Sequential Entry Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Sequential Entry Game

• Players: Two players, Player 1 (Entrant) and Player 2 (Incumbent).
• Strategies: Player 1: Invest / Don’t Invest; Player 2: High Price / Low Price (after entry).

Rules

• Start with an entrant deciding whether to enter a market.
• Players move sequentially: the entrant chooses Invest or Don’t Invest, then the incumbent chooses High Price or Low Price after entry.
• The player who chooses optimally at each reached subgame reaches the subgame-perfect outcome.
• Player 1 moves first
• Player 2 observes and responds

Game Tree

SG1: after (Invest,Invest)

SG2: after (Invest,Don’t)

SG3: after (Don’t,Invest)

• Reduced first-stage payoffs:

Derivation (Best Response Analysis)

SG1

• P1
  • If P2 chooses High, then Low gives $6>2$.
  • If P2 chooses Low, then Low gives $-2>-10$.
• P2
  • If P1 chooses High, then Low gives $6>2$.
  • If P1 chooses Low, then Low gives $-2>-10$.
• Nash equilibrium: (Low,Low) with payoff $(-2,-2)$.

SG2

• P1
  • High gives $14$.
  • Low gives $6$.
• Nash equilibrium / outcome: High with payoff $(14,0)$.

SG3

• P2
  • High gives $14$.
  • Low gives $6$.
• Nash equilibrium / outcome: High with payoff $(0,14)$.

Nash Equilibrium

Orange highlights the SPNE path (Invest,Don’t,High).

Green highlights the SPNE path (Don’t,Invest,High).

Result:

SPNE strategy profiles:

1. SPNE 1 with payoff $(14,0)$

2. SPNE 2 with payoff $(0,14)$

Social Optimum

• Total payoff is $-4$ at (Invest,Invest), $14$ at (Invest,Don’t), $14$ at (Don’t,Invest), and $0$ at (Don’t,Don’t).
• So the socially optimal outcomes are (Invest,Don’t) and (Don’t,Invest).

Insights

Insight:

• Solve the pricing subgames first, then replace them by continuation payoffs in the first-stage game.
• If both firms invest, simultaneous pricing drives both to Low price and payoff $(-2,-2)$.
• The overall game has multiple SPNE because the reduced first-stage game has two asymmetric equilibria.