To understand what I mean by a game and its developments, one would have to classify games a little. A game is basically a game in which you want to determine a winner somehow, in a playful way. The attraction of playing a game lies, among other things, in the fact that, if possible, the winner has not yet been determined. If a game is particularly appealing, exciting, it can even attract spectators. If it is not exciting enough, you can even bet on it.

There are games where you want to eliminate luck factors if possible, and those where you deliberately add them.
There are often favourites and underdogs. But there are also games that are deliberately balanced and fair. Then there is usually an organiser who tries to profit. You can participate in games or watch games. There are children’s games and adult games.

All participants in a game (there are team games and individual games, there is one against one or even 16 starters who all have a chance to win). Nevertheless, there are the possible outcomes, all of which have a certain probability. These add up to 100%.

The chances of winning can be determined before the start of the game or during the game. This does not change the sum of the 100%. Only the distribution can shift. It changes constantly. With each “move” or the element responsible for that game. This goes on until the winner is determined or the whistle is blown. And at the end of the game, one of the possible outcomes is at 100%, all the others at 0%.

Then there are the participants and the spectators. Both parties play a role. The participants have to play the game first. This requires that the game contains moments of suspense. In addition, part of the fascination may come from the uncertainty of a luck factor. The spectators may also have to accept the game by being enticed to watch. Who will win?

My contention then is that the longer the individual outcomes remain away from 100% and the greater the fluctuations, the greater the excitement and the suitability of the game to fascinate both spectators and participants. I will now examine some games and highlight the special features of the individual games on the basis of the diagrams. In this way, I will also examine their suitability as a game, especially as a “money game”.

1) Chess

In chess, the game development diagram depends very substantially on the playing strength of the two participants. However, this does not mean that the tension increases. Chess is the most boring game, in many respects. But that is only in passing. Apart from that, it is in principle only suitable for the players or at best for chess players themselves to watch. Chess will never be accessible to the general public. How is a neutral spectator supposed to feel suspense if he doesn’t understand the moves at all? In tennis, for example, you can tell whether a shot was well played or not, whether it touched the line or was off, even without being able to play. But how are you supposed to explain to a non-chess player that bishop to d3 is nuances better than bishop to c4? He simply turns away and says: “Nah, without me. Play nicely on your own.”

Nevertheless, I can explain the “unsuitability” of chess as a spectator game using the game development diagrams.

There are three typical constellations, each of which yields different types of diagrams. The first constellation is that two weak players play against each other. This constellation will probably result in relatively clearly recognisable and large prongs in the diagram. Nevertheless, this does not necessarily create tension. Maybe just for the participants. The jags that can be clearly seen are all causally related to the gross mistakes that weak players simply make and that would bring about a direct decision among good players.

A neutral spectator might not even notice that and a good player would turn away. Apart from that, one would not want to report or watch too much of bunglers.

The second constellation, “amateur against champion”, will result in a very boring diagram. The master may already be close to 100% before the game. But even if you withhold that from the computer, which only analyses the moves and knows nothing about the playing strengths, the curve will swing relatively quickly towards 100% for the favourite. The amateur simply makes mistakes, the master exploits them. The neutrally analysing computer very soon establishes a clear superiority in the position.

And in the constellation of master against master, the diagram looks like it hardly moves at all. The masters sit there silently, make an incomprehensible move every few minutes and objectively the chances have not changed after this move. And the whole thing goes on for hours. Where is the tension supposed to come from? What’s more, in grandmaster practice 60% of the time it goes towards 100% for the peaceful outcome, the draw.

So here are a few typical diagrams:

Amateur against amateur:

Amateur vs Master:

Master vs Master:

2) Backgammon

With backgammon, things look quite different. The additional element of a random throw increases the tension considerably. Even the best player can’t do anything if the dice play a trick on him. So a typical diagram looks like this: you have a distribution of chances before the roll, then one after the roll and finally one again after the turn. A player’s chances can increase with a good roll of his own dice, with a bad roll of the opponent’s dice or with a bad move of the opponent. The chances decrease if you make a bad roll yourself, if you make a bad move or if your opponent makes a good roll.

Here is a diagram of a backgammon match that was even broadcast on TV. It was the final of London 2008, which my partner Christian Plenz reached against John Hurst. The data and figures provided for the diagram were supplied by the computer programme “snowie”, which is recognised as being extremely good at gambling. For proof: The good players nowadays bet on backgammon positions by still giving their (different) opinions, but as referee the computer program is asked. The opinion is accepted (in the past, so-called propositions were played. The two players whose opinions differed sat down at the board and the position was played until one surrendered). So here is the diagram:

Unfortunately, the match did not go in Christian’s favour. Here are his chances, which in the end suddenly turned out to be 0%. What is curious, as you can see here, is that shortly before the end he was still the clear favourite, with a chance of about 75%. Then came the sudden crash to 42%, then another rise to 67% and then the sudden end.

The reasons for such a development are clear: the match is on the line, the score must be even. The match was played to 17 points. Initially, neither lucky throws nor gross mistakes have much influence. The shifts are always only in the percentage range. What should happen? You make a lucky throw, win a point and have improved your chances by 2 %. Because if you hadn’t made the lucky throw, you wouldn’t automatically have lost the game immediately. In that case, it would have been more like an even game and would have just gone on. But even otherwise you can feel that the change of the match score from 2:2 to 3:2 does not change the overall chances that much.

It is also obvious that a match that is 15:7 does not tend to make gigantic changes. Because if the leading player loses, his chances decrease by only a few percent.

In this match, there must have been a so-called “double match point”. That means that both parties win the match if they win this game. And in this game, Christian had clear advantages until a misfortune befell him. Very often these “misfortunes” are shooting chances that you leave for your opponent and that he also takes advantage of. Obviously, Christian was lucky again and turned the game around. Probably he was able to use the hit piece favourably and bring it around the board again a bit. Then John had to hit the stone again. He succeeded and was finally on the winning track.

Nevertheless, it is interesting to see what information one can still extract from the snowie analysis. First of all, the following diagram:

This requires interpretation: the “Luck John” curve shows how John Hurst’s throws (not the moves) changed his chances. There is, as mentioned, always a distribution of chances before the throw and one after the throw. This is logical, the luck element of dice was deliberately and intentionally built into this game. And if snowie considers the chances after the throw to be better than before the throw, then it is obviously a “lucky throw”. The amount of luck is measurable. It is determined by the increase in the chances of winning as a result of the particular roll. Of course, the reverse is true for a bad roll. It worsens the chances, so it is also bad luck.

The same applies to the “Christian luck” curve. Nevertheless, it is remarkable that both curves of luck increase more or less steadily. This means, translated, that both players were lucky. At the same time, Christian’s luck is obviously strongly limited. Decisively marred by the eventual outcome of the match. He lost (penalty for this form of bad luck, by the way: “only” 15000 euros for second place). In gambler’s jargon, lucky people are also called “newts”. And John was the bigger of the two newts.

The curves “gives away John” and “gives away Christian” indicate the loss of chances due to poor betting technique. At first, snowie’s judgement is seen as objective and correct. So if one has a chance distribution after his throw, then snowie naturally judges this including the associated “best move”. If one does not make this “best move”, one has a loss of chances.

This loss, of course, always increases. The sum of the percentages given away can never become smaller. To do this, you would have to find a better move than the computer, but it is the basis for the analysis. Even if the move were objectively better, the computer would not notice it. Nevertheless, one can see here even more confirmation that Christian was ultimately unlucky. He played better, so he gave away less. Snowie even converts this into percentages. He claims Christian was even 65.42% favourite.

Playing strengths

The insertion of this chapter is necessary sooner or later anyway. There are differences in the quality of the players themselves. In chess this is quite obvious. There are indeed minor coincidences, as explained above. Nevertheless, it is also obvious that one can measure playing strength in the long run, and that there are differences. Even if the outcome of a single game depends on some elements of luck, the stronger player has the much higher chances of winning. In order to make these playing strengths measurable, a Hungarian physics professor by the name of Arpad Elo already thought about this in the 1930s.

I dedicate a separate chapter to this system and its possible applications. At this point I would just like to mention that of course in chess as well as in backgammon (as we have seen in the book there is almost no game, …) where there is not also a certain measure of playing strength. The backgammon diagrams initially assume that these playing strengths are identical. The computer, which has analysed the games and the whole match, does not know the different playing strengths. As we have worked out, Christian played better than John. Of course, that can be a coincidence. There are several possible explanations for this form of coincidence. One would be that John’s decisions were ‘more difficult’ than Christian’s. So he may have to decide more often in more difficult situations. He may also have made the worse decision by chance in many 50-50 situations. Moreover, in backgammon, as in chess, one makes a (right) move for a wrong reason or vice versa. All these elements affect both players. In this respect, Christian’s move choice could also be the better one by chance.

However, if the two were to play many matches against each other, there is no way that this short-term result would be confirmed: Christian is the better player. If that were the case, however (and if I were forced to draw a conclusion, it would only be that Christian is indeed the better player; why should John be, if he made more and bigger mistakes?), then the curve would look quite different from the start. The entry value for Christian would possibly be 56%. And this value would be based solely on the different playing strengths.

The same applies, even to a much greater extent, to chess games. For details on this, however, please read the chapter “The Elo System”.

3) Football

The peculiarities of football are that a goal has such a great influence on the distribution of chances that, in my opinion, the randomness of this circumstance makes the game rather “boring”, at least from the point of view of the development diagram. Nevertheless, here is a diagram that was recorded for the season opener of the second division of the 2008/2009 season between Duisburg and Rostock. The odds were taken from the betfair odds. These odds are changed every second. Here the offered values were recorded once every minute. Here is the diagram:

The blue curve shows the probability of victory for Duisburg, the red the probability of victory for Rostock and the yellow the probability of a draw. The match went like this: Duisburg started the game as favourites (probability approx. 45%). Then after only 3 minutes the score went up to 1:0 and the probability increased to 70%. Then constant further, slight increase. Playing time passed and Duisburg continued to be superior. Then shortly before the break the equaliser, very surprising. The draw rises, the probability of victory falls for Duisburg and that for Rostock rises. Then, shortly after the break, the 2:1. Increase to 80%, because of the later minute of the game. Then, a few minutes later, an (unjustified) penalty including a sending-off. After the penalty is converted, Rostock’s probability of winning the match exceeds Duisburg’s, as the game is now played 11 vs. 10 and the score is even. Then the draw probability rises constantly, while the two win probabilities converge. This went on until the final whistle, the draw increased to 100%, the two win probabilities both went towards 0, and at the final whistle the result was fixed: 2:2.

Here is another diagram of a football match. By the way, the game was as boring as the diagram: 0:0. No shifts in the probabilities or sudden jumps. No special events. The ratio of victories to each other always remains roughly the same. Only this chance that one of the two teams will win at all becomes smaller and smaller as the game progresses, until the draw probability finally reaches 1, the final whistle. It remains a little remarkable that the crash of the win probability for Hanover only really starts at about the 60th minute. Before that, you might say: “There’s still time to score a goal”.

It is important to mention that these estimates for the three possible outcomes were taken exclusively from the betting offers at betfair. This means that they were not only offers, but also “traded” for these values, i.e. bets were placed. It is not possible to give an objective assessment of this. Although one can assume that the market behaves “reasonably” on the whole, i.e. that the assessments correspond more or less to reality, there can still be errors. In particular, the market is based on a basic assessment, so to speak. And this basic assessment is one that I doubt. For I have made my living over a long period of time only by exploiting the errors in the basic assessment.

So Hanover starts here in the diagram with about 62%. That is a fair quota of 1.613. My fair quota, i.e. the computer estimate, which is adjusted by me if necessary according to experience, was 1.59. So the Hanover victory was estimated somewhat higher by me. Stupidly, in this game I failed to get the estimation right. Not only did I estimate it “wrongly”, but I also lost money. I had bet on Hanover to win.

4) Tennis

A tennis match is characterised by special rules. You can “only” win games with single points, win sets with single games and only win matches with several sets. As long as a point is still being played, both players still have a chance. This is different from football in that there is a time limit, a final whistle, and if the score is 0:3 in the last minute of the match, nothing can change about the winner. In tennis, even if you are down 0:2 in a set and the score is 1:5, you can still turn the match around (and I have seen this happen a few times) if you win the next point that is played. Afterwards, a point is guaranteed to be played out again and – depending on the situation – you only have to win the next point often enough to win the whole match in the end. There is no time limit. However, in this respect it does not differ from backgammon.

Recorded here is the semi-final match of the Australian Open 2010 between Andy Murray and Marin Cilic. The situation was that Murray had reached the semi-finals without dropping a single set, i.e. was in outstanding form and, due to the shortness of the matches, was also in a better state of fitness, while Cilic had to go over 5 sets in practically all matches and in some of the very evenly contested games won very narrowly, although he also played very strongly. However, the market had identified a clear favourite before the match. Here is the development chart from start to finish, including the explanation of which events were responsible for the swings and my own behaviour and assessments of this market movement:

The chart shows Murray’s chances of winning before and during the match. These values are also taken from the betting offers on the betting exchange betfair. Before the match, Murray was a favourite of just under 80%. During the first rallies, this probability increased in the sense that spectators obviously “wanted to bet something” and consequently played the favourite. For there was no sign of Murray’s superiority. Both won their service games, for my personal feeling even Cilic was “better in the match”. When the price dropped to 1.22, meaning that the probability of victory in this match phase was “estimated” at 82%, I made a bet myself, on Cilic. My reasoning was: I am convinced that the price will soon rise to Murray, I also discussed this on the phone with my betting partner, because we had only played Murray win before the match, according to our computer, at odds of 1.27. So our live bet was now: “We pay 1.22 on Murray win.” Reasoning: We’ll get a better rate at some point later and can then sell the bet for a profit.

Cilic did indeed manage to break. Logically, my prediction had come true that we would get a better rate. This rate was 1.40 (the inverse of a 70% chance of Murray winning). However, we did not sell the bet then, as we hoped for a further price increase. Theoretical (interim) profits don’t count for anything, only what you actually make counts. But as you can see in the chart, the price continued to plunge on Cilic (and conversely rose on Murray), as Cilic played confidently and even won the first set 6:3 with another break. The market rate at that point was 1.66, so Murray was still favourite. ALer Cilic just didn’t let up in the second set, the market virtually “changed” its mind, with the price on Murray briefly rising to over 2.0 (2.02), making him the underdog for that brief moment. Whether it was coincidence or the famous “6th gear” we give to the better players in Grand Slam tournaments, which Murray then simply engaged, is difficult to judge. We still hadn’t sold the bet (probably out of greed), but did so immediately after the break, as the odds were still 1.52, far higher than the 1.22 we had paid on Murray. Since the match then became one-sided and all the experts, during and after the match, declared that it was Cilic’s obvious exhaustion that tipped the scales in the end (never forget: true prophets wait for events to unfold), we had at least done quite a lot right and achieved a decent plus on the match. Murray’s chances of winning approached 100% pretty steadily after that. Why not always play favourites?