Signalling Entry Game with Modified Low Cost

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Signalling Entry Game with Modified Low Cost

• Players: An incumbent firm (P1), and an entrant firm (P2).
• Strategies:
  • Nature chooses the incumbent’s marginal cost $mc_1\in\{10,15\}$ with probabilities $\alpha$ and $1-\alpha$.
  • The incumbent chooses $p_1\in\{17.5,20\}$.
  • The entrant observes the price and chooses In or Out.
• Rules:
  • The incumbent has marginal cost $10$ 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 $p=17.5$:

Using the notes’ rounded branch label:

• 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 $p=17.5$: $-15<0$, so choose Out
  • After $p=20$: $5>0$, so choose In
• Hence the entrant strategy is

• High-cost incumbent deviation:
  • Following the candidate signal $p=20$ induces In, giving $25+3=28$.
  • Deviating to $p=17.5$ would induce Out, giving $19+25=44$.
• Since $44>28$, the high-cost incumbent has a profitable deviation.
• Therefore the separating strategy is not an equilibrium.

Pooling Equilibrium

• Candidate incumbent strategy:

• If the entrant observes $p=17.5$:
  • The entrant keeps the prior belief:

• Expected payoff from In is

• So staying out after $p=17.5$ 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 $p=17.5$ with no entry: $19+25=44$
  • Deviating to $p=20$, which induces entry: $25+3=28$

• Since $44>28$, the high-cost incumbent does not deviate when the entrant stays out after $p=17.5$.

• Therefore pooling exists when $\alpha\geq\frac{1}{4}$.

Semi-separating Equilibrium

• Terms:

  • $\alpha=P(c_I=10)$ is the prior probability that the incumbent is low cost.
  • $q=P(p=17.5\mid c_I=15)$ is the probability that the high-cost incumbent mimics the low-cost type.
  • $\mu=P(c_I=10\mid p=17.5)$ is the entrant’s posterior belief that the incumbent is low cost after seeing $p=17.5$.
  • $x=P(\text{In}\mid p=17.5)$ is the probability that the entrant enters after seeing $p=17.5$.

• Candidate semi-separating strategy:

  • The low-cost incumbent chooses $p=17.5$ for sure.
  • The high-cost incumbent chooses $p=17.5$ with probability $q$ and $p=20$ with probability $1-q$.
  • For genuine semi-separating, $q\in(0,1)$.

• If $p=20$ is observed, the entrant knows the incumbent is high cost, because only the high-cost type chooses $p=20$ in the candidate strategy.

• Since entry against the high-cost incumbent gives $45-40=5>0$, the entrant chooses In after $p=20$.

• If $p=17.5$ is observed, the entrant updates its belief by Bayes’ rule:

• The numerator is the probability that the incumbent is low cost and chooses $p=17.5$:

• The denominator is the total probability of observing $p=17.5$:

• Therefore

• Entrant indifference pins down $\mu$:
  • If the entrant enters after $p=17.5$, expected payoff is

• If the entrant stays out, payoff is $0$.
• The entrant mixes only if it is indifferent:

• Belief indifference pins down $q$: Substitute $\mu=\frac{1}{4}$ into Bayes’ rule:

so

• Since genuine semi-separating requires $q\in(0,1)$:

• High-cost incumbent indifference pins down $x$:
  • If the high-cost incumbent chooses $p=20$, the entrant enters, giving

• If the high-cost incumbent chooses $p=17.5$, the entrant enters with probability $x$ and stays out with probability $1-x$, giving

• The high-cost incumbent mixes only if it is indifferent:

Nash Equilibrium

Result:

A pooling equilibrium exists when $\alpha\geq\frac{1}{4}$ and is given by

• Incumbent:

• Entrant:

A semi-separating equilibrium exists when $\alpha<\frac{1}{4}$ and is given by

• Incumbent:

• Entrant:

• Posterior beliefs of the entrant are:

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:

• Pooling hides the incumbent’s type at $p=17.5$.
• Semi-separating reveals partial information and makes the entrant mix after the pooled signal.

Insights

Insight:

• Raising the low-cost type from $5$ to $10$ changes the equilibrium set: pooling and semi-separating can now exist.
• Pooling works when the prior probability of the low-cost incumbent is large enough to make entry unattractive after $p=17.5$.
• Semi-separating requires two indifference conditions:
  • The entrant must be indifferent after $p=17.5$.
  • The high-cost incumbent must be indifferent between $p=17.5$ and $p=20$.
• The posterior belief $\mu=\frac{1}{4}$ is pinned down by the entrant’s indifference condition.