Signalling Entry Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Signalling Entry Game

• Players: An incumbent firm (P1), and an entrant firm (P2).
• Strategies:
  • Nature chooses the incumbent’s marginal cost $mc_1\in\{5,15\}$ with probabilities $\alpha$ and $1-\alpha$.
  • The incumbent chooses $p_1 \in \{15, 20\}$.
  • The entrant observes the price and chooses In or Out.
• Rules:
  • The incumbent has marginal cost $5$ or $15$ and fixed cost $0$.
  • The entrant has marginal cost $10$ and fixed cost $40$.
  • If entry does not occur, the incumbent firm has the full market quantity.

• If entry occurs, firms play a Cournot duopoly.

Game Tree

Payoff Details

Round 1

• Low-cost incumbent monopoly profit

• High-cost incumbent monopoly profit
  • If $p=15$:

• If $p=20$:

Round 2

• Considering the profits in a duopoly only.
• Low-cost incumbent duopoly profits

For P1 (incumbent):

Solving for FOC:

For P2 (entrant):

Solving for FOC

Equating the two quantities

• High-cost incumbent duopoly profits

For P1 (incumbent):

Solving for FOC:

For P2 (entrant):

Solving for FOC:

Equating the two quantities:

Derivation (Best Response Analysis)

Separating Equilibrium

• Candidate incumbent strategy:

• Beliefs after observing price:

• Given these beliefs, the entrant compares net entry payoffs:
  • After $p=15$: $-29<0$, so choose Out
  • After $p=20$: $5>0$, so choose In
• Hence the entrant strategy is

• Low-cost Incumbent Deviation:
  • Following the candidate signal $p=15$ induces Out, giving $100+100=200$.
  • Deviating to $p=20$ would induce In, so the low-cost incumbent would not improve.
• **High-cost Incumbent Deviation: **
  • Following the candidate signal $p=20$ induces In, giving $25+3=28$.
  • Deviating to $p=15$ would induce Out, giving only $0+25=25$.

Pooling Equilibrium

• Candidate incumbent strategy:

• If the entrant observes $p=15$:
  • The entrant keeps the prior belief:

• Expected payoff from In is

• So entry after $p=15$ is optimal if

• If the entrant observes $p=20$:

  • The entrant infers the high-cost incumbent.
  • Since entry against the high-cost incumbent gives $5>0$, the entrant chooses In.

• The high-cost incumbent compares:

  • Pooling at $p=15$ with entry: $0+3=3$
  • Pooling at $p=15$ with no entry: $0+25=25$
  • Deviating to $p=20$, which induces entry: $25+3=28$
  • The relevant deviation check is $28>3,\qquad 28>25$

• Hence the high-cost signal prefers to deviate, so pooling is not an equilibrium.

Semi-separating Equilibrium

• A semi-separating equilibrium would require the high-cost incumbent to mix between $p=15$ and $p=20$.
• But for the high-cost incumbent, $p=20$ is strictly dominant.
• Therefore the high-cost signal cannot be indiferent between the two prices.
• So no semi-separating equilibrium exists.

Nash Equilibrium

Result:

The signalling game has a separating equilibrium:

There is no pooling equilibrium and no semi-separating equilibrium.

Social Optimum

• Entry is inefficient when the incumbent is low cost because the entrant’s net payoff is negative.
• Entry is efficient when the incumbent is high cost because the entrant’s net payoff is positive.
• The separating equilibrium implements the efficient entry decision:

Insights

Insight:

• Price acts as a signal because diferent cost signals prefer diferent monopoly prices.
• Separating equilibrium works when beliefs make entry unattractive against the low-cost signal and attractive against the high-cost signal.
• Pooling fails because the high-cost incumbent has a profitable deviation.
• No semi-separating equilibrium exists when one signal strictly prefers one signal.