Paper IX — Social Mathematics

30 QUESTIONS

Scope

Thirty questions. Answer in the idiom of the college: provocations invite a disciplined essay; technical problems require correct, quantitative reasoning; interdisciplinary questions require two fields to be brought together. A question is admissible under this paper only if its answer is a theorem, a proof, a counterexample, or a formally stated open problem; no question of what ought to be chosen, valued, or done is admissible.

1.
Is the market a computer? Say what it computes, and whether the computation is tractable.
2.
Is a price a measurement? If so, say of what, and with what error.
3.
What, exactly, does Arrow’s theorem forbid? State the theorem precisely, and identify which condition each of the standard escape routes abandons.
4.
Prove that majority rule between two alternatives can be profitably manipulated by a voter misreporting a preference, or explain why the naïve extrapolation from the Gibbard–Satterthwaite theorem fails here.
5.
State Condorcet’s jury theorem, and say what becomes of it when the independence of the voters is dropped.
6.
Individually transitive preferences can aggregate to a cyclical majority. Locate exactly where transitivity is lost, and explain why McKelvey’s chaos theorem makes the loss consequential rather than curious.
7.
A committee votes coherently on each premise and on the conclusion, yet the majority position is inconsistent. State the discursive dilemma precisely, and decide whether it is Arrow’s theorem in disguise.
8.
State the revelation principle, and explain why it does not make mechanism design trivial.
9.
Two parties would each gain from trade, yet no mechanism can guarantee that the trade occurs. State the Myerson–Satterthwaite theorem with its exact hypotheses, and identify which hypothesis each known escape relaxes.
10.
The Vickrey–Clarke–Groves mechanism makes truth-telling dominant and the outcome efficient. Explain why it is nonetheless rarely used, on grounds internal to the mathematics.
11.
State the revenue equivalence theorem, and identify the assumption whose failure best explains why real auctions are not all alike.
12.
Prove that a stable matching always exists in the marriage problem, explain in what sense the proof is an algorithm, and say whom deferred acceptance favours.
13.
The top trading cycles algorithm delivers the unique core allocation of the housing market and cannot be profitably manipulated. Prove one of the two properties, and account for the rarity of so clean a result.
14.
Sperner’s lemma guarantees an envy-free division of a cake among any number of claimants. Explain how a combinatorial lemma about labelled triangulations comes to say anything about envy, and why existence here outruns procedure.
15.
The Shapley value is the unique division of a coalition’s surplus satisfying four axioms. State them, decide which is the least innocent, and defend the choice.
16.
Exhibit a cooperative game whose core is empty, state the Bondareva–Shapley condition that diagnoses the emptiness, and say what emptiness means for the players.
17.
Prove that every finite game has a Nash equilibrium in mixed strategies, identifying what each hypothesis of the fixed-point theorem corresponds to in the game.
18.
Computing a Nash equilibrium is PPAD-complete. State what this means, and what it implies for the claim that players, or markets, actually reach equilibrium.
19.
State a folk theorem for infinitely repeated games precisely, and explain why a result that permits almost everything is nonetheless not empty.
20.
Account for the persistence of cooperation among self-interested agents without invoking the word “trust”.
21.
Define an evolutionarily stable strategy, relate it exactly to Nash equilibrium, and say what the refinement buys in a population that does not reason.
22.
Hamilton’s rule, rb > c, is a theorem, an approximation, or a tautology, depending on how its terms are defined. Adjudicate.
23.
Agents who share a common prior and commonly know one another’s posteriors cannot agree to disagree. State Aumann’s theorem, and identify the assumption bearing the most weight.
24.
Distinguish mutual knowledge from common knowledge, and exhibit a coordination problem in which every finite depth of mutual knowledge fails where common knowledge would succeed.
25.
State the independence axiom of expected utility, exhibit the Allais pattern that strains it, and decide whether the pattern refutes the axiom or only the theory’s descriptive ambitions.
26.
A book of bets is Dutch when it guarantees its holder a loss. State the Dutch book theorem, and say whether it grounds the probability calculus or merely reprices it.
27.
The Sonnenschein–Mantel–Debreu theorem shows that aggregate excess demand is essentially arbitrary. State the result precisely, and say what, after it, general equilibrium theory remains a theory of.
28.
Adding a road to a congested network can lengthen every driver’s journey at equilibrium. Exhibit Braess’s paradox, locate the failure — in the network, in the equilibrium concept, or in the drivers — and state the sense in which such inefficiency is nonetheless bounded.
29.
City sizes, word frequencies, and firm sizes obey Zipf’s law. State the regularity precisely, give two generative mechanisms that produce it, and say whether their multiplicity undermines the demand for a single explanation.
30.
Why does this paper contain no macroeconomics? Answer from the mathematics.

Paper IX was added by Examination Expansion v3.0. Papers I–VIII stand verbatim and unrenumbered.

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