📖 博弈论基础：纳什均衡

Economics

I. Lesson Focus

This lesson introduces the fundamental concepts of game theory with a focus on simultaneous-move games, dominant strategies, and Nash equilibrium. Candidates must be able to construct and interpret payoff matrices, identify dominant strategy equilibria, locate all Nash equilibria, and recognize the prisoner’s dilemma in oligopoly settings. These tools are frequently tested in the context of pricing, advertising, and capacity decisions among rival firms.

II. The Problem

Two airlines simultaneously decide whether to add flights on a popular route. If both add capacity, oversupply drives fares down sharply and both earn low profits. If only one adds flights, that carrier gains market share and higher profits while the other suffers. Each airline wants to know: given that it does not observe the rival’s choice in advance, what strategy should it select? Is there a strategy combination in which neither carrier has an incentive to unilaterally deviate once the other’s choice is known? This is the exact question answered by Nash equilibrium and represents a classic CFA-style oligopoly interaction.

III. Basic Concepts of Game Theory

Game theory studies how rational decision makers choose strategies when their payoffs are interdependent. CFA Level I focuses on non-cooperative games, in which players cannot make binding, enforceable agreements.

A game consists of four key elements: - Players: The decision-making entities (usually firms). - Strategies: The complete set of possible actions available to each player. - Payoffs: The profits or utilities received by each player for every possible combination of strategies. - Information: Whether moves are made simultaneously or sequentially.

This lesson examines simultaneous-move games, which are conveniently represented by a payoff matrix.

IV. Dominant Strategies and Dominant-Strategy Equilibrium

A dominant strategy is a strategy that is strictly better for a player no matter which strategy the opponent chooses.

A dominant-strategy equilibrium exists when every player has a dominant strategy; the resulting combination is the equilibrium.

Numerical Illustration: Two firms, A and B, simultaneously choose “High Price” or “Low Price.” Payoffs (in millions of profit) are:

• For A, Low Price yields higher profit regardless of B’s choice (70 > 50 and 30 > 10).
• The same logic applies to B.
• Therefore, the dominant-strategy equilibrium is (Low, Low) with payoffs (30, 30).

Important: Every dominant-strategy equilibrium is a Nash equilibrium, but not every Nash equilibrium is supported by dominant strategies.

V. Definition and Identification of Nash Equilibrium

A Nash equilibrium is a strategy profile in which no player can improve its own payoff by unilaterally changing its strategy, given the strategies chosen by the other players. In other words, each player’s strategy is a best response to the strategies of the others.

Identification Procedure: 1. Examine every possible strategy combination in the payoff matrix. 2. For each combination, check whether any player would benefit by switching to a different strategy while holding others’ strategies fixed. 3. If no player has a profitable unilateral deviation, the combination is a Nash equilibrium.

Key Properties: - A Nash equilibrium need not be Pareto optimal (the classic prisoner’s dilemma). - There may be zero, one, or multiple pure-strategy Nash equilibria. - Mixed-strategy equilibria are possible when no pure-strategy equilibrium exists (beyond CFA Level I scope).

VI. The Prisoner’s Dilemma and Oligopoly Applications

The prisoner’s dilemma is the canonical example of a Nash equilibrium that produces a collectively inferior outcome. Two suspects choose “Confess” or “Deny.” Both rationally choose to confess even though mutual denial would yield a better joint result.

In economics, oligopolists’ decisions on price, advertising expenditure, and capacity expansion frequently exhibit prisoner’s-dilemma characteristics. Although joint profits would be higher under collusion, the inability to make credible commitments leads to the non-cooperative Nash outcome.

Worked Cases

Case 1: Classic Airline Pricing Dilemma

Two carriers, EastAir and SouthAir, simultaneously set “High” or “Low” fares. Payoffs (in billions of profit):

Solution: - EastAir’s best response: Low when SouthAir chooses High (12 > 8) and Low when SouthAir chooses Low (4 > 2). Low is therefore dominant for EastAir. - The identical logic holds for SouthAir. - Nash equilibrium: (Low, Low) with payoffs (4, 4). - Interpretation: Although both would prefer (High, High) with (8, 8), each has a unilateral incentive to undercut the rival. The cooperative outcome is not sustainable without enforceable commitment.

Case 2: Advertising Game Without Dominant Strategies (Modified)

Two beverage firms decide whether to launch a massive advertising campaign. Payoffs (in millions of profit):

Analysis: - For Coke, Advertise is better both when Pepsico advertises (60 > 30) and when it does not (80 > 50). Advertise is therefore dominant for Coke. - The same holds for Pepsico. - Nash equilibrium: (Advertise, Advertise) yielding (60, 60). - Note that even though mutual non-advertising yields (50, 50), the dominant-strategy logic still drives both to advertise. Slight changes to the matrix can eliminate dominant strategies and produce multiple pure-strategy Nash equilibria.

Case 3: Capacity Expansion Game

Two steel producers simultaneously decide whether to expand capacity. Annual profit payoffs (in billions):

Step-by-Step Solution: 1. Identify best responses. - If B expands, A’s best response is Expand (20 > 10). - If B does not expand, A’s best response is still Expand (35 > 30). - Expand is therefore dominant for A; the same holds for B. 2. Nash equilibrium: (Expand, Expand) with payoffs (20, 20). 3. Economic insight: Both firms rationally expand, creating industry overcapacity and lower profits than the mutually non-expansion outcome (30, 30). This illustrates why oligopolists often fail to achieve the jointly best result.

Traps

Key Formulas

• Dominant strategy for player i: strategy $s_i^$ such that for all opponent strategies $s_{-i}$ and all $s_i \ne s_i^$, $u_i(s_i^*, s_{-i}) > u_i(s_i, s_{-i})$.
• Nash equilibrium condition: strategy profile $(s_1^, s_2^)$ satisfies $u_i(s_i^, s_{-i}^) \ge u_i(s_i, s_{-i}^*)$ for every player i.
• Dominant-strategy equilibrium $\Rightarrow$ Nash equilibrium.
• Nash equilibrium is not necessarily Pareto optimal.
• In a payoff matrix the first number belongs to the row player, the second to the column player.

Practice Questions

Q1. In the airline pricing game above, is (High, High) a Nash equilibrium?
A. Yes, because joint payoffs are maximized
B. No, because either carrier can improve its payoff by switching to Low
C. Yes, because both carriers have dominant strategies
D. Cannot be determined

Q2. Which of the following is necessarily a Nash equilibrium?
A. A dominant-strategy equilibrium
B. A Pareto-optimal outcome
C. The cooperative solution
D. The subgame-perfect equilibrium of a sequential game

Q3. Two firms each have a dominant strategy to cut price, resulting in (Cut, Cut) with lower profits than mutual high prices. This situation is best described as:
A. A Pareto improvement
B. A prisoner’s dilemma
C. A mixed-strategy equilibrium
D. A Stackelberg equilibrium

Q4. In the steel capacity game (Case 3), if regulators credibly forbid any capacity expansion, the most likely equilibrium profit pair becomes:
A. (20, 20)
B. (30, 30)
C. (35, 10)
D. (10, 35)

Q5. In a simultaneous-move payoff matrix, if player A always prefers strategy X no matter what the opponent does, strategy X is A’s:
A. Nash strategy
B. Dominant strategy
C. Dominated strategy
D. Mixed strategy

Q6. Which statement about Nash equilibrium is least accurate?
A. Each player chooses a best response to the other players’ strategies
B. Multiple Nash equilibria may exist
C. It always maximizes the sum of the players’ payoffs
D. It can be a non-cooperative outcome

Q7. A game has two pure-strategy Nash equilibria: (High Ad, Low Ad) and (Low Ad, High Ad). This implies the game:
A. Contains dominant strategies for both players
B. Is not a prisoner’s dilemma
C. Has both players preferring the same equilibrium
D. Suffers from a coordination problem

Q8. In the airline game (Case 1), if the two carriers can sign and enforce a binding “maintain high fares” contract, the equilibrium outcome would most likely become:
A. Still (Low, Low)
B. (High, High)
C. Indeterminate
D. A mixed-strategy equilibrium

Answers

Takeaways

• The defining test of Nash equilibrium is whether any player can improve its own payoff by a unilateral deviation.
• Every dominant-strategy equilibrium is a Nash equilibrium, but the converse is not true.
• In a prisoner’s dilemma the Nash outcome is non-cooperative and Pareto inferior.
• Always label the row player’s payoffs first in each cell of the matrix.
• CFA exams frequently test simultaneous pricing, advertising, and capacity games; practice constructing matrices and locating every Nash equilibrium.
• Oligopolists often fail to sustain collusion precisely because the Nash incentive structure rewards cheating.