23506 Chap.8 SPE Applications: Stackelberg, First-Mover Advantage and Nature's Move
SPE Applications: Stackelberg, First-Mover Advantage and Nature's Move
Stackelberg quantity competition uses the Cournot demand and cost primitives but changes timing. Firm A commits first; Firm B observes and responds. The follower’s strategy is not one output but a response after every possible leader quantity. For leader quantities 0,1,2,3, the complete best-response tuple is (3,2,2,1). The leader substitutes that reaction schedule into its own profit and chooses output 3, inducing follower output 1.
The complete SPE is (3,(3,2,2,1)), while the equilibrium path is (3,1). Price is 6 and profits are (4.5,1.5), compared with Cournot profits (3,3). Commitment shifts profit toward the leader even though the follower observes more before acting. This first-mover advantage depends on credible irreversible commitment and strategic substitutes; chronology alone does not guarantee advantage. Off-path response entries validate the leader’s deviation comparison.
Nature introduces fixed exogenous probabilities rather than strategic choices. Before a state is observed, players compare expected continuation payoffs; after observation, they use realised payoffs. Insurance is weakly preferred when accident probability α is at least 2/9. In the oil-well application, the follower’s no-prior-investment cutoff is 1/5 and the leader’s initial-investment cutoff is 2/7. Equality creates tied best responses and potentially several complete SPE profiles, so parameter intervals must preserve boundary cases.
What this chapter covers
- 01Simultaneous continuations inside subgames
- 02Stackelberg timing and commitment
- 03Follower response tuple (3,2,2,1)
- 04Leader output 3 and path (3,1)
- 05Nature, expectation and insurance
- 06Oil-well thresholds 1/5 and 2/7
Complete Stackelberg SPE
- 2Calculate follower profit for every response after each leader quantity.
- 2Record the complete follower best-response tuple.
- 2Substitute each induced response into leader profit.
- 2Choose the leader argmax and trace the path.
- 2Report complete SPE, price and both profits.
Key terms
- Stackelberg leadership
- Sequential quantity competition with observed, credible leader commitment.
- Reaction plan
- A follower action specified after every possible leader choice.
- First-mover advantage
- A higher equilibrium payoff arising here from commitment that changes the follower response.
- Nature
- An exogenous random mechanism with fixed probabilities and no preferences.
- Probability threshold
- A parameter value at which two actions yield equal expected payoff.
- Boundary multiplicity
- Additional best responses and equilibria supported by exact indifference at a cutoff.
SPE Applications: Stackelberg, First-Mover Advantage and Nature's Move FAQ
Why is (3,1) incomplete?
It is only the path. The follower must also specify responses after leader deviations.
What is the follower plan?
In leader-action order 0,1,2,3 it is (3,2,2,1).
What is the full Stackelberg SPE?
(3,(3,2,2,1)) under that tuple convention.
How does it compare with Cournot?
Cournot gives outputs (2,2) and profits (3,3); Stackelberg path (3,1) gives profits (4.5,1.5).
Does moving first always help?
No. The advantage here uses credible commitment and the model’s reaction structure.
Does Nature have a strategy?
No. Nature follows fixed probabilities and has no best response.
When should expected payoff be used?
At decisions made before the relevant state is observed.
What is the insurance cutoff?
Insurance guaranteeing 80 is weakly preferred to the 100/10 lottery when α≥2/9.
What are the oil cutoffs?
Firm 2’s relevant entry cutoff is 1/5; Firm 1’s initial investment cutoff is 2/7.
What happens at equality?
Both tied actions are best responses; combine them with other optimal contingencies and check all supported SPEs.
How do commitment and chance interact in a complete SPE solution?
Start by separating strategic and exogenous branches. For every observed leader action, calculate the follower argmax and record a complete reaction tuple, retaining ties. Substitute each follower response into the leader payoff and choose the best induced outcome; report both the full profile and the realised path. Explain first-mover advantage through irreversible commitment and the reaction schedule, not chronology alone. At a Nature node, use the fixed stated probabilities rather than an indifference condition. A player choosing before the state observes only an expected continuation value; a player acting afterward uses the realised state payoff. Solve each probability inequality from its original positive and negative outcomes, intersect the result with [0,1], and preserve exact equality as a set of best responses. Combine tied boundary actions with all other optimal contingencies, then recheck the earlier mover. Finish with profile, state-contingent paths and expected payoff at the pre-state decision. This keeps observation, commitment, strategic mixing and Nature’s probability conceptually distinct.
Exam move
Rebuild the follower tuple from the profit function and state its node order. Create a leader table with chosen quantity, induced follower response and leader profit; this separates path from profile. Compare Cournot and Stackelberg while holding demand, cost and feasible quantities fixed, then explain commitment rather than chronology. For Nature games, annotate each choice as before or after state observation. Derive every probability threshold from its expected-payoff inequality, substitute the exact cutoff to verify equality and report actions below, at and above it. Practise multiplying boundary best-response sets into complete contingent strategies instead of choosing a convenient tie silently.
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