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*"fu**ndamental 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:**? 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!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!