ECON90034 Chap.6 Sequential Games and Subgame Perfection
Sequential Games and Subgame Perfection
When players move in turn, the game is drawn as a tree and solved by backward induction. A strategy must name an action at every node a player owns, even nodes never reached, which is why counting strategies and building the strategic form are standard exam tasks.
Some Nash equilibria of a sequential game rest on incredible threats, actions a player would not take if the node were reached; subgame perfection removes them by requiring a Nash equilibrium in every subgame. The review lecture lists entry deterrence and a two-period battle of the sexes, the difference between simultaneous and sequential games, and drawing a simultaneous game as a tree with information sets.
Its worked example is an entry game with a second chance to quit, solved both by backward induction and through its full strategic form.
What this chapter covers
- 01
Game trees and the order of payoffs
- 02
Backward induction and the equilibrium path
- 03
Complete strategies and counting them
- 04
Representing a sequential game in strategic form
- 05
Finding all Nash equilibria and identifying the subgame perfect ones
- 06
Subgames and incredible threats
- 07
Entry deterrence and commitment
- 08
Simultaneous games drawn as trees: information sets
Worked example · free
Backward induction in a pricing game
- 1After High, Firm 2 compares 4 with 6 and chooses Low.
- 1After Low, Firm 2 compares 2 with 3 and chooses Low.
- 1Firm 1 compares High, which now pays 1, with Low, which pays 2, and chooses Low.
- 1SPNE: Firm 1 plays Low; Firm 2 plays (Low after High, Low after Low). Path Low then Low, payoffs (2, 3).
- 1Firm 1 has 2 strategies; Firm 2 has two nodes with two actions each, so 2 × 2 = 4.
Key terms
- Backward induction
- Solving a game tree from the last moves to the first, choosing the best action at each node given what follows.
- Incredible threat
- A promised action that the player would not actually carry out if the node were reached, because another action pays more there.
- Information set
- A group of nodes that a player cannot tell apart when choosing, drawn joined by a dashed line; it turns a tree into a simultaneous-move game.
- First-mover advantage
- The gain a player obtains by committing to an action before the others, so that their best replies work in the first mover's favour.
- Subgame
- The part of a game tree that starts at a single decision node and includes everything that follows it; the whole game is also a subgame.
Sequential Games and Subgame Perfection FAQ
Why must a strategy include actions at nodes that are never reached?
Whether a node is reached depends on the other players' strategies, so a complete plan must say what the player would do everywhere. Those off-path actions are exactly where incredible threats hide.
Can a Nash equilibrium of a sequential game fail to be subgame perfect?
Yes. A profile can be a Nash equilibrium of the whole game while prescribing an action in some unreached subgame that is not a best reply there; backward induction rules such profiles out.
How do you count a player's strategies in a tree?
Multiply the number of actions at each node the player owns. A player with two nodes and two actions at each has four strategies, and the strategic form has one row or column for each.
What is entry deterrence?
An incumbent threatens to fight an entrant. If fighting after entry pays the incumbent less than accommodating, the threat is not credible and the entrant enters; commitment that changes those payoffs can make it credible.
What is the difference between an equilibrium, a path and an outcome?
The equilibrium is the full list of strategies, including actions at unreached nodes. The path is the sequence of moves actually played. The outcome is where the path ends, and the payoffs are the numbers attached to it. Exam answers should state each one separately.
Exam move
For every tree, do three things in order: solve it by backward induction and circle the chosen action at every node; count each player's strategies and write the strategic form; then find all Nash equilibria and say which are subgame perfect and why the others fail. Practise writing the answer as strategies, path and payoffs, because the exam asks for all three.
Time yourself: a two-level tree should take under five minutes from backward induction to the full list of Nash equilibria.
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