Sequential Entry-Deterrence Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Sequential Entry-Deterrence Game

• Players: Entrant and Incumbent.
• Strategies: Entrant chooses Out or Enter; Incumbent chooses Accommodate or Fight after entry.
• Rules: Players move sequentially and optimal play is found by backward induction.

Game Tree

Payoff Details

• If Out: $(3,0)$.
• If Enter and Accommodate: $(2,2)$.
• If Enter and Fight: $(1,-1)$.

Derivation (Backward Induction)

• If entry occurs, incumbent compares $2$ (accommodate) vs $1$ (fight), so chooses Accommodate.

• Anticipating accommodation, entrant compares $2$ (enter) vs $0$ (out), so chooses Enter.

Nash Equilibrium

Result:

The subgame perfect Nash equilibrium strategy profile is:

The equilibrium outcome payoffs are $(2,2)$.

Nash Equilibrium vs.\ SPNE

• The strategy profile {Out,Fight} is also a Nash equilibrium.
  • Given Fight after entry, the entrant prefers Out because $0>-1$.
  • Given Out, the incumbent’s action after entry is off the equilibrium path, so deviating from Fight to Accommodate does not change the realised payoff.
  • Hence no player gains from a unilateral deviation at that strategy profile, so Nash equilibrium allows {Out,Fight}.
• But it is not subgame perfect because, in the subgame after Enter, the incumbent prefers Accommodate since $2>1$.

Result:

The associated equilibria are:

• $\{\text{P1}, \text{P2} \} \mapsto \{ \text{Enter}, \text{Accommodate} \}$ with a payoff of $(2,2)$.
• $\{\text{P1}, \text{P2} \} \mapsto \{ \text{Out}, \text{Fight} \}$ with a payoff of $(0,3)$.

Only {Enter,Accommodate} is SPNE.

Insights

Insight:

A threat to Fight may appear in a Nash equilibrium even when it is not credible.

SPNE removes such equilibria by requiring optimal play in every subgame, not just along the realised path.