Subgame Perfect Equilibrium

students, in dynamic games, players often move one after another instead of all at once. That means a strategy can promise one thing at the start and something different later depending on what actually happens. 🎯 The big question is: which threats and promises are believable? In this lesson, you will learn how subgame perfect equilibrium helps us remove non-credible behavior from equilibrium analysis.

What you will learn

By the end of this lesson, you should be able to:

define a subgame perfect equilibrium;
check whether a Nash equilibrium is subgame perfect;
apply this refinement to dynamic games.

Think of a business negotiation, a pricing war, or a take-it-or-leave-it offer. In these situations, a player may threaten to do something costly just to influence another player’s choice. Subgame perfection helps us ask whether that threat would really be carried out once the game reaches that point. 🧠

Dynamic games and the problem of credibility

In a static game, everyone chooses at the same time. In a dynamic game, players move in sequence, and later players observe earlier actions before choosing their own. Because of this, a strategy must tell a player what to do not just at the start, but after every possible history of play.

Here is the key issue: a Nash equilibrium only requires that no player wants to deviate from the equilibrium path overall. It does not always require that the planned actions remain optimal after every possible turn of events. That can lead to non-credible threats.

For example, suppose a player says, “If you enter my market, I will start a price war even though it hurts me.” If starting the price war would be bad for that player once entry has happened, then the threat is not credible. A smart entrant should not believe it. Subgame perfection fixes this by requiring optimal behavior in every part of the game where decisions still matter.

A good way to remember this is: a believable strategy must still make sense after the game gets to any later decision point. ✅

What is a subgame?

A subgame is a part of a dynamic game that begins at a decision node and includes everything that happens afterward. The subgame must contain all the later actions and information sets that follow from that node. In simple words, it is a “game inside the game.”

If a player must make a decision at a certain point, we can look at the rest of the game starting there. The strategy should be optimal not just at the beginning, but also in that smaller game.

This matters because future choices can depend on past choices. A strategy that seems acceptable from the beginning may become unreasonable once the game reaches a later stage. Subgame perfection checks every such stage.

Definition of subgame perfect equilibrium

A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium in the original game and in every subgame of that game.

More formally, a strategy profile is subgame perfect if, for every subgame, the restriction of the strategy profile to that subgame is a Nash equilibrium of that subgame.

This means:

each player’s strategy must be best response behavior after every history that starts a subgame;
every promised action must be credible because it must also be optimal later on.

Subgame perfection is a refinement of Nash equilibrium. That means every subgame perfect equilibrium is a Nash equilibrium, but not every Nash equilibrium is subgame perfect. 🔍

How to check whether a Nash equilibrium is subgame perfect

students, there is a practical method for checking subgame perfection:

Find the subgames. Identify every decision point where a subgame can begin.
Solve the game from the end backward. Start with the last decision and determine the best action there.
Work backward step by step. At each earlier decision, compare payoffs from the available actions, knowing how later players will behave.
Compare with the candidate Nash equilibrium. If the proposed strategies are optimal in every subgame, then the equilibrium is subgame perfect.

This backward approach is called backward induction. It works especially well in finite games with perfect information, where players move one at a time and everyone sees previous moves.

Backward induction is useful because it forces each decision to be believable at the moment it is made. If a player would not actually carry out a threat later, then that threat should not shape the earlier play.

Example 1: entry and deterrence

Imagine a market entry game.

Player 1 is an incumbent firm.
Player 2 is a potential entrant.
First, the entrant decides whether to enter.
If the entrant stays out, both players get moderate payoffs.
If the entrant enters, the incumbent chooses whether to fight with a price war or accommodate the entrant.

Suppose the incumbent claims: “If you enter, I will fight.” The entrant stays out because of this threat. Could that be a Nash equilibrium? It might be, if the entrant’s best response to the threat is to stay out.

But now look at the subgame after entry occurs. Once entry has happened, the incumbent compares payoffs from fighting and accommodating. If accommodating gives the incumbent a higher payoff than fighting, then fighting is not optimal. That means the threat to fight is not credible.

So even if staying out was part of a Nash equilibrium, the profile is not subgame perfect if the incumbent’s post-entry action is not a best response in the subgame. The equilibrium fails because it depends on behavior that the incumbent would not actually choose later. 🚫

This is a classic reason subgame perfection is important: it removes threats that only work because of an unrealistic promise.

Example 2: backward induction in a simple sequential game

Consider a game with two players.

Player 1 chooses $A$ or $B$.
If Player 1 chooses $A$, then Player 2 chooses $C$ or $D$.
If Player 1 chooses $B$, the game ends.

Suppose the payoffs are:

if $A$ then $C$, payoffs are $(2,1)$;
if $A$ then $D$, payoffs are $(0,3)$;
if $B$, payoffs are $(1,0)$.

To find a subgame perfect equilibrium, start with Player 2’s decision after $A$. Player 2 compares $1$ from $C$ with $3$ from $D$, so Player 2 chooses $D$.

Knowing this, Player 1 compares:

choosing $A$ leads to payoff $0$ for Player 1 because Player 2 will choose $D$;
choosing $B$ gives payoff $1$.

So Player 1 chooses $B$.

The subgame perfect equilibrium is therefore: Player 1 chooses $B$, and Player 2 chooses $D$ if $A$ is reached. This outcome is believable because Player 2’s action after $A$ is optimal in the subgame.

Notice something important: if Player 2 had threatened to choose $C$ after $A$ even though $D$ is better, that threat would not be credible. Subgame perfection rules it out. ✅

Why subgame perfection matters in economics and strategy

Subgame perfection is widely used because many real situations involve sequential decisions.

Examples include:

firms setting prices one after another;
countries deciding on trade retaliation;
negotiators making offers and counteroffers;
auctions with sequential bidding;
courts, regulators, or political actors responding after earlier choices.

In all these settings, people care about what others will really do after seeing earlier actions. If a threat is not credible, it should not affect rational choices.

Subgame perfection improves predictive power because it helps eliminate equilibria that survive only because of empty threats. It also helps explain why some outcomes are unstable once players think carefully about later incentives.

For instance, a firm may announce harsh retaliation to scare competitors away. But if retaliation would hurt the firm more than accommodation, a rational competitor should realize the threat is empty. Subgame perfection captures this logic precisely.

Relationship to Nash equilibrium and limitations

Every subgame perfect equilibrium is a Nash equilibrium, but the reverse is not true.

Why? Because a Nash equilibrium only checks whether any player wants to deviate from the proposed strategy profile at the start of the game. It does not fully check what happens after every possible history. A player may have no incentive to deviate in the overall game, but still have an incentive to choose differently in a later subgame.

Subgame perfection is a stronger requirement, so it removes more equilibria. That is helpful, but it also means some Nash equilibria are excluded even if they seem plausible at first glance.

There is one important limitation to remember: subgame perfection works best in games where subgames are well defined. In games with information sets that mix several decision nodes, some parts of the game may not form proper subgames. In those cases, other refinements may be needed.

Still, for many finite sequential games, subgame perfection is the standard tool for testing credibility. 📘

Conclusion

students, subgame perfect equilibrium is a powerful way to analyze dynamic games because it requires rational behavior everywhere in the game, not just at the beginning. It helps you identify whether threats and promises are believable, and it is usually found using backward induction.

The main idea is simple but important: if a player would not actually follow through later, then that action should not shape earlier decisions. By checking every subgame, you refine Nash equilibrium into something more realistic and more useful for predicting strategic behavior.

Study Notes

A subgame is a part of a dynamic game that starts at a decision node and includes all later play.
A subgame perfect equilibrium is a strategy profile that is a Nash equilibrium in every subgame.
Subgame perfection is a refinement of Nash equilibrium, so every subgame perfect equilibrium is a Nash equilibrium.
It removes non-credible threats, meaning promises that would not be optimal when the time comes.
Backward induction is the standard method for solving finite sequential games with perfect information.
To check subgame perfection, solve from the end of the game backward and verify that every action is optimal in its subgame.
In market entry, bargaining, pricing, and retaliation games, subgame perfection helps determine whether threats are believable.
A strategy can be part of a Nash equilibrium and still fail to be subgame perfect if it relies on later actions that are not rational.
Subgame perfection is especially useful in dynamic games where players observe earlier moves and respond strategically.
The core lesson: credible behavior must be optimal not only at the start, but also after every possible history of play.