponents, introducing the concept of mixed strategy equilibrium (where a player assigns probabilities to different actions rather than picking one for certain).

Matching Pennies

Two players simultaneously reveal a coin. Player 1 wins if both coins match (both heads or both tails); Player 2 wins if they differ. There's no single choice that always works.

• No dominant strategy exists. Choosing heads beats an opponent who chose heads (for Player 1), but loses to tails. Every choice can be exploited if it's predictable.
• Mixed strategy equilibrium requires 50/50 randomization. Each player flips with equal probability, making the opponent unable to gain an edge.
• Models zero-sum competition. One player's gain exactly equals the other's loss. Think of penalty kicks in soccer: the kicker and goalkeeper must randomize to avoid being predictable.

Chicken Game

Two drivers speed toward each other. Each can swerve (back down) or stay straight (escalate). If both stay straight, they crash. If one swerves and the other doesn't, the one who swerved loses face while the other "wins."

• Each player prefers to swerve if the other won't, but prefers to stay straight if the other swerves. Your best response depends entirely on what the other player does.
• Two pure-strategy Nash equilibria exist (one swerves, the other doesn't), plus a mixed strategy equilibrium. The outcome often hinges on credible commitment and reputation.
• Models brinksmanship situations. Nuclear standoffs, labor negotiations, and firms competing for market dominance all share this structure: escalation is rewarding only if the other side backs down.

---

Games Where Coordination Beats Individual Optimization

These games show that dominant strategies don't always exist, and success depends on trust, communication, or social norms that align individual choices. The challenge isn't resisting temptation to defect; it's making sure both players end up on the same page.

Stag Hunt

Two hunters can pursue a stag (which requires both of them working together) or a hare (which either can catch alone). Stag is the bigger prize, but if your partner chases hare instead, you get nothing.

• Cooperation yields the highest payoff. Hunting stag together is the best outcome for both players, but only if both commit to it.
• Risk dominance vs. payoff dominance creates the central tension. Hunting hare is safer (you get something no matter what), but hunting stag together pays more. The "safe" choice and the "best" choice pull in different directions.
• Models network effects and technology adoption. Everyone benefits from using the same platform, but switching requires trust that others will switch too.

Battle of the Sexes

Two people want to spend the evening together but prefer different activities (say, one prefers a concert, the other a sporting event). Going together to either event beats going alone to your preferred one.

• No dominant strategy for either player. Each prefers their own activity but values coordination over going alone.
• Two Nash equilibria exist (both at the concert, or both at the sporting event), requiring negotiation or a focal point to resolve which one they'll pick.
• Illustrates coordination problems in markets. Standardization decisions, compatible technologies, and industry conventions all involve this structure: agreement matters more than which specific option is chosen.

---

Games Revealing Social Preferences and Fairness

Standard game theory assumes players maximize material payoffs. These games consistently show that real humans care about fairness, reciprocity, and social norms, challenging the pure rationality assumption.

Ultimatum Game

A proposer splits a sum of money (say, $10). The responder can accept the split (both keep their shares) or reject it (both get nothing).

• Under strict rationality, the proposer should offer the minimum (say, $1), and the responder should accept any positive amount rather than walk away with nothing.
• Empirically, low offers get rejected. Offers below about 20-30% of the total are frequently turned down, even though rejection is "irrational" in pure payoff terms.
• Fairness concerns override material incentives. This matters for understanding wage negotiations, pricing, and why people punish behavior they perceive as unfair even at a cost to themselves.

Dictator Game

Same setup as the Ultimatum Game, except the responder has no choice: they must accept whatever the proposer gives. The proposer has all the power.

• Keeping everything is the dominant strategy. The recipient can't punish or reward, so there's zero strategic reason to share.
• Most dictators share anyway (typically giving 20-30% of the total), revealing altruism, guilt, or internalized social norms.
• Isolates pure altruism from strategic fairness. Since there's no threat of rejection, any sharing that occurs can't be explained by self-interest.

---

Games Modeling Resource Competition and Public Goods

These games explain why markets fail to provide public goods and how common resources get overexploited. Dominant strategy analysis reveals the logic behind free-riding and the tragedy of the commons.

Public Goods Game

Each player in a group decides how much to contribute to a shared pool. Contributions are multiplied (representing the social benefit of public goods) and then split equally among all players, regardless of who contributed.

• Free-riding is the dominant strategy. Contributing costs you directly, but the benefits are spread across everyone. Your best move is to let others pay while you enjoy the results.
• Collective contributions maximize group welfare, but individual incentives undermine provision. If everyone free-rides, the public good doesn't get funded.
• Directly models public goods problems. National defense, clean air, and public broadcasting all face this structure, which is why government intervention (taxes, regulation) is often the proposed solution.

Hawk-Dove Game

Two players compete over a resource. Each can play Hawk (fight for it) or Dove (back down). If both play Hawk, they fight and both suffer costly damage. If one plays Hawk and the other Dove, the Hawk takes the resource. If both play Dove, they split it.

• No pure dominant strategy. Being a Hawk wins against Doves but leads to destructive conflict against other Hawks. Being a Dove is safe but gets exploited by Hawks.
• Equilibrium depends on population composition. In evolutionary game theory, a stable mix of Hawks and Doves emerges as a mixed strategy equilibrium.
• Models common resource conflicts. Territorial disputes, competitive advertising wars, and arms races all follow this pattern: aggression pays only when opponents are passive.

---

Quick Reference Table

---

Self-Check Questions

1. Which two games both have dominant strategies that lead to Pareto-inefficient outcomes, and what distinguishes their strategic structures?

2. If given a payoff matrix where Player A's best response is the same regardless of Player B's choice, what term describes Player A's strategy, and which classic game best illustrates this?

3. Compare and contrast the Ultimatum Game and Dictator Game: what does the difference in observed behavior tell us about the source of fairness in economic decisions?

4. An FRQ describes two firms deciding whether to advertise aggressively or maintain current spending, where aggressive advertising is costly but captures market share from passive competitors. Which game does this most closely resemble, and what outcome would you predict?

5. Why does the Stag Hunt have two Nash equilibria while the Prisoner's Dilemma has only one, and what does this difference imply about the role of trust in strategic situations?