Zeta and the St. Petersburg paradox

March 21st, 2014

The St. Peterburg paradox is a hypothetical game with infinite expectation. This means: If you are offered this game: Bet everything you have.

The game is played by a bank that will throw a fair coin. The game starts by the bank putting a sum, say $1 into the pot and a player can enter the game by waging money on the game. If the coin lands Tails, the player wins the pot. If it lands Heads, the bank will double the pot. Now what is the fair price for the player.

Usually such games can be treated with simple probability theory. We compute the “average outcome” of the game.

For the first round there is a 50% chance that the bank will pay $1, so the expected value (EV) is 50cts. For the second throw the bank will pay $1 in 25% of the cases (the “other” 50% from the first throw and the 50% chance that Tails comes this round).

So as long as the game goes, the bank will pay 50cts on average. Now the problem: It can happen that Tail never comes and so the average payout for the bank will be

50cts +50cts +50cts +50cts +50cts +50cts +50cts +50cts +50cts +50cts …. = ?

This is a divergent sum, most of us have learned that this goes to “infinity” and that this is not a proper value.

So no matter how much you are asked to be allowed to play: Math says you should accept, because on average you will win an infinite amount of money. This doesn’t make sense in the real world, there is no infinite amount of money.

Lets modify the game a bit. Lets say that you can bet on the event that Tails comes only at the k-th throw. So you will win if and only if the k-th throw is Tails and all throws before where Head. What would be the fair price for such a bet: $2-k    

Now we can hedge our bets to ensure we break even: We bet all possible moves, 50cts for the first throw, 25cts for the second, 12.5cts for the third, etc. All together we pay $1 for our infinite number of bets and we will win $1 all the time. Boring. Actually very boring, because it can take an infinite time until we win our Dollar. Imagine they throw once a minute and it falls 3000 times Heads in a row - you have spent two days in the casino.

We haven’t solved the paradox yet, be just eliminated the infinite winnings, but this wasn’t the problem.

To get to big winnings, we have to place the same bet for all bets, lets calculate what we have to pay the cashier if decide to pay $x for each:

$x +$x +$x +$x +$x +$x +$x +$x +$x +$x +$x +$x +$x +$x + … = 

Again infinity. This makes sens as the expectation is infinite, but careful: Infinity is not a number, it makes no sens to check equality to two infinite values.

What would you say that the value of the sum above is -x/2 ? This you get from the Zeta function: The sum

1+1+1+1+1+1+1+1+1+… = ζ(0) = -1/2 ( any positive number raised to the 0th power gives 1)

Multiply this by x and you get -x/2. Lets do this for the sum of the payout and we get -25cts = ζ(0) * 50cts.

A fair bet is when the expectation matches the wager. So we have -25cts = -x/2 and hence 50cts for the wager.

OK, this is rather shaky math, but does it perhaps make sense?

So we enter the game for 50cts. If Tails fall we get $1 and we go home with the good feeling that we doubled our money. If we don’t win (50% of the time), we have a shot at even higher winnings. Sounds good? It looks like we got into this game far too cheaply.

But we can look at the game in a different way: Actually we always win, but only if Tails falls we are allowed to take our winnings. In other cases we are playing with doubled stakes the game from the beginning. This is similar to the Gambler’s ruin scenario: Instead of taking our winnings, we are waging them again. The twist here is that the game could go on forever and the loss is never realized, but neither is the win. If the bank would offer “double?” if we win it would be obvious that we would play on forever, building an infinite bankroll without ever cashing in (the same principle is used in the Who wants to be a Millionaire TV show, but this game is finite).

If we now limit the game to 10 throws and the player loses if all come out Heads, the expectation is $5, so it makes sense to enter this price, because you can win up to $512. If now the game is limited to 100 throws it, the expectation is $50 and you could get very, very rich. But also you have to survive the first 5 throws to get into the money. In any case: You lose in the majority of games you play.

The interesting fact is that if the game is unlimited, the game degenerates back to the single throw case


Not only open source

December 27th, 2007

I pulled out an old issue of brand eins which remained still unread in the pile of magazines. If you don’t now this magazine yet and you have some interest in off-beat economics and you can read german - get it.

An article with interview with Don Tapscott on his Wikinomics I found quite interesting. The point -in short - is that openess will make you advance quicker and be more successful. I don’t buy everything he says, but he is right in many aspects.

Take the simple example of open source and closed source. Where do you think hides the worst coding and design? It cannot be generalized of course, but closed source has better to get away bad coding. There are exceptions in any domain, Axis for example (at least the 1.4) is a kludge and some closed source stuff I worked on had been beautifully engineered. Anything has to stand the test of time, thus in an active open source problems will be addressed - if not god-like committers reject these changes.

And here I lost interest in continuing until I found Why giving away your code is not dangerous.

Read the rest of this entry »


June 24th, 2007

Just read an interesting article about ravens and their intelligence.

A scientist was cited that ravens developed their intelligence from competition on food within their own species. Most mammals don’t do this, large herds don’t compete within each other. Are they prey because of that because they haven’t had any need to develop their mind? Most predators only focus on their rather limited (in an intellectual sense) prey and neither develop a higher intelligence.

What did I learn from this that made me writing this: We are the same even today in the scale of enterprises. Large company outsource to get large, easy to manage workforces (in hardware we even talk about server farms). In conclusion this means that large companies becomes stupid prey and if you are (want to be ?) part of it your are either prey or will become so sooner or later.

That sums up nicely what I wrapped up a while ago with: “Whenever I leave for another enterprise, it will be one with a total staff of 15 or less.” In such a group you always have a fierce competition, there is no herd that protects the weaker until they fall behind hopelessly.

You can say now “I hope you will die before you get old”, but when I extrapolate what I thought about myself ten years ago and what I know now - I am looking forward to get older.

Finally, it is getting late and I want to watch some film before going to bed: The enterprise I have in mind (this time it will be me with some others) will carry the raven in its name and logo. So you shepards, watch out.

Das macht Sinn

March 14th, 2006

Wer gerade selbst eine Website entwickelt hat, guckt sich manchmal auch gerne die Websites der anderen an. Gestern stieß ich dabei auf die Seite der Übersetzungsagentur Semantis: www.semantis.be. Dort machte es ganz schön Sinn:

“Eine Übersetzung ist dann gut, wenn man nicht merkt, dass es eine ist.”
Sagen wir mal so, ich halte das zwar für eine notwendige, keinesfalls aber für eine hinreichende Bedingung. Es gibt nämlich sogar Übersetzungen, die dermaßen schlecht sind, dass man noch nicht einmal mehr merkt, dass es Übersetzungen sind.

“Die Leser sollten den Inhalt verstehen, ohne die Form erraten zu müssen.”
Hä?! Welche Form jetzt eigentlich?

“Unsere erste Priorität gilt der Qualität.”
Nee, das glaube ich Euch irgendwie einfach nicht. Früher hieß es außerdem immer “höchste Priorität”. Sonst könnte man ja mehrere Prioritäten nebeneinander haben, und das wäre doch ein Widerspruch.

“Semantis Translation ist die gesammelte Erfahrung zweier Profis, die total seit 15 Jahren in der Branche arbeiten und die sich entschieden haben, dass ihre gebündelten Fähigkeiten dem Kunden noch mehr nützen.”
Das macht natürlich wirklich Sinn. Ich arbeite mit meiner gesammelten Erfahrung ja auch total gebündelt in der Branche.

“Ausbildung: Übersetzerlizenz”
Hätte ich auch gerne, wo gibt es die?

“Ihnen einen Service zu bieten, der Sie rundum befriedigen wird.”
Das kommentiere ich jetzt mal lieber nicht.

Nach diesem rundum gelungenen Opening füllte ich selbstverständlich das Formular für kooperationswillige Übersetzer aus. Die Felder “Spezialitäten” und “Beilagen” haben mich ehrlich gesagt etwas verwirrt. Ich würde sagen: Jungs, da müsst ihr noch total nachbessern. Sonst muss man ja erst die Form erraten, um den Inhalt verstehen zu können. Und dass das keinen Sinn macht, habt Ihr ja selbst schon gemerkt.

À la bonne heure

February 28th, 2006

Wie TV-Sportlern wahrscheinlich bereits aufgefallen ist, verwenden deutsche Sportreporter seit einiger Zeit den Ausruf à la bonne heure!, häufig auch: Chapeau! À la bonne heure! (Eventuell auch: À la bonheur!)

Wie es dazu kam, was der Ausdruck im Französischen und im Deutschen (vielleicht) bedeutet, wie er zu übersetzen ist - diesen Fragen ist Creative Translations für Sie nachgegangen:

1. Vermutete Entstehungsgeschichte
Nachdem seit ungefähr 2003 die Ausrufe Chapeau! und Chapeau bas! zunächst während der Tour de France, dann auch bei Sportarten ohne direkten Frankreichbezug nicht mehr ins Deutsche übersetzt worden waren, hatten sie sich im Jahr 2005 abgenutzt und mussten durch einen weiteren französischen Begriff verstärkt werden. Die Wahl fiel auf à la bonne heure!

2. Bedeutung im Französischen
Im Französischen bedeutet à la bonne heure! einfach nur rechtzeitig und wird meistens sogar abwertend im Sinne von endlich, das war höchste Zeit! verwendet.

3. Bedeutung im Deutschen
Die Bedeutung von à la bonne heure! im Deutschen konnte bisher nicht vollständig geklärt werden. Es handelt sich jedoch um einen Ausruf der Anerkennung.

4. Übersetzung
Vom Französischen ins Deutsche: rechtzeitig oder endlich.
Bei der Übersetzung vom Deutschen ins Französische sind zwei Fälle zu unterscheiden:
- Steht à la bonne heure! im Deutschen alleine, rät Creative Translations zu Chapeau! oder Chapeau bas!
- Wird à la bonne heure! im Deutschen zusammen mit Chapeau! verwendet (also: “Chapeau! À la bonne heure!“), empfehlen wir:

“Oh là là là là là là là là là là là là là là!”

mit freundlicher Genehmigung von creative-translations