So far, we've looked at games where players make their moves simultaneously, without knowing what the other will do. We use a payoff matrix to map out these one-shot decisions. But many strategic situations aren't like that. Decisions often unfold over time, with one player acting, and others responding. These are called sequential-move games.

To analyze these, we move from a flat matrix to a branching structure called a game tree, or an extensive form game. A game tree shows the sequence of moves, the information available to each player at each decision point, and the final payoffs. The root of the tree is the starting point, the branches are the possible actions, and the nodes are the decision points.

How do we solve this? We use a powerful and intuitive technique called backward induction. Instead of starting at the beginning, we start at the end. We look at the final decisions in the game and figure out what the rational player would do in each case. Then, we work our way back up the tree, assuming that players will make these rational choices at every step.

Backward induction helps us find a clear path through the game. It also highlights a critical issue in sequential games: non-credible threats. A non-credible threat is a move that a player threatens to make, but which they would have no rational incentive to actually carry out if the situation arose.

Think of it this way: some outcomes that might appear as Nash Equilibria in a matrix don't hold up under the scrutiny of sequential logic. A player might threaten a disastrous move to scare another player into a certain action. But if that threat would also harm the threat-maker, it's not believable. Backward induction systematically eliminates these empty threats.

This brings us to the Subgame Perfect Equilibrium (SPE). An SPE is a refined version of a Nash Equilibrium. It's a set of strategies for each player that is a Nash Equilibrium not just for the overall game, but for every single subgame within it. In essence, an SPE requires every player's strategy to be optimal at every stage of the game, no matter what has happened before. It's a plan that's always rational, with no room for non-credible threats.

Let's apply this to a classic business scenario: market entry. Imagine an incumbent company, 'MonoCorp', is the sole provider in a market. A potential competitor, 'EntraCorp', is deciding whether to enter the market.

EntraCorp moves first: it can either 'Enter' or 'Stay Out'. If it stays out, MonoCorp continues to enjoy its monopoly profits. If EntraCorp enters, MonoCorp must then decide how to react: it can either 'Accommodate' the new rival (and share the market) or 'Fight' by starting a price war.

Before EntraCorp makes its move, MonoCorp might issue a threat: "If you enter, we will start a price war and drive you out!" Let's use backward induction to see if this threat is credible.

Start at the end, with MonoCorp's decision. If EntraCorp has already entered, MonoCorp can either 'Fight' and get a payoff of -2, or 'Accommodate' and get a payoff of 5. The rational choice for MonoCorp is to Accommodate.

Now, let's go back to EntraCorp's decision. EntraCorp, being rational, anticipates MonoCorp's rational response. It knows that if it enters, MonoCorp will choose to accommodate. Therefore, EntraCorp compares the payoff of 'Staying Out' (0) with the payoff of 'Entering' (5). It will choose to Enter.

By analyzing sequential games and focusing on subgame perfection, we can make much more realistic predictions about strategic outcomes. We move beyond what players say they will do and focus on what they have an incentive to do when it's their turn to act.