It's once again time for the European Football Championship. And as is tradition around here it's also time for betting pools with friends and colleagues.
The usual bets are set up like this: 3 points for the correct score, 2 points for the correct difference, 1 point for the correct winner (not cumulative, you either get 0,1,2 or 3 points). Ie. betting 2:2, with the actual result being 0:0 will get you 2 points (correct difference of 0).
Frankly I know nothing about football, so I've always resorted to odds by bookmakers to help me decide on my bets. But I've often wondered about the best betting strategy. Here's what I got so far:
First of all, betting the right result is usually very trick, so I believe we are better off focusing on betting on the correct difference.
Let's assume there's a maximum of 12 goals which can be scored in ~94 minutes (a goal every 7-8 minutes). That leaves us with 91 (1+2+...+13) possible results. If you bet 0:0 you can either be correct or you can bet the correct goal difference. There are 7 possible combinations with the same goal difference (0:0, 1:1, ..., 6:6). If you bet 1:0 then there are 6 possible combinations with the same goal difference (1:0, 2:1, ... 6:5). If you bet 2:0 you get 6 possible combinations and if you bet 3:0 you get 5 possible combinations. So we can say that if you increase the goal difference in your result you either get less or the same amount of combinations (never more), see  for the math.
Assuming you have no idea who the better team is you should always bet 0:0, leaving you with the highest number of possible combinations. And you probably should bet 0:0 rather than 1:1 because I believe it has happened more often (no source for this).
Luckily we usually have an idea, or rather a probability of who might be the better team. Online bookmakers can provide you with pretty good probabilites via the odds they offer. One company could offer the following odds: Team 1 winning: 1.53, Draw: 4.33, Team 2 winning: 7.00. This gives us the probability of Team 1 winning: 65.4%, Draw: 23.1% and Team 2 Winning: 14.3% .
Now what should you bet? We now have two sources of information, a purely mathematical probability and a probability created by the bookmaker. To decide on which result we bet, we simply multiply the mathematical probability with the bookmaker probability in order to create our own score. Betting 0:0 results in a score of 7.6% (7/91) * 23.1% = 1.7% . 1:0 on the other hand results in a score 65.4% * 6.5% (6/91) = 4.2%, which is clearly superior. So we haven't learned very much so far, except that you should never bet more than a 2 goal difference.
There is one exception though; assume we have the following odds: Team 1 winning: 3.0, Draw: 3.0, Team 2 winning: 2.9 and the respective probabilities: Team 1 winning: 33.3%, Draw: 33.3%, Team 2 winning: 34.5%. Clearly you should bet on Team 2, right? Let's calculate our own score again: Score for a draw: 33.3% * 7.6% = 2.56% and score for Team 2 winning: 34.5% * 6.5% = 2.27%. Surprise! According to our score system, it still makes more sense to bet on a draw rather than Team 2! In other words the increase in winning probability for Team 2 (created by the model of the bookmaker) cannot offset the reduction of the mathematical probability in our score .
Now, I know that I've simplified things quite a bit by only attempting to bet on the correct difference. So if you have a better strategy for betting against your friends and colleagues please let me know in the comments!
 The left column is the difference you bet and the right column is the number of possible combinations:
There's a total of 49 combinations, minus 7 for a difference of 0, yields 42 combinations. Since those 42 combinations are symmetrical, we get 42*2 + 7 possible combinations, yielding again a total of 91 combinations.
 You might notice that these probabilities add up to more than 100%, that's the so called bookmaker margin, ie. the profit a betting company is expected to make.
 This is not a (joint) probability. The two probabilities are not independent, multiplication is simply a way to rank the various bets against each other.
 This score is obviously useless if you assume that bookmakers have a perfect model, ie. they know the actual probability of team x winning. But that's unlikely to be the case, especially in a tournament like the Euro Cup, where some teams play very inconsistently. Hence the need for our own score (or strategy) which gives equal weights to mathematical and bookmaker probabilities.