The Fundamental Theorem Of Betting

I used two popular websites to get betting odds for the today's match between Egypt and Nigeria. Here are some odds from two example sites:

Site 1: Egypt (2.90) - X (3.20) - Nigeria (2.35)

Site 2: Egypt (2.65) - X (3.10)- Nigeria (2.65)

Which site of the two should be preferred over the other? Generally, how to compare the attractiveness of different betting odds? In what follows, I am presenting what I call as the "fundamental theorem of betting".

Consider just the single game "Egypt vs Nigeria" which has 3 possible turnouts: "Egypt wins", "nobody wins" and "Nigeria wins". Let us also consider that we have a budget of Δ dollars and that we split this sum to all three turnouts, betting Δ1, Δ2 and Δ3 respectively. Finally, for every event, we expect to win some money, say W1, W2 and W3. In general, we are just interested in winning more money than we spent ( Δ dollars), so for every event that might take place (e.g. "Egypt wins" ) we might want W1 > Δ or W2 > Δ or W3 > Δ. But we might want to ask: Can all of these inequalities hold? The obvious answer is no. Otherwise, the booker would go bankrupt. Next, we have to ask: What values do the W's satisfy? It turns out that this is a linear function, with coefficients determined by the betting odds of the booker!

Consider that we have a budget Δ and we want to split this in all k possible turnouts. Thus, we will have to pay Δ1, Δ2,Δ3,..., and Δk respectively. For every turnout i the booker provides a betting odd Mi. This means that we will win Wi = Mi * Δi, in case of this turnout. Finally, every profit has a specific ratio of the whole budget Δ. This is the real profit since we are interested in making more money than we spent. For every turnout the profit will be denoted as: Ai = Wi / Δ. One can easily see that:

(A1/M1)+(A2/M2)+...+(Ak/Mk) = 1

For clearer inspection:

This is what one could call the "Fundamental Theorem Of Betting".

Remember that the μ's are the booker's betting odds and the α's are the ratios of profit over the overall budget. For example, if we invested $200 in total and Α1 = 2.0 then if turnout #1 happened we would win 200x 2.0 = $400 and so on. When an α is greater than 1 this means that we won more than we invested in total and so we are interested in establishing at least one α greater than 1.0. This means that it would be useful to consider one single quantity:

M = (1/M1)+(1/M2)+(1/M3)+...+(1/Mk)

The following cases now exist:

1) M > 1.0

If this is the case then the booker is going bankrupt. Players can construct a set of α's such that all of them are greater or equal to 1.0. This means that every player either wins more than he spent in total or he gets his money back (zero loss)

2) M = 1.0

In this border case, the players can set all α's to 1.0 and never suffer a loss. This is not very meaningful and players might try to disturb the ratios by increasing and decreasing some of the ratios.

3) M < 1.0

This is expected to be much more frequent in real world betting odds. Players will have to decide how to distribute their bets.

What's more important, the quantity M can be used as a metric of how attractive the betting odds are. The greater the quantity M is, the better the odds are for the players. Going back to the example in the beginning we have:

Site 1: M = (1/2.90)+(1/3.20)+(1/2.35) = 1.0828

Site 2: M = (1/2.65)+(1/3.10)+(1/2.65) = 1.0773

One can see with a single arithmetic comparison that Site 1 gives slightly better margins for profits (1.0828 is bigger) and it should be preferred to Site 2. Even within the same book house, players should prefer games with higher M factors than others. The first step to a complete "Theory of Betting" has been made!

The Most Influential Person On Computer Science

Who was the most influential person on computer science (or computing if you prefer)? A web search is not particularly helpful. Most of the times, the pages I got were confusing CS with programming. However, to quote Edsger Dijkstra, "Computer Science is no more about computers, than astronomy is about telescopes". Some names may come quickly to

mind: Alan Turing, Church, Kleene etc, were all pivotal to the development of the computing science. But a deeper search can reveal the person who had the most critical impact: David Hilbert.

Hilbert was a German mathematician who worked on an incredible range of subjects: number theory, logic, geometry and over to special relativity theory. Yet, it was not his direct contribution that matters, but the influence he had to others on working on the fundamentals of computation. And it all started by his zealous stance towards a deep philosophical debate of the "ignoramus et ignorabimus".

And now the story begins.

Back in 1872 Emil du-Bois Reymond was a well-respected physiologist with considerable work on electro-physiology, studying the electric "properties" of human and animal tissues. At this year, he publishes a paper in which he claims his philosophical belief of "ignoramus et ignorabimus" (=we don't know and we won't know). This was a slightly more pessimistic version of the well known Socratic view "I know one thing; that I know nothing" but it was gaining acceptance through the philosophical circles at the time. The main point was that there are some stuff that humans do not know and they will never know (e.g. the origin of life or the universe) One might agree that it seems like a yet-another harmless philosophical idea. Some people were supporting it, others would be skeptical, but all in all, life would go on. But science was meant to take a dramatic turn because a man would take this debate to the extreme.

At that time David Hilbert was 10 years old. In subsequent years, through his work, he would confront deep philosophical problems, even within mathematics. However, the "ignorabimus" movement had really hit his nerve. In 1900, in the International Mathematics Conference in Paris, he addresses a call to arms to mathematicians by claiming: "in mathematics there is no ignrorabimus!". He uses mathematics to exterminate once and for all the pessimistic view, and claims that there should be a finite process of inference rules, that when applied, they would infer any correct statement. He then asks from mathematicians to help him in this direction (the 2nd problem in the problem list he announces at this speech was related)

Almost 28 years later, in 1928, Hilbert poses yet another challenge: The Entscheidungsproblem asks specifically for an algorithm that will be able to decide if a specific statement (e.g. 1+1=2) is true or not. During the 30's great things just start happening. In 1931, Kurt Godel breaks the first problem with his ground-breaking theorem of incompleteness. In short, most interesting systems, can either infer contradictory statements or they cannot infer some correct statements. Later, in 1936 and 1937 Church and Turing independently solve the last puzzle by proving that there are statements that cannot be proved or disproved. Hilbert's dream was shuttered but a new scientific field was born out of the proofs of these very problems! It is important to note here that, 5 years before his proof, Alonzo Church was in the famous Gottingen University during the years 1929-1931, visiting...David Hilbert.

But wait, there is more! It was not only theoretical computer science that was catalyzed by the influence of David Hilbert. The creation of the first general-purpose electronic computers would be pioneered by his assistant. Even in 1942, one year before the ENIAC project, Hilbert's former assistant in the University of Gottingen, would start an ambitious project of creating a general-purpose computer in the Institute of Advanced Studies at Princeton University, that ended in 1952. His work all these years marked the beginning of this era and his architecture is preserved more or less even in today's computers. His name was ...John Von Neumann!

Other than Hilbert, who else can claim of having a bigger impact on computer science?

If Philosophers Were Programmers

Although not obvious, philosophy actually has a strong relation with programming, at least for me. If you think about it, software code reflects much of how the developer perceives the problem and its solution. Before starting to program, developers spend some time thinking over the problem, identifying important properties and their underlying connections, a process that reveals their philosophy as the way they perceive real-word situations. Likewise, philosophers are constantly trying to identify the most important properties of the issues they reflect on, like life, conscience or God.

Under this perspective one might be able to make a consistent mapping of the ideas behind programming languages and the ideas that philosophers have come up over the years. It is perfectly reasonable to consider the programming languages as the different philosophies of a virtual world, in which entities do exist and interact with each other. To this respect, even the fundamental philosophical questions receive an interesting transformation: For example "What is self-conscience?" can be rephrased as "What is reflection?".

To the fun part, one might ask: "What if philosophers were programmers? What programming language they would use?". Well, here are my answers!


Socrates : The Hardcore Assembly Programmer
Socrates was one of the founders of philosophy but this is not where the connection ends. Socrates had devised a clever methodology to win every debate. He kept asking questions until a contradiction was reached. So, when someone would claim "morality is important", Socrates would ask "How do you define morality?".


In a similar manner, everything in Assembly begs for a question. There is nothing pre-assumed (at least in pure Assembly, not the distros filled with pre-processed libraries and other junk) and everything has to be as succinct as possible to have a meaning. If you were to work with the programmer Socrates and shared something like "var x = null;", your partner would start by asking "What is var?" !



Aristotle : The Influential C Programmer
Aristotle had a huge impact on Western philosophy, founding many scientific areas, from physics to biology. He was the first to closely examine real entities as the real essence of everything, in contrast to Plato's abstractions. His philosophy is driven by the golden mean as the key to reaching morality or understanding life (matter and form).

The C programming language was equally influential to the design of all other "programming philosophies", most obviously in the syntactical level. In addition, by the time of its writing in the early 70's, C was supposed to be the golden mean between the so-called high-level languages and the Assembly language, combining the capability to write machine-independent code combined with the power of low-level access.



Plato : The Idealistic C++ Evangelist
Plato is a huge figure in philosophy, student of Socrates and teacher of Aristotle. That said, I owe you an explanation about the obvious anomaly: How come that C++ is coming after C? Let me explain. Plato is famous for his Forms or Ideas, that refer to the archetypical versions of the things around us. So, the cup in your desk has is a shadow of a similar oval-shaped archetype in the world of Ideas. In programming words, it is an instance of the Cup class.

Similarly, C++ , as an extension of C, is the first language that tries to capture this idea of forms by giving the developers the capability to abstract the problem before doing anything else. This is a major step by itself, since even if no actual code solving the problem is provided, the classification and the problem modelling are evident and valuable to others. You might wonder, why Plato would not program in Java. Well he could, but there is another parameter to the story: Plato is not so confident how symbols can represent his Forms, and clearly prefers the spoken dialogue (as mentioned in Phaedrus). In a similar manner, C++, not entirely confident in its direction, remains a superset of C, being fully backwards-compatible with the more non-ideal syntax of C.



Stoics : The Happy Perl Community
Stoics and their philosophy (Stoicism) had silently, a far-reaching impact not only to Western philosophy but to the philosophy and the global culture as a whole. Interestingly enough, there is no single man behind it, but it was actually a collaborative intellectual achievement. Stoicism denies anything immaterial and tries to explain the world through propositional logic. So, Stoics reject everything Ideal and concenrate in morality, in which they call us to get free from anything we can't control, but rather appreciate the freedom to self-introspect and reach true wisdom. Stoicism rejects political systems and other formalities, and promotes Socrates' citizen of the world for everyone. People are meant to be brothers, away from distinctions, aiming to contribute happily to a society of friendship and love (jus commune gentium). You should already notice the influences to most widespread religions, like Christianism and Buddism.

Most interestingly, Perl was created in the 80's, a decade in which finally logic/functional programming had found its place in the programming languages world. However, the Perl community (and language) shares much more striking similarites with the Stoics and their philosophy. Perl as a language is to the best possible extent, free of form. Actually the most common phrase in the Perl world is "there is more than one way to do it" or TIMTOADY for short. The philosophy behind Perl rejects syntactical constraints, giving the freedom to its programmers to code in their style, but at the same time encouraging sharing and contribution to the community. Perl's power lies to a great extent to the existence of CPAN, the archive of modules and software happily shared by Perl programmers all around the globe. The language's influence to the programming world has been silent, but much more far-reaching than what is immediately observable. One could mention its strong influence to scripting, dynamic typing and functional programming, but it could be summarized to a joke which is familiar to Perl fans: The next market's crash will be triggered by a bug in someone's Perl script.



Rene Descartes : The True Java Guru
Descartes was the first philosopher of the Western culture to stand up against the Classical Ancient Greek philosophy. His core philosophy as mentioned in his famous Article 7 of the "Les principles de la philosophie" is based on the concept of cogito (=intellectual ego). Descartes believes that doubt is a proof of existence, and cogito is the cause of doubt, arriving to the famous "cogito ergo sum" (=I think therefore I exist). The cogito is not just another process we do, but actually all we do. So, what we want, imagine or feel is directly accessible through it. Descartes nearly 'proves' the existence of God, by the fact that we are able to think about the necessity of his existence. In fact any Ideal or Form can be directly accessible by our cogito. Descartes also marks another landmark in the history of philosophy: Beginning from his work, philosophy is trying to avoid confusing abstractions and to establish a succinct, almost geometrical form. Descartes presents his ideas nearly in the form of theorems.

Descartes would be the perfect Java guru. Java was the first strongly-typed language, in which everything must have a type (or share a Form) before it is being used, matching perfectly the Descartes' efforts to be always exact for what he is talking about. Descarte's cogito is in fact a revisit of Plato's Forms, with a slight variation in which ideals exist because we think about them and not in another universe. To that respect, his philosophy is purely object-oriented, as the solutions in which we arrive, are direct products of our intellects.




Immanuel Kant: The First Python Programmer
Kant found the 'easy' way to the pantheon of philosophy by rejecting two prevailing and opposing methodologies, Descarte's cogito and the empiricism, by shouting 'It's both!'. Kant investigated how humans reason, claiming that experience offers the truth, but which has already been filtered by intellectual judgement (a priori). At his mature years, he examined aesthetics, and the theory trying to explaining the way we perceive beauty. Kant was an extremely concise personality, being obsessed with tideness and exactness, doing the same things, exactly at the same time every day, to the extent that his acquaintances were 'using' him to calculate time!

Similarly, Python is a programming language that tries to combine different solutions and promote it as a new one. As a language it accepts multiple programming paradigms, from object-oriented to contract-based programming. Python programmers reject the free formats of languages such as Perl, and although they borrow several features from it, they emphasize on simple and explicit code. Python becomes so 'obsessive' that imposes whitespace identation as delimiters for code blocks to its users. In the "Zen of Python", the first out of the 19 commandments, the first one is "Beatiful is better than ugly". Kant's obsession to beauty and aesthetics, makes him triumphantly the first Python programmer ever.



Ludwig Wittgenstein: Natural Born Haskell Programmer
Wittgenstein reformed Western philosophy going as deep as to examine Socrates' 'recipe' for debate success. His monumental work, the Tractatus Logico-Philosophicus, can be compared to a hard graduate mathematical book in Logic. Wittgenstein identifies the semantic and symbolic forms as the root of all philosophical problems, leaving the rest that can be explicitly defined as the subject of science. Using pure logic, he deducts that language inherent ambiguity is what makes philosophy repeat itself, and closing his book with the famous 'What we cannot speak of, we must pass over in silence', claims to have solved,..., all philosophical problems.

Wittgenstein is a natural born Haskell programmer. Haskell was not the first functional programming language in town, but from late 80's and onwards, it has prevailed as the most important among the group. Haskell is not meant to be accesible by anyone, and just like the austere and succinct Tractatus, as Wikipedia states, it has a strict mathematical and logical form. Haskell, being purely functional, goes as deep as redefining the way we treat abstract data types, the same way Wittgenstein goes back to Socrates' dialectic to reform modern philosophy.


These all may sound weird, but for programmers, it is easy to realize these deeper connections. I am not quite sure if the same holds for philosophers. Anyway, at least by now, it should make much more sense why in every article in Wikipedia, presenting a programming language, there is special section named "Language philosophy".

My Daily Encounter With Orion

Programming is happening mostly at nights, at least for my case, because this guarantees silence and concentration. I barely can stand 2-3 hours of constant typing (unless I am really absorbed) and so I am frequently making some quick breaks by hanging around my balcony to get some good, fresh air.

I have the great privilege to be able to watch a big deal of the sky, since my house is not located at the center but in a quiet semi-suburban area near the sea, but never actually bothered to know more about the things I was gazing at. This is when I decided to set up Stellarium on my computer.

The first lesson was about the big bright triangle that I was seeing more and more often lately. It's base has three stars and it turns out it is the most known constellation on Earth. It is the Orion constellation (photo). If I was less ignorant, I'd know the basics, i.e. that Orion is visible to almost all of the Earth and in Greece it moves from East to West during the winter nights.

Wikipedia states it is the longest observable constellation, since it was formed 1.5 M years ago and it will last for the next 1M-2M years, thus having a great bond with human civilization. Orion is to be found on all cultures. In Ancient Greece, Orion was a Hunter who questioned Gods' power on Earth. Near Orion, on can find many other sky landmarks, such as Sirius on the east, the brightest star of the sky, the Taurus on the West and the Pleiades, the little 7 stars who look a little cloudy, which are next to it. Ancient imagination sometimes linked these stars together by picturing a fight of Orion with Taurus or sometimes Orion chasing the Pleiades.

Once an overwhelming experience that excited the imagination, then a great navigation tool, the stars seem to have lost their impact on human civilization (well at least the sky stars). Astromoners have computed their behaviour to the final digit to begin with.

However this is still a rewarding activity and altough I don't know if I ever forget anything that I have learned so far, since it is already a daily routine I will expect that at 4-5 am of every winter morning, Orion will be there pointing to the North, making some good company during my quick and cold homeworking breaks.

Donald Knuth And The Complexity Of Songs

Donald Knuth is a living legend among computer scientists. His monumental work-of-life "The Art Of Computer Programming" (I was never able to fully follow it) is a standard reference/textbook/work of art for computer science. Mr. Knuth took a new, immature geek-only hobby and transformed it into a solid and complete scientific discipline.

Among his contributions, the systematic analysis of the complexity of algorithms, really stands out. As a concept complexity is very familiar to developers. It refers to how a specific algorithm or algorithmic solution to a problem scale to the 'problem size'.

If his work is a reason to admire him, his humor is a reason to love him. Recently, I had the most splendid time reading his hilarious paper titled "The Complexity Of Songs" ! Here is the story.

A song has some lyrics which we have to remember in order to sing it. Humans are trying to learn lyrics of length s (space complexity) to sing a song of length n. You would expect mathematically that s ~ n, but Knuth investigates all human 'inventions' to reduce the song space complexity!!

After a series of 'theorems', Knuth proves the existence of songs with s= a.n, a<1, s="O(sqrt(n))," s="O(logn)">and finally.....s=O(1)!! Yes, you read well. 

Knuth argues the first step was the invention of the refrain, the repeated part of a song. If the song has m verses of length V and the refrain has length R, then the song length is roughly n = V.m+R.m and the complexity s = m.V+R. Hence we have a reduction of V/V+R in song complexity. So, a refrain is also a tool to save some memory space to remember the song!

Remember the old song "Old Mc'Donald had a farm, Ei-gh,Ei-gh,Oh!" ? Well this has a complexity s = O(sqrt(n)). It is much like the Greek song "Οταν θα πας κυρα μου στο παζαρι...". In this pattern, each verse includes all previous verses. For m verses n = o(m^2) and s = O(m), so s = O(sqrt(n)). This means that it is much easier to remember! That's why they are most suitable for children! A same pattern can yield a log complexity.

Now to the best part, Theorem 2 is the real killer! The introductory text goes as follows:"However, the advent of modern drugs has led to demands for still less memory, and the ultimate improvement of Theorem 1 has consequently just been announced:

Theorem 2 (Donald Knuth)
There exist arbitrarily long song of complexity O(1)

PROOF. Define U = 'uh huh','uh huh' and the k-th verse Vk = 'That's the way', U, 'I like it', U . This is a constant V. Then the song V^k, completes our proof!! Oh dear!

This last one really cracked me up!!!   :)

A One-Line Code on How To Parse a URI

Uniform Resource Identifiers(URI) is perhaps synonymous to the Internet itself. As the name implies it is used to identify something on the net. Typically a URI is a different than a URL(Uniform Resource Location) but in most cases we use them interchangeably.

Mathematically URI=URL+URN, which means an entity can be identified either by its location(URL) or its name(URN). See the relevant Wikipedia page for details.

Every URI has the following syntax:

(scheme)://(authority)(/path)?(query)#(fragment)

with query and fragment being optional. So in a URI like http://www.example.com/comment?id=1#1123, we have:

scheme="http"
authority="www.example.com"
path="/comment"
query="id=1"
fragment="1123"

A very common task is when we have to parse a URI into its components. The most usual case is URL normalization, which is essential for search crawlers. During this process, a URI is trinsformed into an equivalent canonical form, such that 2 different canonical forms cannot refer to the same resource. For example the URL http://www.example.com and the URL http://www.example.com:80 , are different but refer to the same location on the net. This greatly helps spiders retrieving every time pages they have not visited before. You can find a good article about URL normalization here.

To parse a URI using a regular expression is really easy. In fact the method to do this is given by Tim Berner's Lee (photo) himself in the RFC for the URI standard. Using Perl, if $uri has the URI then by applying

$uri =~ /^(([^:\/\?#]+):)?(\/\/([^\/\?#]*))?([^\?#]*)(\?([^#]*))?(#(.*))?/;

Then:
scheme=$2
authority=$4
path=$5
query=$7
fragment=$9

That's it! All languages (C#, Java etc) has an almost identical syntax and it is straightforward to port it there given that $n refers to the n-th group.

Wi-Not Mobile: A Different Pocket PC Freeware Application

[This article discusses a new freeware application for Windows Mobile devices, in the creation of which I feel excited to have participated. It is called Wi-Not Mobile and can be found in this site]

You may have noticed for some time the link on the right. It points to a new mobile platform we built, a limited number of talented professionals and me. Starting with silly little ideas, it quickly grew to integrate some research results of our personal projects. The result was Wi-Not Mobile, a mobile platform for information, communication and entertainment. Wi-Not Mobile is a freeware pocket pc application and you can find it here. It is on air for a week now and with limited or now advertising has far exceeded 1000 registered users and many more downloads. Here is a small tribute to this one-year full time effort and some technical details for those who are interested.

For a quick introduction here is a Youtube video, which tries to describe Wi-Not Mobile in 4 minutes:



We have received so much feedback, both good and bad and I feel really intrigued in trying to understand how others view this application and we are so enthusiastic we really see criticism as an opportunity. It may sound typical but it is true. Here is my understanding.

First of all, Wi-Not mobile is not an application. It is a platform and there is a perfect reason for that. The platform is a more general concept and solves much more difficult problems than an application. Wi-Not's platform defines a basic API which we later use to create all different functions that exist inside. Let's iterate through the various features to show how.

Wi-Not Mobile offers free web TV streams for your Pocket PC, but also the client to consume it. If we were limited to this (like tons of other popular TV playing programs) then Wi-Not would be an application. But we are not. TV streams in Wi-Not Mobile are updated automatically whenever there is an available new version of the TV channels. The same holds for the web radio streams found in Wi-Not Mobile. This alone does make an excuse for the term 'platform' but the story is not over yet.

The Instant Messaging module of Wi-Not is a clean and efficient way to exchange messages. Of course it was not built to compete with already popular mobile IM clients (MSN Mobile and others). But there is something special inside that is actually a function that the underlying platform can support. It enables for automatic translation. For example, a user can define that he writes in Spanish, his friend define that he reads in Italian, and still be able to communicate in their native languages, a feature that at least for now cannot be found in any other IM mobile client.

Moving on to entertainment portion, we arrive to what we tenderly call "Wi-Lol". There is another trick here. What it can do? You simply type the game you want to play, and the module (using the platform's middeware) will locate free and available flash games to play in your mobile. The difference with other similar functions is that we do not store a single byte of game content. So how is it done? The platform defines a search API, which can have many "flavors" (inheritance in programming words), and these can be a web search, game search, or a music search. Each of these kinds share some common properties (base class) but in the meantime have many differences. The game search module of the Wi-Not locates flash files and reports back available links. With the embedded flash player it is possible to play it on the fly!

There is also a great option for getting music on your pocket pc. We have put there a special media finder, which can accept keywords for your search and locate audio files. This is entirely new for a mobile application. Again, we do not store a single byte of audio content. With a hit rate of around 90% you will able to get the mp3 file you were looking for. Again the platform here plays the most crucial role. It is able to maintain places that are more probable to provide with an answer and also to dynamically find more.

I could really talk for hours about this stuff but I am sure it will bore most of the people. There are certainly many more posts coming about this effort. We are really excited about the outcome and ready to do more. There are also many hard lessons from releasing into public your creations and perhaps I should talk about it later.

Till then you are more than welcome to download Wi-Not Mobile from here and I would be grateful if you could provide with your thoughts about it. Have fun!