← Game Theory
第 3 章 · 22 分钟
Nash equilibrium: a consistency condition
What it guarantees, and what it never guarantees.
概念地图
Nash equilibrium
A profile in which nobody gains by moving alone.
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Locus01
延伸阅读
Cournot · 1830s · the duopoly quantity model
Wrote down, inside one concrete model, the condition later called equilibrium: each firm's quantity is optimal against the other's. He argued against taking price as exogenous; the cost is that the step lived only inside that model and produced no general concept.
Foundational
Nash · around 1950 · the definition and existence
Abstracted Cournot's condition into a solution concept for any finite game and proved by fixed point that it exists in mixed strategies. This argued against 'only zero-sum games are solvable'; the cost is that equilibria are usually not unique, prising existence apart from predictive power.
Turn
Schelling · 1960 · choosing among equilibria
Argued that when equilibria are many, what decides is shared salience and commitment rather than finer calculation. The most persistent counterexample to reading equilibrium as prediction.
Counterexample
Bonanno · the chapters on Nash equilibrium
The subject of this course. The textbook separates existence from uniqueness cleanly; this chapter pulls out the step 'existence is not prediction', because that is where the concept is most often misused.
Subject of this course
机制02
Nobody moves alone
Nash's solution concept, from around 1950, imposes exactly one condition, and this chapter is about its three silences. First the name: a Nash equilibrium. Note the two qualifiers in the definition — 'given the others' and 'alone'. It says nothing about how the profile came about, nothing about how the parties reached agreement, and nothing about whether it is good for anyone. Those three silences are what this chapter delivers, and they are the three places the concept is most misused.
机制A strategy profile, through leaving every unilateral deviation unprofitable, becomes a self-consistent point.
可迁移性测试
Move it to a road network: a traffic assignment where no driver can go faster by switching routes is self-consistent — which does not make it the assignment with the shortest total travel time. The isomorphism breaks in that drivers adjust day by day and observe the result, while a one-shot game has no such adjustment.
机制03
Self-consistent is not best
Equilibrium and the quality of the collective outcome are separate matters. Tucker attached a two-person story to that separation in the 1950s and it has been the standard case ever since: the prisoner's dilemma. It has exactly one use in this course — as a counterexample: the unique solution is Pareto improvement-dominated by another feasible outcome, and that is no objection to the definition, which never contained a condition about the collective at all.
机制A consistency condition, through constraining only unilateral deviation, gives no guarantee about the quality of the collective outcome.
可迁移性测试
Move it to antibiotic prescribing: each clinician prescribing a little more loses nothing while everyone else holds steady, and rising resistance leaves everyone worse. The isomorphism breaks in that resistance is a cumulative physical process, while the dilemma's payoff table is identical in every round.
Derivation04
From best response to equilibrium, and what it does not entail
What is being proved: an equilibrium is an intersection of best responses, and whether that intersection exists, is unique, is good, or gets reached are four things none of which implies another.
Each player's best-response correspondence is
A hypothesis requiring the to be non-empty; chapter 1's completeness hypothesis is used here.
Strategy sets are finite (or compact convex with continuous quasi-concave payoffs)
A hypothesis. The existence theorem needs it; without it an equilibrium may not exist.
Equilibrium is defined as a profile with for all
A hypothesis, and the definition itself.
推导 · 0 / 4
裂缝05
The boundary of this chapter's claims
争议地形06
本书主张
Nash equilibrium is the basic solution concept for non-cooperative games: it gives every self-consistent profile and always exists in mixed strategies.
另一种看法
The evolutionary and learning line holds that equilibrium should be seen as the resting point of an adjustment process rather than a one-shot product of reasoning; the resulting stable set is sometimes a strict subset of the Nash set (evolutionarily stable strategies) and sometimes contains cycles outside it.
分歧扎在
The disagreement is rooted in whether equilibrium is the product of reasoning or of a process. The book treats it as the consistency condition of rational reasoning, the other side as the limit of repeated adjustment, so they handle non-uniqueness quite differently.
什么证据能裁决
What would settle it is the trajectory of repeated play: in a game with several equilibria, record the path and see which one it settles on, if any. If the resting point is set by initial conditions and the adjustment rule with no role for salience, the process account is stronger; if it settles on the salient one, the reasoning account is.
The evidence points to both operating: salience shapes the first round, adjustment decides long-run convergence. What is missing is a design that separates them — in most experiments salience and initial conditions are confounded.
Boundary and counterexample07
The 'outcomes land on equilibrium' commitment as a testable specification
The modelling commitment under test: a real interaction's outcome lands on a Nash equilibrium, that is, beliefs are mutually correct.
Hypothesis dropped: each party's belief about the others' strategies is correct
The definition requires each to be optimal against a correct expectation; with incorrect beliefs the profile need not be an equilibrium.
The counterexample: in a coordination game both sides play their own preferred equilibrium strategy and the outcome is a non-equilibrium profile
Both are acting on an equilibrium, and together they are not at one.
The observable: the frequency distribution over strategy profiles in an experiment with real payments
Payoffs must be fixed by real payment and not fitted afterwards.
What counts against it: the frequency of non-equilibrium profiles exceeds a pre-registered threshold, or play jumps irregularly between equilibria
The threshold is fixed in advance on how much deviation costs equilibrium its predictive meaning.
Pre-registered failure condition: non-equilibrium frequency above threshold, or the convergence path depending on salience cues outside the model
Either counts as the commitment failing.
推导 · 0 / 3
接口08
Which slot it hangs on
挂在哪个槽位The conclusion you already hold is probably 'find the equilibrium and you know what will happen' — a solution concept read as a prediction. This chapter goes straight at it.
Does this chapter (a) replace 'equilibrium is prediction', or (b) open a new slot for consistency conditions?
慢变量Register two slow variables: how many of your analyses have multiple equilibria, and in how many of those you supplied a selection basis. Look at the second first.
小结09
本章小结
01Equilibrium constrains only unilateral deviation and promises nothing about the collective outcome.
02'Equilibrium equals a fixed point of best response' is a rewritten definition; existence is an applied fixed point theorem.
03Existing, being unique, being good and being reached are four things none of which implies another.
04In pure strategies an equilibrium may not exist; the existence theorem lives on mixed strategies.
05If payoffs may be fitted afterwards, 'the outcome lands on an equilibrium' stops carrying observational risk.
提取练习 · 合上书，先自己答一遍。
?Is 'an equilibrium always exists in mixed strategies' (a) an applied theorem or (b) a modelling commitment about reality?
?Does the prisoner's dilemma show (a) a flaw in the definition or (b) that the definition never had a condition about the collective?
Forced choice10
B · Identify the tag
'By Kakutani's theorem an equilibrium must exist' — what kind of step is that?
二选一
Forced choice11
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。