← Game Theory
第 4 章 · 22 分钟
Order and commitment: extensive form and backward induction
Why a strategy must cover nodes that are never reached.
概念地图
Extensive form
Order written into a tree; a strategy becomes a plan for every node.
深色为本章概念，浅色为其他章
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Locus01
延伸阅读
Zermelo · 1910s · solvability of finite perfect-information games
Proved that games like chess have a determinate result in principle. He argued against treating such play as an unanalysable craft; the cost is that 'in principle' and 'computable' are separated by combinatorial explosion, a gap still open.
Foundational
Selten · around 1965 · subgame perfection
Pointed out that a Nash equilibrium can rest on a non-credible threat, and proposed the stronger condition of holding in every subgame. This argued against treating Nash equilibrium as the end of the story for sequential games; the cost is that under imperfect information the refinement needs further conditions on beliefs.
Turn
The centipede experiments · from the 1990s
In games where backward induction predicts stopping immediately, participants often cooperate for several rounds. The most direct counterexample to reading backward induction as a prediction about people.
Counterexample
Bonanno · the chapters on extensive form and backward induction
The subject of this course. The textbook is explicit that a strategy specifies an action at every node, and this chapter explains why that formal requirement carries all the reasoning about credibility.
Subject of this course
机制02
Order written into a tree
Strategic form discards order, and order is the whole basis on which a commitment can work. The notation that puts it back, from the Kuhn line, is the extensive form. Once the representation changes, chapter 1's line about covering every situation acquires its weight: there are many positions off the equilibrium path, a strategy must take a position on them too, and the next block explains why the entire chapter rests on those positions.
机制Order, through being written as the levels of a tree, makes 'what the later mover saw' part of the model.
可迁移性测试
Move it to an approval workflow: only once you draw who signs first and whether the later signer sees the earlier comment can you discuss whether the first signature moves the second. The isomorphism breaks in that a workflow diagram usually shows the paths that occur, while an extensive form must draw the branches that do not.
机制03
Unreached nodes decide credibility
The warranty clause in the previous block's setting has a name in game theory: a non-credible threat. It was precisely to filter this class out that Selten proposed subgame perfection. What matters is where the filtering criterion lives: entirely at the nodes that are never reached — so chapter 1's formal requirement is not fastidiousness but the load-bearing wall of this reasoning.
机制Off-equilibrium nodes, through deciding whether a threat would be carried out, filter out some Nash equilibria.
可迁移性测试
Move it to a warranty clause: any delay triggers a claim, but when delay comes the cost of claiming exceeds the loss, the other side reads that, and the clause does nothing. The isomorphism breaks in that a third party can enforce a contract, turning non-credible into credible.
Derivation04
From finite perfect information to the backward-induction solution
What is being proved: on a finite tree of perfect information, backward induction yields a subgame perfect equilibrium — and which hypotheses that uses.
The tree is finite: finitely many nodes and finitely many branches at each
A hypothesis. Drop finiteness and the induction has no base, so the process cannot start.
Perfect information: every information set is a singleton
A hypothesis. With non-singleton information sets the subgame partition is broken and backward induction cannot proceed node by node.
An optimal action can be selected at each node (no ties, or ties broken arbitrarily)
A hypothesis. With ties the solution is not unique, but at least one exists.
推导 · 0 / 4
Worked derivation05
Two Nash equilibria, only one credible
Incumbent and entrant: the entrant chooses to enter or not, and after entry the incumbent chooses a price war or accommodation. Why is 'threaten a price war' a Nash equilibrium but not subgame perfect?
Compute the incumbent's optimal action when the turn actually arrives, then look back at the threat.
裂缝06
The boundary of this chapter's claims
争议地形07
本书主张
Backward induction follows directly from sequential rationality: take the optimal action at every node and the resulting profile rules out all non-credible threats.
另一种看法
The bounded-reasoning and belief-revision line holds that backward induction requires believing the opponent will be rational from here on even after observing behaviour that contradicts the theory, and that this belief condition is internally strained: on a long tree one deviation invalidates the premise of the reasoning.
分歧扎在
The disagreement is rooted in how beliefs should be updated after a deviation, not in the algorithm. Backward induction's reasoning defaults to a deviation not changing your judgement about the future; the other side treats the deviation itself as evidence about the opponent's type.
什么证据能裁决
What would settle it is behaviour after a deviation: in the centipede game, record what one side does after the other departs from the prediction. Continuing to follow backward induction supports the default belief condition; a marked shift toward cooperation shows the deviation was read as type information.
The evidence points to deviations being read as information: participants cooperate more after the other cooperates once. What is missing is a design separating 'read as information' from plain imitation or reciprocity — their behavioural consequences overlap in most experiments.
Boundary and counterexample08
The 'people act by backward induction' commitment as a testable specification
The modelling commitment under test: real people complete the whole induction and believe the other will too.
Hypothesis dropped: induction depth is unbounded and mutual rationality survives a deviation
The prerequisite for backward induction to correspond to reality; more steps strain both computation and belief at once.
The counterexample: in a six-node centipede game the share stopping at the first step is far below the predicted whole
The prediction is 'stop immediately', and the number of cooperative rounds observed is directly comparable.
The observable: the distribution of stopping steps in first play with real payments
First play, to keep learning out; fixed payments, to keep payoffs from being fitted afterwards.
What counts against it: the share stopping at step one falls below a pre-registered threshold
The threshold is fixed in advance on how low costs backward induction its predictive meaning.
Pre-registered failure condition: the first-step share below threshold, or the stopping point moving monotonically later as the tree lengthens
The second counts too: it shows induction depth is bounded.
推导 · 0 / 3
接口09
Which slot it hangs on
挂在哪个槽位The conclusion you already hold is probably 'moving first is an advantage' — a judgement about order. This chapter goes at its condition.
Does this chapter (a) replace 'moving first is an advantage', or (b) constrain it: the advantage comes from commitment, and commitment means changing your own payoffs?
慢变量Register two slow variables: how many of your recent commitments changed your own payoff structure, and how many were only stronger wording. Look at the second first.
小结10
本章小结
01The extensive form writes order into a tree, which is why a strategy must cover unreached nodes.
02'Backward induction yields a subgame perfect equilibrium' is the definition unpacked plus induction, not a finding about people.
03A non-credible threat can support a Nash equilibrium but not subgame perfection; the whole test lives off the equilibrium path.
04Non-singleton information sets break the subgame partition; an infinite tree has no base for the induction.
05What carries observational risk is the commitment that people complete the induction and believe the other will.
提取练习 · 合上书，先自己答一遍。
?Is 'backward induction terminates on a finite perfect-information tree' (a) established by induction or (b) a modelling commitment about reality?
?Is 'threaten a price war' shown not to be subgame perfect by checking (a) incentives on the equilibrium path or (b) incentives at unreached nodes?
Forced choice11
A · Identify the mechanism
Which of these is a mechanism?
二选一
Forced choice12
C · Locate the crack
In which situation does this framework give a confident and wrong answer?
二选一
读到这里，把它记为已读。