- Content map: SMU H3 Game Theory Map
Chapter 6: Combining Sequential and Simultaneous Moves
Games with both Sequential and Simultaneous Moves
Definition:
A sequential game can contain proper subgames, and some of those subgames may themselves be simultaneous-move games written in normal form.
- The key tasks are to identify subgames, solve them, and then work backward through the overall tree.
- Timing matters because changing the order of play can change which equilibrium is selected.
Subgames
Definition:
A subgame is a part of an extensive-form game that begins at a single decision node and contains that node together with all its successors. The full game is a subgame of itself.
- A subgame must be a self-contained decision problem.
- It cannot cut across an information set or omit later branches.
- Counting subgames is the first step before solving the game.
Insight:
Subgames isolate continuation problems: once a node is reached, players face a smaller game that must still be solved rationally.
Subgame Perfect Nash Equilibrium
Definition:
A subgame perfect Nash equilibrium is a pair of strategies, one for each player, that forms a Nash equilibrium in every subgame.
- SPNE is the natural refinement of rollback equilibrium.
- It requires sequential rationality after every possible history, not only along the realised path.
- Every SPNE is a Nash equilibrium, but some Nash equilibria fail this stronger test.
Result:
SPNE rules out equilibria supported by non-credible continuation behaviour.
Backward Induction over Subgames
Definition:
When a later stage is itself a game, backward induction replaces that subgame by its Nash-equilibrium outcome and then moves backward to earlier decisions.
- Solve each terminal subgame first.
- Replace each solved subgame by its equilibrium payoff.
- Use the reduced game to analyse the earlier stage.
- If a subgame has multiple Nash equilibria, the whole game may have multiple SPNE.
Insight:
Rollback with subgames is the same logic as ordinary backward induction, except the continuation object is a game rather than a single move.
Writing Strategies in Sequential Games
Definition:
In an extensive-form game, a strategy is a complete contingent plan specifying an action at every node a player could face.
- SPNE is a profile of full strategies, not just an equilibrium path.
- Off-path actions still matter because they determine whether continuation play is rational in every subgame.
- As the tree grows, the number of complete strategies can become very large.
Normal Form versus Extensive Form
Definition:
The normal-form representation of an extensive game lists all complete contingent strategies and the payoff associated with every strategy profile.
- Normal form is useful because it reveals all Nash equilibria of the game.
- Extensive form is sharper because it shows which of those equilibria are credible in every continuation.
- A large normal form can contain many Nash equilibria even when only a few are subgame perfect.
Insight:
Normal form is complete about strategies; extensive form is sharper about credibility.
Changing Order of Play
Definition:
Changing the order of play changes who can commit first and who gets to respond after observing earlier actions.
- Sequential play can select a different outcome from the simultaneous version of the same strategic problem.
- First-mover advantage may arise when early action credibly shapes later incentives.
- Second-mover advantage may arise when waiting preserves flexibility.
- Hybrid games can also inherit multiple SPNE when the continuation subgame has multiple equilibria.
Result:
Timing is strategically relevant because equilibrium depends not only on payoffs, but also on who moves when and what they observe.