# 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 .

# Michael Schapira: Internet Routing, Distributed Computation, Game Dynamics and Mechanism Design II

This post is authored by Michael Schapira. (It is the second in a series of two posts.)

In thse two post, I outline work on Internet routing and sketch important areas for future work, both on routing itself and, more broadly, on mechanism design, game theory and distributed computation.

A brief reminder from post 1:

The Internet comprises administrative domains known as Autonomous Systems (ASes), which vary in size, from large (multi)national networks to small networks servicing schools or small businesses. The task of establishing routes between ASes, called interdomain routing, is currently handled by the Border Gateway Protocol (BGP)—the core routing protocol of the Internet. BGP is one of the most critical pieces of the Internet’s infrastructure and can be regarded as the glue that holds the Internet together.

We brefly mentioned two main challenges in this area. The first is the quest for network stability, and the second was giving  agents incentives to “behave well”. We will discuss here these challenges in more detail.

## Challenge I: The Quest for Network Stability

Informally, a “stable state’’ is a global routing state that, once reached by BGP, remains unchanged. Formally, a stable state is an assignment of routes R1,…,Rn to the n source nodes (d is assigned the “empty route’’ Rd=Ø) such that for every node i (1) there is a node j such that Ri=(i,j)Rj, where (i,j) Rj is the route that has (i,j) as a first link and then follows Rj to d; and (2) for every node j such that Ri≠(i,j)Rj, and (i,j)Rj is simple, it holds that Ri >i (i,j)Rj.

# Futures Trading as a Game of Luck

A recent interesting article by Ariel Rubinstein entitled “Digital Sodom” (in Hebrew) argues that certain forms of  futures trading (and Internet sites where these forms of trading take place) are essentially gambling activities.

The issue of “what is gambling” is very intereting. In an earlier post entitled “Chess can be a game of luck” the interesting question regarding “games of luck” and “games of skill” was discussed. It was argued that for a betting game, if, over time, for all players (or even for most players), the expected gains are negative, then we can regard the game as primarily a game of luck. (The claim is that this is a reasonable interpretation of the current law, even if not the only possible interpretation.)

# Michael Schapira: Internet Routing, Distributed Computation, Game Dynamics and Mechanism Design I

This post is authored by Michael Schapira. (It is the first in a series of two posts.)

In this post, I’ll outline work on Internet routing and sketch important areas for future work, both on routing itself and, more broadly, on mechanism design, game theory and distributed computation.

The Internet comprises administrative domains known as Autonomous Systems (ASes), which vary in size, from large (multi)national networks to small networks servicing schools or small businesses. The task of establishing routes between ASes, called interdomain routing, is currently handled by the Border Gateway Protocol (BGP)—the core routing protocol of the Internet. BGP is one of the most critical pieces of the Internet’s infrastructure and can be regarded as the glue that holds the Internet together.

Over the past decade, routing with BGP has been studied from engineering, computational, game theoretic and economic perspectives. Combining algorithmic and economic considerations in the study of interdomain routing is natural, because the many separate domains that make up the Internet are independent economic agents that must jointly execute a distributed algorithm in order to route traffic. In addition, interdomain routing motivates exciting new questions in computer science, game theory, and economics.

A (Simplified) Model of Interdomain Routing

The following is a simplified model of interdomain routing, based on the seminal work of Thimothy Griffin and Gordon Wilfong, and on the game-theoretic model of Hagay Levin, Michael Schapira, and Aviv Zohar   Continue reading

# Impossibility Result for “Survivor”

Consider a set of $n$ agents and a directed graph where an edge $(i,j)$ means that agent $i$ supports or trusts agent $j$. We wish to choose a subset $C$ of size $k$ of trustworthy agents. Each agent’s first priority is to be included in $C$. We want to find a function from directed graphs of trust, to k-subsets $C$ of the vertices. We require two axioms

Axiom 1: If exactly one player is trusted by some other player then this player will be selected to the set $C$ of trustworthy players.

Axiom 2: A player cannot prevent not being selected to $C$ by misreporting his truthful judgement of his fellow players (even if he knows the way other players are going to vote).

Axiom 2 is referred to as “strategy proofness“.

Impossibility theorem (Alon, Fischer, Procaccia and Tennenholtz): For any $k$, $1\le k\le n-1$ there is no mechanism for choosing the $k$-subset $C$ of trustworthy agents that satisfies Axioms 1, 2.

A special case of the theorem, when $k=n-1$, can be described in terms of the TV reality game “survivor”. Continue reading

# Chess can be a Game of Luck

Can chess be a game of luck?

Let us consider the following two scenarios:

A) We have a chess tournament where each of forty chess players pay 50 dollars entrance fee and the winner takes the prize which is 80% of the the total entrance fees.

B)  We have a chess tournament where each of forty chess players pay 20,000 dollars entrance fee and the winner takes the prize which is 80% of the the total entrance fees.

Before dealing with these two rather realistic scenarios let us consider the following more hypothetical situations.

C) Suppose that chess players have a quality measure that allows us to determine the probability that any one player will beat the other. Two players play and bet. The strong player bets 10 dollars  and the waek player bets according to the probability he will win. (So the expected gain of both player is zero.)

D)  Suppose again that chess players have a quality measure that allows us to determine the probability that any one players will beat the other. Two players play and bet. The strong player bets 100,000 dollars and the weak player bets according to the probability he will wins. (Again, the expected gain of both players is zero.)

When we analyze scenarios C and D the first question to ask is “What is the game?” In my opinion we need to consider the entire setting, so the “game” consists of both the chess itself and the betting around it. In cases C and D the betting aspects of the game are completely separated from the chess itself. We can suppose that the higher the stakes are, the higher the ingredient of luck of the combined game. It is reasonable to assume that version C) is mainly a game of skill and version D) is mainly a game of luck.

Now what about the following scenarios:

E) Two players play chess and bet 5 dollars.

Here the main ingredient is skill; the bet only adds a little spice to the game.

F) Two players play chess and bet 100,000 dollars.

Well, to the extent that such a game takes place at all, I would expect that the luck factor will be dominant. (Note that scenario F is not equivalent to the scenario where two players play, the winner gets 300,000 dollars and the loser gets 100,000 dollars.)

Let us go back to the original scenarios A) and B). Here too, I would consider the ingredients of luck and skill to be strongly dependant on the stakes. The setting of scenario A) can be quite compatible with a game of skill where the prizes give some extra incentives to participants (and rewards for the organizers), while in scenario B) it stands to reason that the luck/gambling factor will be dominant.

One critique against my opinion is: What about tennis tournaments where professional tennis players are playing on large amounts of prize money? Are professional tennis tournaments  games of luck? There is one major difference between this example and examples A and B above. In tennis tournaments there are very large prizes but the expected gain for a player is positive, all (or at least most) players can make a living by participating. This changes entirely the incentives. This is also the case for various high level professional chess tournaments.

For mathematicians there are a few things that sound strange in this analysis. The luck ingredient is not invariant under multiplying the stakes by a constant, and it is not invariant under giving (or taking) a fixed sum of money to the participants before the game starts. However, these aspects are crucial when we try to analyze the incentives and motives of players and, in my opinion,  it is a mistake to ignore them.

So my answer is: yes, chess can be a game of luck.