ForMaRE Formal Mathematical Reasoning in Economics Manfred Kerber - - PowerPoint PPT Presentation
ForMaRE Formal Mathematical Reasoning in Economics Manfred Kerber Christoph Lange Colin Rowat University of Birmingham Computer Science Economics www.cs.bham.ac.uk/research/projects/formare/ ARW 2013 Dundee 12 April 2013 supported by
Manfred Kerber Christoph Lange Colin Rowat University of Birmingham Computer Science Economics
www.cs.bham.ac.uk/research/projects/formare/
ARW 2013 Dundee – 12 April 2013 supported by EPSRC grant EP/J007498/1
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 1/34
Motivation: Proofs in economics use often undergraduate level maths Proofs in economics are error prone (just as in any other theoretical fields) Formalization should be achievable – not just for computer scientists, but also for economists (?) Understand problems with the usage of theorem proving systems (!)
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 2/34
Motivation: Proofs in economics use often undergraduate level maths Proofs in economics are error prone (just as in any other theoretical fields) Formalization should be achievable – not just for computer scientists, but also for economists (?) Understand problems with the usage of theorem proving systems (!) Outline Related Work Pillage games Auction theory Matching problems Financial risk Summary
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 2/34
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 3/34
A constitution respects UN if society puts alternative a strictly above b whenever every individual puts a strictly above b. The constitution respects IIA if the social relative ranking (higher, lower, or indifferent) of two alternatives a and b depends only on their relative ranking by every individual. The constitution is a D by individual n if for every pair a and b, society strictly prefers a to b whenever n strictly prefers a to b. [Geanakoplos 05]
Theorem (Arrow – 3 Proofs by Geanakoplos 2005)
(For two or more agents, and three or more alternatives,) any constitution that respects transitivity, IIA, and UN is a D.
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 4/34
“Social choice theory turns out to be perfectly suitable for mechanical theorem proving. . . . However, it is unclear if this will lead to new insights into either social choice theory or theorem proving.” [Nipow09] “we form an interesting conjecture and then prove it using the same [mechanized] techniques as in the previous proofs. . . . the newly proved theorem . . . subsumes both Arrow’s and Wilson’s theorems.” [Tang-Lin09] “When applied to a space of 20 principles for preference extension familiar from the literature, this method yields a total of 84 impossibility theorems, including both known and nontrivial new results.” [Geist-Endress-11]
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 5/34
Gea01 John D. Geanakoplos. Three brief proofs of Arrow’s impossibility theorem. Discussion Paper 1123RRR. New Haven: Cowles Foundation, 2001. Gea05 John D. Geanakoplos. “Three brief proofs of Arrow’s impossibility theorem”. In: Economic Theory 26.1 (2005), pp. 211–215. Nip09 Tobias Nipkow. “Social choice theory in HOL: Arrow and Gibbard-Satterthwaite”. In: Journal of Automated Reasoning 43.3 (2009), pp. 289–304. Wie07 Freek Wiedijk. “Arrow’s impossibility theorem”. In: Journal of Formalized Mathematics 15.4 (2007), pp. 171–174. Wie09 Freek Wiedijk. “Formalizing Arrow’s theorem”. In: S¯
adhan¯ a
34.1 (2009), pp. 193–220.
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 6/34
TaLi09 Pingzhong Tang and Fangzhen Lin. “Computer-aided proofs
Intelligence 173.11 (2009), pp. 1041–1053. GrEn09 Umberto Grandi and Ulle Endriss. “First-Order Logic Formalisation of Arrow’s Theorem”. In: Proceedings of the 2nd International Workshop on Logic, Rationality and Interaction (LORI-2009). Ed. by X. He, J. Horty, and E.
2009, pp. 133–146. GeEn11 Christian Geist and Ulle Endriss. “Automated search for impossibility theorems in social choice theory: ranking sets
(2011), pp. 143–174.
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 7/34
AgHoWo09 Thomas Ågotnes, Wiebe van der Hoek, and Michael
Artificial Intelligence 173.1 (2009), pp. 45–79. VeLeOn06 René Vestergaard, Pierre Lescanne, and Hiroakira Ono. The inductive and modal proof of Aumann’s theorem on
Advanced Institute of Science and Technology, 2006.
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 8/34
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 9/34
Given a resource allocation X ≡ {{xi}i∈I |xi ≥ 0,
i∈I xi = 1}, the following
axioms can be defined. A power function π satisfies WC (weak coalition monotonicity) if C ⊂ C′ ⊆ I then π (C, x) ≤ π (C′, x) ∀x ∈ X; WR (weak resource monotonicity) if yi ≥ xi∀i ∈ C ⊆ I then π (C, y) ≥ π (C, x); and SR (strong resource monotonicity) if ∅ C ⊆ I and yi > xi∀i ∈ C then π (C, y) > π (C, x).
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 10/34
WC (weak coalition monotonicity) if C ⊂ C′ ⊆ I then π (C, x) ≤ π (C′, x) ∀x ∈ X Definition[“WC”, any[π, n], bound[allocationn[x]], WC[π, n] :⇔ n ∈ N ∧
∀
C1,C2 C1⊂C2∧C2⊆I[n]
∀
x π[C2, x] ≥ π[C1, x]] ]
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 11/34
WIPπ[C, x] :=
xi t1 t2 q3 = r t3 s23 D
D
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 12/34
WIPπ[C, x] :=
xi t1 t2 q3 = r t3 s23 D
D
Stable Set: S =
(0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 1
2, 1 2), ( 1 2, 0, 1 2), ( 1 2, 1 2, 0),
( 1
4, 1 4, 1 2), ( 1 4, 1 2, 1 4), ( 1 2, 1 4, 1 4),
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 12/34
Formalization: Theorema I. Represent the main definitions and results Proofs: Prove some theorems in Theorema Pseudo Algorithm: Summarize the results in a Theorema algorithm with
proved in Theorema. Presentation at ICE 2012 (Initiative for Computational Economics,
ice.uchicago.edu/) look into other areas.
We organized a symposium at this year’s AISB convention on Do-Form: Enabling Domain Experts to use Formalised Reasoning
www.cs.bham.ac.uk/research/projects/formare/events/aisb2013
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 13/34
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 14/34
Auctions
Auctions are a mechanism to distribute resources (e.g., eBay, ICANN, possibly High-Frequency Trading [Peter Cramton]) Given: a set of individual bids for a good (not necessarily the same as the value an individual ascribes to the good!) Goals: give the good to the bidder who values it most determine prices maximize revenue Auctions are designed and some properties are proved.
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 15/34
Second-price auction: a highest bidder wins, pays highest remaining bid.
Theorem (Vickrey 1961)
In a second-price auction, “truth-telling” (i.e. submitting a bid equal to one’s actual valuation of the good) is a weakly dominant strategy. The auction is efficient. earliest result in modern auction theory simple environment in which to gain intuitions
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 16/34
A definition
Given some auction, a strategy profile b supports an equilibrium in weakly dominant strategies if, for each i ∈ N and any ˆ b ∈ Rn with ˆ bi bi, ui
ˆ
b1, . . . , ˆ bi−1, bi, ˆ bi+1, . . . , ˆ bn
ˆ
b
be better off by deviating from the original bid bi.
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 17/34
A flavour of Isabelle:
definition equilibrium_weakly_dominant_strategy :: "participants \<Rightarrow> real vector \<Rightarrow> real vector \<Rightarrow> allocation \<Rightarrow> payments \<Rightarrow> bool" where "equilibrium_weakly_dominant_strategy N v b x p \<longleftrightarrow> valuation N v \<and> bids N b \<and> allocation N b x \<and> vickrey_payment N b p \<and> (\<forall>i \<in> N. (\<forall>whatever_bid. bids N whatever_bid \<and> whatever_bid i \<noteq> b i \<longrightarrow> (let b’ = whatever_bid(i := b i) in payoff (v i) (x b’ i) (p b’ i) \<ge> payoff (v i) (x whatever_bid i) (p whatever_bid i))))"
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 18/34
theorem vickreyA: fixes N :: participants and v :: "real vector" and x :: allocation and p :: payments assumes card_N: "card N > 1" assumes val: "valuation N v" and spa: "second_price_auction N x p" defines "b \<equiv> v" shows "equilibrium_weakly_dominant_strategy N v b x p"
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 19/34
proof - have defined: "maximum_defined N" using card_N unfolding maximum_defined_def by (auto simp: card_ge_0_finite) from val have bids: "bids N b" unfolding b_def by (rule valuation_is_bid) from spa bids have allocation: "allocation N b x" unfolding b_def second_price_auction_def by simp from spa bids have pay: "vickrey_payment N b p" unfolding b_def second_price_auction_def by simp { fix i :: participant assume i_range: "i \<in> N" let ?M = "N - {i}" have defined’: "maximum_defined ?M" using card_N i_range unfolding maximum_defined_def by (simp add: card_ge_0_finite card_Diff_singleton)
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 20/34
Isabelle/HOL (CL with Markarius Wenzel): higher-order logic (typed), interactive theorem proving environment, document-oriented IDE (DONE) Theorema 2.0 (Wolfgang Windsteiger): FOL + set theory, textbook-style documents (Mathematica notebooks), proof management GUI (in progress) Mizar (Marco Caminati): FOL + set theory, text editor, proof checker (DONE) Hets/CASL/TPTP (CL with Till Mossakowski): sorted FOL, text editor, proof management GUI, front-end to local or remote automated provers (in progress)
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 21/34
Representation is non-trivial, since it is partly not easy to understand the theorems, partly it is easy to make mistakes. Find mistakes by use and proof. Notice hidden assumptions Often proofs that look simple, are still non-trivial for theorem provers. First rationalize proofs (e.g., here 4 cases rather than 9 from the
HOL vs FOL, automated vs interactive ATPs differences are not that relevant after all (but the complexity of the argument).
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 22/34
Representation is non-trivial, since it is partly not easy to understand the theorems, partly it is easy to make mistakes. Find mistakes by use and proof. Notice hidden assumptions Often proofs that look simple, are still non-trivial for theorem provers. First rationalize proofs (e.g., here 4 cases rather than 9 from the
HOL vs FOL, automated vs interactive ATPs differences are not that relevant after all (but the complexity of the argument). Next steps: Redo the proof constructively for algorithm extraction. Prove more recent theorems/properties
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 22/34
[Picture from http://upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Gerrit_van_Honthorst_-_De_koppelaarster. jpg/350px-Gerrit_van_Honthorst_-_De_koppelaarster.jpg] Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 23/34
[The following slides are adapted and abbreviated from Utku Ünver’s talks
Examples: House allocation problem (agents to houses) House market problem (match pairs (agent, house) to each other) match living kidney donor-receiver pairs Students and schools Marriage matching
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 24/34
(A, H, >) with A = {a1, . . . , an} agents, H = {h1, . . . , hn} houses, and
preferences >a
µ : A → H is a solution
1
µ is Pareto efficient if there is no better ν so that νa(a) ≥ µa(a) for all
a and νa(a) > µa(a) for some a.
2
Mechanism is incentive compatible if it is best for each agent to tell the truth.
3
A mechanism is non-bossy if an agent can influence the allocation of houses for other agents only by getting a different house.
4
A mechanism is neutral if it is invariant under permutations. 1 and 2 may be incompatible.
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 25/34
Randomized serial dictatorship (RSD) means that one agent can choose their preferred choice, then a second theirs and so.
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 26/34
Randomized serial dictatorship (RSD) means that one agent can choose their preferred choice, then a second theirs and so. “RSD” is incentive compatible and Pareto efficient.
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 26/34
Randomized serial dictatorship (RSD) means that one agent can choose their preferred choice, then a second theirs and so. “RSD” is incentive compatible and Pareto efficient.
Theorem (Svensson 1998)
A mechanism is incentive compatible, non-bossy, and neutral iff it is serial dictatorship. Possible contribution: Find counter-examples of other characterizations.
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 26/34
Randomized serial dictatorship (RSD) means that one agent can choose their preferred choice, then a second theirs and so. “RSD” is incentive compatible and Pareto efficient.
Theorem (Svensson 1998)
A mechanism is incentive compatible, non-bossy, and neutral iff it is serial dictatorship. Possible contribution: Find counter-examples of other characterizations. Question: Can other characterizations be found?
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 26/34
incompatible incompatible compatible compatible
Algorithm of Roth, Sönmez, Ünver from 2005 for 2 way exchange. Question: Can the correctness proof be made formal?
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 27/34
[Picture from http://www.flickr.com/photos/cau_napoli/4554437754/]
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 28/34
[The following slides are adapted and abbreviated from the talk of and discussions with Neels Vosloo from the Bank of England 4/5 April 2013 at the Do-Form symposium.] Unilever plc 10 April 2013 2,792.50p 3 month libor 10 April 2013 1.02% contract size 1000 sell/buy sell strike price 3,200.00p maturity 9 July 2013 The VaR model of the bank computes from many of such assets an overall risk (taking into account 5,000 to 10,000 different risk factors, based on statistical models. FSA’s task (team of 8): test the VaR models of 18 banks.
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 29/34
Challenge
Identify patterns that are consistent with certain deficiencies in models or their implementation Question: Can we help to provide good tests? (e.g., [Raymond Reiter AIJ, 1987, “A Theory of Diagnosis from First Principles”]?)
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 30/34
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 31/34
On paper, a lot of contextual information is implicit: e.g. types and scopes of variables Auctions: “N = {1, . . . , n} is a set of participants, often indexed by i”
⇒ n ∈ N, Ni ∈ N.
Unfamiliarity with the writing style: e.g., Currying Irrelevant Information: Not all information given on paper is relevant for the formalization difficult information: “Let it be common knowledge that each vi [valuation] is an independent realizations of a random variable ˜ v, whose distribution is described by density function f. Then a strategy for bidder i is a mapping gi such that bi = gi(vi, f) ≥ 0, where bi is known as i’s bid.”
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 32/34
Misleading illustrations: Geanakoplos’ 1st proof of Arrow: “by changing his [preference order] at some profile [set including his preference and the other voters’ preference] he can move [alternative] b from the very bottom of the social ranking to the very top” – but there are no state transitions, just two different preferences profiles! Computers hate creative (ab)use of notation and sloppy typing e.g. the loser of an auction was originally characterized as xj = pj = 0 (xj ∈ {0, 1} but pj ∈ R) excessive usage of sets. sums involving zeros: “An auction is efficient if it maximizes
function), i.e. “efficiency states vi ¯ v ⇒ xi = 0”.
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 33/34
Computer Science in general, Theorem Proving in special can support economists. We have to work towards adjusting our methods to economics problems. Specialist knowledge is required, the systems in its current form are still difficult to use by non-experts. There are many challenging problems. Join in:
www.cs.bham.ac.uk/research/projects/formare/
Manfred Kerber, Christoph Lange, Colin Rowat ForMaRE – Reasoning in Economics ARW 2013 Dundee – 12 April 2013 34/34