Subgame Perfect Nash Equilibrium: Theory and Cournot Insights

Nash equilibrium can contain non‑credible threats—strategies that would never be optimal if the game ever reached the threatened node. Backward induction resolves this problem in perfect‑information games, but it fails when information is imperfect or the game is infinite. Subgame Perfect Nash Equilibrium (SPNE) extends backward induction by demanding optimal play in every subgame, thereby removing non‑credible threats while agreeing with backward induction in games of perfect information.

Defining Subgames

A subgame consists of a single decision node together with all its successor nodes. The initial node must lie in a singleton information set, ensuring that no information set is split between the inside and outside of the subgame. By definition, the entire game itself qualifies as a subgame.

Solving for SPNE

The standard “recipe” for finding an SPGE is:

1. Identify every subgame in the extensive‑form representation.
2. Work backward from the smallest subgames, solving each as a separate Nash game.

If a subgame yields multiple Nash equilibria, each candidate must be tested in the larger game because every SPNE must induce a Nash equilibrium in every subgame. Consequently, every SPNE is a Nash equilibrium, but not every Nash equilibrium satisfies the subgame‑perfection requirement.

Application to Cournot Competition with Entry Costs

Consider a two‑stage industry model.

Stage 1 – Firms simultaneously decide whether to incur a fixed entry cost (F).

Stage 2 – Firms that entered compete in a Cournot quantity game.

For a given number of entrants (k), the Cournot equilibrium yields profit per firm

[ \pi_k^{*}= \left(\frac{1-c}{k+1}\right)^{2}, ]

quantity per firm

[ q_i^{*}= \frac{1-c}{k+1}, ]

and total industry output

[ Q = \frac{k}{k+1}(1-c). ]

Entry equilibrium occurs when the fixed cost (F) lies between the profit that the ((k+1)^{\text{st}}) firm would earn and the profit of the (k^{\text{th}}) firm:

[ \pi_{k+1}^{} < F \le \pi_{k}^{}. ]

If (F) is high, only a few firms find entry worthwhile, leading to monopoly or oligopoly structures. When (F) is low, many firms can profitably enter, pushing the market toward more competitive outcomes. This logic illustrates how SPNE captures credibility: firms will not threaten entry or exit unless the associated profit calculations make the threat optimal.

Mechanisms Behind Credibility

Subgame perfection forces each player’s strategy to be a Nash equilibrium in every subgame, which eliminates any threat that would be irrational to carry out at the relevant node. In the Cournot entry example, the “threat” of additional entrants is credible only when the profit from being the next entrant exceeds the entry cost.

“Subgame perfect Nash equilibrium… is going to capture the idea of credibility, but it will apply to a larger class of games than backward induction did.”

“Industries that have large fixed costs will tend to have fewer firms in them.”

These statements underscore that SPNE refines equilibrium predictions by embedding the requirement of optimal behavior at every possible contingency.