**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)*

**attractiveness**of different betting odds? In what follows, I am presenting what I call as the

*"fu*

*ndamental theorem of betting*".*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:

**? The obvious answer is**

*Can**all of these**inequalities hold***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!**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**

**"Fundamental Theorem Of Betting"**.

**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)**

**1) M > 1.0**

**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**

**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**

**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*

*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!