Eyal Sulganik: Towards a Theory of “Mathematical Accounting”

The following post was kindly contributed by Eyal Sulganik  from IDC (Interdiciplinary Center)  Herzliya. Eyal was motivated by our poll on certainty “beyond a reasonable doubt,” which is related to several issues in accounting.

Mathematicians, I believe, are always looking for new areas where their models and concepts can make a difference. Physics, Economics, CS, Biology are just some examples, surely not exhausting a longer list of such areas. Although the origins of accounting emanate from mathematicians (for example: L. Pacioli, and even  A. Cayley found interest in it), it is a fact,  though not unexplained,  that these days  (almost) no mathematicians are interested in accounting and there is no field of “mathematical accounting”.  In the following few paragraphs I would like to thus draw attention to accounting as a possible field for mathematicians. Surprisingly, a possibly “profitable field”. I believe that accounting can be subject, inter-alia, to use of theories of Formal Systems, Information Theory, Voting  theory, Fuzzy Logic, Graph theory and even Catastrophe theory. In brief, accounting (Financial Reporting) deals with the measurement and reporting of economic events .  As such, it is a measurement system interlaced with an information system. (Results such as Blackwell theorem on the comparison of information systems are of relevance).

Financial reporting of firms, the lifeline of the Capital Markets, is dictated by “reporting standards”. Those reporting standards are determined by “standard boards” (according to voting rules and procedures, which are very interesting for analysis)  and interpreted and evolve over time (as is the case with other languages).   The reports are audited by accounting firms.  Auditing theory became more sophisticated  but even fairly standard tools like Benford Law are not yet routine.

Moreover, a huge debate centers on whether to adopt a rule-based system where “every” possible scenario is prescribed in advance or whether to adopt a principle based system which gives” freedom” to every reporting entity in reflecting the economic substance of an event.

It is well known that a rule-based systems provide greater comparability (between firms), but at the same time, as they are more rigid and make use of “bright lines”,  can be more easily forced to reflect form over substance . Indeed, Bright line Accounting rules are not continuous functions and hence small changes in the description or design of an event can lead to enormous differences in the reported values. For example, given that the definition of “CONTROL” was based on a “50% legal test” ,until recently it was the case that  if company A was holding 50.01% of the shares of company B (other holders being each  much smaller)  and sold only  1.02% it could recognize a profit, due to “loss of control”,  as though it sold the whole holding and bought back 49.01%. Needless to say that although holding “only” 49.01% , A CONTROLs B (Danny Ben Shahar, Desmond Tsang and Myself are in the process of demonstrating that “accounting theory of control” is inconsistent with Shapley Value).

Principle based systems, on the other hand, must make sure that its principles are common knowledge. For example, if a provision for loss regarding a claim against a firm depends on the chances of loss being “Probable” or  “remote” or “reasonably possible” a question arises what the preparers and users of the financial statements think about those terms. Many years ago, me and my colleague Yossi Aharoni found out-through questionnaires- that different types of agents have different probabilistic interpretations to those terms and we explained the mis-communication it can cause. I attach two simple papers (one co-authored with Danny Ben Shahar, the other one in hebrew),  that can shed some more light on the above point of view and I dare state a wish that a new field of “mathematical accounting” will be created .

Knighted for Services to Mathematics

The Birthday and Diamond Jubilee Honours 2012 was released on 16 June 2012 in the United Kingdom and Tim Gowers was knighted for “services to mathematics”! So I suppose Tim is now becoming “Sir William.”

It is possible that the Queen mainly thought about Tim’s wonderful research contributions. (Indeed, I was told, lately she took the relevant volumes of GAFA with her to bed.) But certainly Gowers’s services to mathematics include also the Princeton companion and the very short introduction, general-public presentations, Gower’s blog and Tim’s frequent

comments on other blogs, Polymath, Contributions to MathOverflow (the first user to get eight golden badges, among other things), Fighting Elsevier (which is now referred to as the “academic spring“) and more. Congratulations!

What does “beyond a reasonable doubt” practically mean?

(Motivated by two questions from Gowers’s How should mathematics be taught to non mathematicians.)

Celebrations in Bar-Ilan, HU, and the Technion; A new blog: Windows on Theory; Turing’s celebration on “In Theory”; Graph Limits in Princeton

Last monday we had the annual meeting of the Israeli Mathematical Union (IMU) that took place this year in Bar-Ilan University in Ramat Gan. (IMU is famously also the acronym of the International Mathematical Union but in this post IMU will stand for “Isreali Mathematical Union.”) The first IMU meeting that I participated was in 1972 and I try to participate every year.  One year, Nati Linial and me were the treasurer and secretary of the IMU and were assigned (by the IMU president at the time, Yisrael Aumann) the task of organizing the meeting. The IMU meetings have different formats and usually they are very successful and this year was no exception. So I will tell you a little more about it later. Five more short items for today:

1. Last Wednesday two new research centers were inaugurated. Here at the Hebrew University of Jerusalem the new Quantum Information Science Center  had its kick-off workshop on Wednesday. Here are the slides of the lecture I gave  devoted to my debate with Aram Harrow. (As always, remarks and corrections are most welcome.)
2. In Haifa at the Technion , the newly founded Technion-Microsoft Electronic-Commerce Research Center held a one day workshop on electronic-commerce .
3. There is a new research blog Windows on Theory devoted to research in theoretical computer science. It is run by theory people from Microsoft Research, most from the Silicon Valley. The last post mentioned a nice story of Ehud Friedgut about hitting a rotating sphere from a 1997 paper of mine.
4. Last month, Luca Trevisan was the Erdos’ Lecturer of the center for Discrete Mathematics and Theoretical Computer Science, here at HU. Luca is one of the world leaders in theoretical computer science and also the author of a famous blog: In Theory.   Luca Trevisan’s blog’s celebration of Alan Turing’s 100th birthday, spanned over several posts,  will be devoted to a big part of Turing’s life  – being gay.
5. Starting on monday: A conference on graph limits (“Graphs and Analysis“) at IAS, Princeton. Many great talks are expected. Our blog is negotiating with Itai Benjamini for letting us post the slides of his lecture on Euclidean versus graph metrics.

The Bar-Ilan Meeting

David Kazhdan gave a talk about the classic master equation.

The abstract of his lecture read:

We formalize the construction by Batalin and Vilkovisky of a solution of the classical master equation associated with a classical action, a regular function on a nonsingular affine variety. We introduce the notion of stable equivalence of solutions and prove that a solution exists and is unique up to stable equivalence. A consequence is that the associated BRST cohomology groups are independent of choices and are uniquely determined up to unique isomorphism by the classical action.

David, however, decided to give a talk about the background to a talk described in the abstract. He gave a quick mathematical introduction to classical and quantum mechanics, leading to the Faddeev-Popov method, described in as elementary and jargon-free as possible  differential geometry language. Beautiful talk!

Let’s glimpse at the lecture: Quantum mechanics is about calculating integrals of the form $\int_X e^{is(x)/h} \nu (x).$ While classical mechanics is the study of the stationary points of $s(x)$. (In cases where the stationary point is unique, the derivative at a stationary point $x_0$ is well-defined and the Hessian is non-zero, and other pleasant things occur  we can approximate the integral by a normalized value of $s(x_o)$.)

When there is a group G (gauge symmetry) acting on X  then we can try to make the computation on the quotient X/G . (This computation leads to expansion in terms of Feynman diagrams.) Faddeev-Popov method is based on the idea: don’t take quotients but rather enlarge your space! So you replace X by the Cartesian product of X with the associated Lie algebra of G. (Here, a structure of a “supermanifold” emerges.) At this point a) some differential geometry enters into the picture, b) this allows to the master equation promised in the title to emerge, and c) The computation has some cohomological nature.

After David’s talk Jonathan Wahl gave a talk  on Topology versus geometry of complex singularities. There is a famous book by Milnor on this topic and a lot have been learned since. Then there was a brief prize awards session. Our annual  Erdos’ prize for a mathematician under 40, was awarded to Irit Dinur, and the Haim Nessiyahu prize for excellent doctoral thesis was awarded to Alexander Sodin on his work on random matrices.  Ran Raz gave a talk about PCP and, in particular, on Irit’s contribution.  This is the 20th anniversary of the PCP Theorem. Heartily Congratulations to Irit and Sasha.

And then we separated into ten parallel sessions: Algebra and number theory; Analysis; Discrete Math and Computer Science; Dynamics and Ergodic Theory;  History and Philosophy of Mathematics; Homotopy Theory;  Mathematical Education; Topology and Geometry; Probability; Nonlinear Analysis and Optimization. To avoid being torn between too many good options my policy is usually to stay with the combinatorics session. But this year since my flight was brutally delayed and I landed only at five in the morning I had to choose the option of going home to sleep.