Now that we have learned the method with which one can check the quality of one’s own predictions even in the long term and neither have to bet money nor have to consult the assessments of other prophets, it is time to think about how to compare the assessments of two to several players, tipsters, prophets, bookmakers. And in this case, even to determine an optimal, mathematically flawless, correct accounting method.

For this I mention again the problem with the previous example, the one about the quality check of predictions: We can definitely compare in the long run the predictions of two prophets with each other by the given method. We also get a statement. But the statement is still open to interpretation (but in this context, please remember general statements that statisticians are able to make: They provide their statements with a probability of error). So you could still interpret these results in different ways. Who was actually better?

For this I will give you an example of two prophets who, after a number of, let’s say, 1000 games, have achieved the following results: One of them expected a determination, i.e. an average probability of 40.5%. But he only achieved 39.9%. The other, however, had a determination of 38.95% on the same selection of games, but achieved an average probability of 39.2%. If we now consider the criteria that are considered favourable for us: Expecting as high a probability as possible and being as close to it as possible with the probability that occurred, then the question arises with these two results: who predicted better?

The one expected a higher one, but did not quite achieve it. The other expected a slightly lower one, but also underachieved. The deviation is somewhat higher for player 1, but he achieved the other goal of predicting as high as possible better. Should we now consider the absolute deviation? Or should the level of the average expected probability be interpreted in the first player’s favour? And if so, by how much?
In any case, it is not entirely simple. Apart from the fact that there would of course be luck and bad luck, even if we set clear criteria.

So we are looking for a financially or points-wise correct calculation. The way the betting market works at the moment is this: someone offers odds (traditional bookmaker, Asian betting market, betting exchanges). The player, who can also be the provider (betting exchange), accepts or refrains from an offer or asks for the odds he would like to have. But what if one were to let the assessments “bet” against each other? How would one have to settle them?

First of all, we leave out the provider or organiser of the bets, who theoretically has to earn something for making the betting offer available. In this respect, all bets offered are not exactly calculated at 100%. They leave the provider with a theoretical profit, as explained elsewhere. But let us first try to make the “game” absolutely fair and, in the simplest case, let two prophets compete against each other.

I will write down two different assessments and we will then consider how they would have to be settled with each other for each possible outcome of the game so that everyone comes into their own. Here are the assessments for now. I come back to the Bundesliga season opener 2008/2009 Bayern Munich – Hamburger SV. I have already noted down my assessments, here once again and then with an alternative assessment:

First of all, these are simply two different forecasts. Taking into account the vanity of the prophets, let us assume that everyone is convinced of the truth of his odds, his assessment, and would bet on it. Now one wonders how a bet should come about? Each offers the other his pay rate (columns BMQuoten) and asks the other if he is willing to bet on it. The second prophet would even have a good reason to play Bayern to win, given his assessment. This is because the pay odds are above the fair odds. Still, that would not be universal.

But isn’t there a correct settlement system in which it is enough for each of the two to note only the probabilities (and thus in the inverse the fair odds) and thus all bets are already placed? A bet directly on the assessment?

There is the system. Sure there is. You just have to set it up well so that there is no workaround for one of them to write down an incorrect estimate and still obviously profit. I first present the correct system and then demonstrate why a simpler system fails.

The correct procedure (whether playing for money or only for “points”) is: Both bet on all three chances at the odds offered by the opponent. The decisive factor is now the calculation of the betting amount. The amount bet on the fair odds of the opponent is the sum of the two probabilities of the event occurring. By this condition it is clear that both make exactly the same bets, on each chance this is nevertheless different. How much you yourself are willing to bet depends on how high you think the chance is, according to the translated reasoning. The more percentages you give to a chance, the more money you are willing to bet and bet. That makes sense, doesn’t it?

Mathematically, it works out quite correctly. The statement of each participant (here only two for the time being) who writes down an assessment must be: “My probabilities — and thus also the fair odds — are correct. Any deviation from them that someone else assumes is a mistake. I am willing to bet on that deviation.” Such are the conditions of participation. As I said, you can also play it initially just for points (as a betting game) and without wagering any money. But desirable it sounds, doesn’t it? I’m writing down the truth. Anyone who deviates from it, i.e. has a different truth, must pay for it in the long run. Because: if mine is right, I emerge victorious. It is also worth mentioning here: One is powerless against luck and bad luck in all such games. Because: even God (who for this example only knows the true probabilities and then rolls the dice; luckily I have the chapter “Paradoxes” with me. Because: how else could one call the term “true probability”? Oh yes, the opposite of paradox is “tautology”; a self-proving statement) could also lose after a given number of games, events, if he were unlucky. A beautiful nonsense.

You add up the probabilities of the two prophets and so you have the stake per event. In the concrete example, it looks like this:

FC Bayern HSV 1.2080 0.4997 0.2922

These are the sums of the probabilities. This sum is of course exactly 2. We add up the chances of two participants whose individual sum is 1 each. To give an example of the calculation: Player 1 has assumed 56.60% for Bayern to win, Player 2 has assumed 64.20%. This results in 0.566 + 0.642 = 1.208. The player who has the higher chance bets, so to speak, a higher share of this total amount on his assumed higher chance.

These numbers can be converted into monetary amounts by agreeing on an amount per unit. So, for example, 1000 euros per point would result in bets of 1208 euros each on a home win, 500 euros on a draw and 292 euros on the 2, win HSV. Because the amounts are bet on each other, they sound high, but they are not really. All that remains is what one of the two pays more in odds than the other.

This is not necessary in a betting game without money stakes. You play for units.

The point is this: In the spirit of fairness, I am willing to bet on the opponent’s assessment. I am also prepared to bet on each of the odds. At the same time, of course, I am willing to pay my own fair odds. In doing so, I have no disadvantage. I get the advantage from the fact that the other person may pay more, a higher odds. If not on this event, then on the one I consider more probable than he does, and that is in my interest. But only on one condition: He also bets with me, also on my assessments. Then my advantage of a better assessment will already make itself felt. Because: Somewhere he has paid a higher price than I have. And on this event he has paid the rate wrongly. Because the assessment is wrong, too high. So there can only be a dispute about the amount of the stakes. I play with him, he plays with me. That is fair. But how high do we have to play? These components are linear. It is enough to simply add them up.

For intuitive reasoning:
The higher I rate an event, the higher the chance of occurrence, the more money I am willing to bet on it. I control the amount I bet by the probability I assign to the event. If I consider an event to be unlikely(er), I also bet less. Absolutely linear. The lower the chance, the lower the bet. The same goes for my opponent.

But now let’s first look at what happens and what it means from the point of view of the individual player. These stakes calculated above. Let’s assume that we convert it into money for today. Then we agree beforehand how many euros we play per stake point, otherwise it’s just units. But let’s say we play for 1000 euros per stake point today. That results in a stake of 1208 euros for each player on the other’s fair odds if Bayern wins, about 500 euros stake for each on the other’s draw odds and 292.2 euros on the underdog’s win. Now we have to calculate the result, first of all the possible result. What happens now if Bayern wins?

Player 1 must pay player 2 1208* 1.77, the stake * of his own fair odds that he was prepared to pay. But he only gets back 1208* 1.56, the fair odds of the other player. So he loses a total of 0.21 units (the difference between 1.77 and 1.56, 1.56-1.77 = -0.21), making 210 euros. If the draw comes, the X, then player 1 must pay out his fair odds, i.e. 3.59 * 500, but would get back 4.52 * 500 euros. He would win 0.93 * 500 = 465 euros. If Bayern wins away, the calculation is as follows: Player 1 pays out 6.44 * 229.2 and gets back 7.30 * 229.2, wins 0.86 * 229.2 = 197.11 Euro. Player 1 loses if Bayern wins, but wins on X and 2. This seems logical as far as it goes, because his chances of winning on 1 are lower than the other player’s, and his chances of winning on X and 2 are higher.

The amounts he would win are also in proportion to the probability of occurrence. He gets more money back on the smaller chances, the estimates on X differ the most, so the payout amount is highest here.

Now I will show what a correct calculation sheet with all possibilities would look like before the match:

In this way, game after game can be settled correctly. Whether in euros or in points. And with more than two participants in the game, not much changes. All probabilities are added up, everyone bets against everyone this stake at each participant’s fair odds.

In addition, I have included the winning expectations of the two players, assuming that the individual’s estimates are correct. And, as expected, miracles of mathematics, but a prerequisite all the same: both have a positive expectation of winning, from their own point of view. If my estimates are correct, I will also win in the long run. Both think the same.

The calculation is the same as always: simply multiply the probabilities of occurrence by the payouts that will occur and you get the equity. The only special feature here is that because there is no truth, but only two competing assessments in which each participant is convinced of the correctness, everyone multiplies their own probabilities (of whose correctness they are convinced) by the result that then occurs. Of course, there are still all kinds of coincidences. For example, this one: The result turned the course of the game upside down.
In addition, both assessments can be wrong, i.e. the truth can lie in the middle or somewhere else, perhaps with the third participant taking part later? Maybe with you?

Investigating alternative accounting methods (and refuting them).

If one should ask why one has to use such a complicated procedure (it isn’t, once understood, is it?) and whether there wouldn’t be a simpler one, I must of course have an answer ready. I would like to present it. The first procedure that would come to mind would be to simply always bet the same amounts on the different odds. You with me at my odds, I with you at yours. And the amounts are always the same. So 100 euros on each chance, for example. This sounds reasonable and there is no problem with it, as long as both sides make an effort to honestly write down the probabilities and thus the fair odds.

The problem arises like this: One of the participants does not know the correct odds and has no idea at all. He makes it easy for himself and writes down 33.33% for each chance. Here, too, a miracle of mathematics: he cannot lose with this method. The apparent miracle is easily explained by the fact that the procedure is wrong. The reasoning here:

We write down the same as in the previous example, only we replace the amounts calculated in the first example with the amounts 100 – 100 – 100.

As we can see, player 1 gets no reward for his efforts. Even if he is right and everything is calculated exactly, he is only exactly par, not a positive equity (the €0.03 is due to rounding error, it should be exactly 0). This approach is therefore refuted: The amounts have to be different to get a reward for deviating from the equal distribution — if you are right about that (there are certainly games where 33.3% — 33.3% — 33.3% vote for 1-X-2)….

Small note: If player 2 was right in his assessment, he has a positive equity. That is legitimate. On the other hand, it means that every small mistake that player 1 makes in his assessment would immediately have a positive effect on player 2’s equity. Against player 1’s procedure, there are only two ways to get his equity to 0: Either hit the truth or copy player 2’s procedure.

Let’s now take a quick look at what happens to player 1’s equity if we use my (correct) procedure, in the same example. So player 2 sticks to his equal distribution. Now player 1 must come into the profit area after all. You can see:

By grading the amounts on the individual chances in the correct proportion, player 1 receives his correct reward in the event that the event he considers more probable occurs. Of course, it remains the case that he loses on the other chances. But he gets a positive equity if his estimates are correct. If player 2 is right, he naturally also has a positive equity, that remains unchanged.

A second method to determine the amounts to be paid out would be another obvious one: One simply takes the difference in the probability that occurred for the two. If one estimates the chance at 70%, the other at 60%, the event occurs, then it is quite simple: you pay out the 10% to the winner. And you agree on amounts that you bet per percentage point. It’s the same with the betting game. Each winner, a better prophet, receives his reward through the difference in percentage points that he has gained through his better assessment. What is the problem here?

The catch is this: One player good-naturedly writes down his estimate (which he thinks is correct). The other player uses an antidote. Just like in the other procedure described above.

Here the antidote looks like this: I simply write 100% on the (generally accepted) favourite event. Or, for my sake, if that is objected to, I take 98% and 1% respectively on the other two chances. Just to satisfy the basic mathematical understanding that any event lying in the future must have a probability between 0 and 1. It would not be necessary for accounting purposes. Let us see how the equity of the two players develops in this case.

In contrast to the other (correct) procedure, when betting on percentage points, one would agree on a euro amount per percentage point won. For the sake of illustration, I have assumed that the two players have previously agreed on an amount of 10 euros per percentage point. This then results in the following picture:

Here we see the devastating result: Even with an absolutely correct assessment, if player 1 made it or knew it, he already has a negative equity. If you are looking for reasons: With the help of the antidote that one simply estimates the greatest probability even higher, one gets the big winnings on the event with the greatest probability. Player 1 himself admits that the home win has the highest probability. And he still has to pay out a gigantic amount on this (favourite) event. He does not get anywhere near this amount back on the events with the smaller probabilities. Conclusion: it is an unfair system to bet on the difference in percentage points, at least guaranteed when one of the two refuses to write down the real estimates but instead writes down “fictitious” ones, justified only by the (messy, incorrect) accounting method.

Final comment:

To apply this method to a betting game, I imagine that there is a website where all participants are given the opportunity to submit their assessments. Initially, you limit it to the Bundesliga. For example, one could even show the current average assessment in each case. Whether this is an advantage for the player making the prediction remains at least questionable. If he decides to copy the majority opinion, he will definitely not be in the lead but only in the midfield (which could be a good strategy in the long run).

If I sense concerns now, they are certainly not entirely groundless: weekly winners would have to be chosen, and overall winners as well. The person who now tries to torpedo the process by writing down abstruse assessments in order to possibly come out on top just once could be out of the running in the long run due to a bad result. Of course, there are still a few things to consider in order to make the betting game perfect. Especially how exactly. Nevertheless, in my opinion it is a worthwhile alternative. Especially among friends, where I have even been able to use it for some world championships.