Game Theory · GT-29
A Nash equilibrium that also holds up in every possible sub-scenario of the game — not just along the path actually played.
A refinement of Nash equilibrium requiring that players' strategies constitute a Nash equilibrium not only for the whole game, but for every possible subgame (every point the game could reach, even ones that end up unplayed) — ruling out equilibria that only 'work' because they rely on threats or promises nobody would actually follow through on if called.
Introduced by Reinhard Selten in his 1965 paper, work that later contributed to his shared 1994 Nobel Memorial Prize in Economic Sciences alongside John Nash and John Harsanyi.
The Mechanism
A threat that looks credible in the whole game, but fails in one branch
The subgame beginning at the incumbent's decision node reveals the flaw: if entry actually happens, fighting yields -5 while accommodating yields +2 — accommodation is strictly better for the incumbent at that point, so the 'always fight' threat is not credible, even though it deters entry as a Nash equilibrium of the overall game.
01 · IT RULES OUT INCREDIBLE THREATS AND PROMISES
The core refinement over plain Nash equilibrium
A strategy profile can be a Nash equilibrium of the whole game purely because a threat, if believed, successfully deters the other player from ever reaching the subgame where the threat would need to be carried out — subgame perfection asks the harder question: would the threatened action actually be optimal if that subgame were reached anyway? If not, it's not subgame perfect, however well it deters in the full game.
02 · IT'S FOUND VIA BACKWARD INDUCTION IN FINITE GAMES
The two concepts are operationally linked
In finite, perfect-information games, backward induction is precisely the method used to find the subgame-perfect equilibrium — solving from the last node backward automatically filters out any strategy that wouldn't actually be optimal if that later subgame were reached, which is exactly what subgame perfection requires.
03 · IT REVEALS WHY COMMITMENT DEVICES ARE STRATEGICALLY VALUABLE
The 'incredible threat' problem is exactly what commitment solves
If the incumbent could somehow make fighting genuinely costly to avoid (a public reputation commitment, an inflexible pricing policy locked into a contract), fighting would become the actual best response in that subgame, converting an incredible threat into a credible, subgame-perfect one — this is precisely the strategic logic behind commitment devices.
Where It Fails / Inversion
Where it fails / inversion
Subgame perfection assumes players can correctly anticipate and reason through every possible subgame, including ones vanishingly unlikely to ever be reached — in practice, opponents sometimes genuinely can't distinguish a credible threat from an incredible one in real time, meaning an 'incredible' threat can still deter in practice even though formal theory says it shouldn't, simply because the threatened party isn't confident enough to call the bluff.
How To Use It
Worked example · calling out an incumbent's bluff
A market incumbent threatens to slash prices and fight any new entrant into its territory. A subgame-perfect analysis asks: if entry actually happens, is fighting really the incumbent's best move, or would accommodating (given the entry has already occurred and can't be undone) actually preserve more of the incumbent's profit? If fighting would genuinely hurt the incumbent more than accommodating once entry has occurred, a sophisticated entrant should recognize the threat as non-credible and enter anyway — unless the incumbent has taken some prior, verifiable step (excess capacity investment, a public pricing commitment) that makes fighting the incumbent's actual best response once entry occurs.
How to use it
Before letting any threat or promise from a counterpart change your strategy, don't just ask whether it would work if believed — ask whether the other party would actually want to follow through on it if you called their bluff and forced the scenario to occur. If the answer is no, the threat isn't subgame perfect, and rational analysis says you can safely ignore it.
See Also