23506 Chap.9 Limits of SPE: Forward Induction, Trust Game, Chain-Store Paradox, Centipede
Limits of SPE: Forward Induction, Trust Game, Chain-Store Paradox, Centipede
Subgame perfection gives disciplined continuation play but can leave selection or prediction questions open. In the outside-option Battle of the Sexes, Study gives (2,2), while Date leads to a subgame with equilibria (S,S) and (F,F). The first continuation gives Boy 1 and supports Study; the second gives Boy 3 and supports Date. Both complete profiles are SPE because both continuations are Nash equilibria of the proper subgame.
Forward induction uses the observed sacrifice of the outside option. In normal form, DateS is strictly dominated by studying. After it is removed, Girl’s S is weakly dominated by F among surviving plans. Observing Date therefore supports the inference that Boy intends F. This selection argument does not replace the two-SPE backward calculation; it interprets an earlier costly move. In the trust game, selfish monetary payoff gives receiver response y(x)=0 for every transfer and sender choice x=0. Positive observed transfers can reveal reciprocity or other preferences absent from that benchmark.
The chain-store paradox similarly preserves the formal answer while diagnosing a missing informational channel. After entry, cooperation pays the branch 4 rather than zero from fighting, so fighting is not credible in the complete-information finite game. Reputation can matter if later entrants are uncertain whether the store is an aggressive type and update from observed conduct. In the centipede, asset value is 3t+3; the standard finite game stops at every node because stopping pays 2t+2 while continuing to a next-period stop pays t+2. The four-period path is (4,2), even though terminal cooperation would give (7.5,7.5).
What this chapter covers
- 01Multiple SPEs with an outside option
- 02Forward induction through dominance
- 03Trust game and selfish-payoff SPE
- 04Chain-store non-credible deterrence
- 05Reputation through type uncertainty
- 06Four- and 100-period centipede induction
One hundred-period centipede proof
- 2Calculate the period-100 asset and final mover’s Stop and Continue payoffs.
- 2State the induction hypothesis that the next mover stops.
- 2Calculate the current mover’s payoff from continuing one period.
- 2Compare it with immediate stopping for general positive t.
- 2Conclude complete strategies, path and initial payoff.
Key terms
- Forward induction
- Inference from an observed earlier choice about which later intentions remain rational.
- Outside option
- An action ending the strategic continuation at a specified payoff.
- Trust game
- A sender transfer is multiplied before a receiver decides how much to return.
- Chain-store paradox
- Finite complete-information SPE rejects deterrence that can seem reputationally plausible.
- Reputation
- Belief about a player’s type or future conduct shaped by observed actions.
- Centipede game
- Alternating Stop/Continue decisions over a growing asset with a finite endpoint.
Limits of SPE: Forward Induction, Trust Game, Chain-Store Paradox, Centipede FAQ
Does SPE choose one outside-option outcome?
No. Two subgame equilibria generate two continuation values and two SPEs.
What does forward induction select?
The stated dominance sequence supports Date followed by F because the costly DateS plan is ruled out.
What is the trust-game SPE?
The sender chooses x=0 and the receiver’s complete strategy returns y(x)=0 for every x.
Does positive trust disprove the calculation?
No. It suggests preferences, beliefs or repeated value not represented by selfish monetary payoffs.
Why will the chain store cooperate after entry?
Cooperation pays 4 at that node, while fighting pays zero.
How can reputation matter?
With uncertainty about an always-fight type, early fighting can update later entrants and deter them.
What is the four-period centipede SPE path?
Stop immediately at period 1, producing payoff (4,2), with Stop prescribed at every later node.
What is the period-100 asset value?
3×100+3=303; final Stop gives 202 and terminal Continue gives 151.5 to the mover.
Why does the induction step work?
Immediate Stop gives 2t+2, while Continue followed by next-period Stop gives t+2.
Do modified centipedes have the same answer?
Not necessarily. A changed terminal award or stopper share can reverse the base or recursive comparison.
How should I explain a gap between SPE and intuitive or observed behaviour?
Keep formal calculation and interpretation in separate stages. First derive the complete SPE under the displayed payoffs and information, including every off-path action. Second identify the exact tension: multiple credible continuations in the outside-option game, positive transfers under a trust benchmark, seemingly useful fighting in a finite chain store, or efficient continuation in a centipede. Third name one precise extension. Forward induction uses a costly observed action to eliminate an irrational intended continuation. Trust or reciprocity changes utility beyond displayed monetary payoff. Reputation requires uncertainty about type, observation of conduct and belief updating. Centipede cooperation may reflect social preferences, uncertain horizons or other assumptions, but the standard finite induction remains correct. Do not say that intuition simply defeats equilibrium. State which primitive would have to change and what new evidence would support it. For modified centipedes, recompute the terminal base case and generic predecessor rather than inheriting immediate stopping. This disciplined explanation preserves both mathematical validity and behavioural insight.
Exam move
Keep a two-column page headed Formal benchmark and Interpretive extension. For each example, solve SPE first and then identify the narrow tension: multiplicity for the outside option, social preference for trust, type uncertainty for reputation, or efficiency and horizon length for centipede. Reproduce the forward-induction deletion order rather than quoting its survivor. For centipede variations, calculate the terminal comparison before writing a generic predecessor; track final-player parity and exact shares. Avoid saying SPE is wrong. Say what it predicts under stated payoffs and information, then name the additional assumption that could explain another pattern.
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