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!

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.