Roland Dürre
Wednesday November 10th, 2010

Game Theory – The Prisoner’s Dilemma

Is there anyone among you who has never heard of it? The prisoner’s dilemma (Gefangenendilemma) as a central component of the Game Theory (Spieltheorie)?

For those who do not know what it is about, here is a short description of the so-called cover story:

Two gangsters (G1 and G2) committed a crime together. They can, however, hot be proved guilty of the crime itself. But some smaller offenses can be pinned to them. Now they are isolated from each other. Both prisoners are offered the same thing:

They can either

admit to the crime and incriminate the partner in doing so

or

deny it.

The admission may eventually mean that the “chief witness” goes home free, but only if his partner denies. Said partner will then get five years for lying. If both admit, both get four years (reduction by one year because of the admission). If both deny, the crime cannot be pinned to them. Both will then get one year because of the minor offenses.

There are four alternatives:

1. G1 admits, G2 denies: G1 = 0, G2 = – 5, G1 + G2 = – 5
(The one who admitted as chief witness will walk free, the one who denied will get five)

2. G1 denies, G2 denies: G1 = -1, G2 = -1, G1 + G2 = – 2
(Both deny, nobody can prove them guilty, both get one year in jail because of minor offenses)

3. G1 admits, G2 admits: G1 = -4, G2 = -4, G1 + G2 = -8
(Since both parties admit, they both get four years, which means one year less because of being penitent)

4. G1 denies, G2 admits: G1 = -5, G2 = -0, G1 + G2 = -5
(Same as 1., but the other way around)

Since, in game theory, you usually calculate with gains (profit) but years in jail are not really gains, I gave those years a negative algebraic sign. Consequently, the profit can be written as the positive difference between, for instance, -2 on -5, which equals +3.

I must admit that I am not really a fan of game theory in general and the prisoner’s dilemma in particular.

The entire affair is a nice example from the year 1950. Basically, it only shows that sometimes different decisions might make sense, dependent on whether the interests are egotistical or mutual. In my opinion, that is not quite enough “central content of a theory”. Here are some more

weaknesses of the model:

  • You can take random numbers, and the effects will be just as random.
  • The cover story is artificially constructed and oversimplifies a highly complex situation. Consequently, you can interpret all manner of things into it. All of those are hugely dependent on social situations and personal structures.
  • Difficult decisions, and here particularly the ethical ones, can never be generalized.
  • Every individual situation differs from the next. There is no measurable objectivity.
  • The probabilities of a decision are never taken into consideration, yet they are always intuitively there (gangster’s honour, consequences of the wrong decision,…).
  • Taking the example of the prisoner’s dilemma, just imagine how the discussion would change if the death penalty were the maximum penalty in the model. That would leave room for plenty of speculation. And besides, I really cannot imagine how such a situation might be objectively simulated in a field test.
  • For instance, if you attempt to give the prisoner’s dilemma general validity by changing the cover story, you will notice that it cannot be done. You can see it best in an entirely neutral model: two creatures have to decide pro or con totally independent from each other. For the consequences, the success measurements are a for pro against con and vice versa. You get b for both pro instances and c for both con. And the probability that the other party will decide pro or con is appropriated as 50%, or at another value. Using this kind of model, you can easily build up both theories that make sense and theories that do not. And then you can experimentally prove all manner of things.

From a scientific point of view, the prisoner’s dilemma, along with the entire game theory, is probably useless both mathematically and sociologically. Not to mention the economical usefulness, which today is always at the forefront.

RMD

P.S.
🙂 As you can see, I am not really very much in favour of the game theory. Incidentally, the same is true of Operation Research, Profiling or NLP for humans. I tend to get sick in my stomach easily when confronted with them. To me, they often seem like “scientific esoteric” with a high potential for danger in various directions. Therefore, I intend to write a few more articles on game theory. You might call them “criticism of game theory”. The game theory is defined both as part of mathematics (Mathematik) and sociology (Soziologie). And it seems to me that it is just not really possible to combine the two areas.

6 Kommentare zu “Game Theory – The Prisoner’s Dilemma”

  1. Chris Wood (Wednesday November 10th, 2010)

    Dearest Roland, I am no expert, but I do know that game theory leads to useful insights in science, as well as interesting mathematical developments. For instance, it was unclear for decades whether there should be other infinities between the number of integers and the number of real numbers. It could be shown that either way no contradiction arose in set theory. Now it has been found “useful” for game theory to have just one such infinity. I put useful in quotes, because it often takes centuries for some branch of mathematics to show practical use, (other than making a few people happy).
    Game theory explains various aspects of behaviour of humans and other living things. (Evolution can be regarded as a big game with winners and losers). For instance in a game with more or less honest players, it can be shown that the optimal strategy is to be honest, but to retaliate against those who cheat. (I omit various details of the “game”). Animal behaviour can often be shown to fit very closely to such optima. Human behaviour is more complex, but also often fits. Various laws of justice, such as your “crown witness” example, are based on such considerations.
    Of course, to deal with the Mafia, one needs something more than simple game theory. But remember that Newton’s gravity is still really useful, although we now know that it is only an approximation. I sincerely hope that good games theorists will be involved in deciding how to cope with the instability of international finances. But I suspect they are just busy getting rich.

  2. six (Wednesday November 10th, 2010)

    Lieber Roland,

    ich möchte Dir intuitiv recht geben, obwohl ich weit davon entfernt bin, so tief in die Materie blicken zu können wie Du. Habe allerdings miterlebt, dass bei der Erwähnung des Nash-Gleichgewichts (Analyse der Entscheidungssituation) und des cut and choose (Lösungsstrategie) Menschen wohlig aufstöhnen. Als Imponierwissen funktioniert die Spieltheorie also schon einmal gut. Würde mich aber freuen, wenn im Sinne der Lösung von Dilemmata mehr für mich drin wäre. Warte also gespannt auf Deine weiteren Ausführungen und hoffe, sie zu verstehen.

  3. rd (Thursday November 11th, 2010)

    Lieber Detlev,
    mit Erwähnen von Pareto-Verteilung, Paretoprinzip, auch Paretoeffekt oder 80/20-Regel, Pareto-Optimierung, Pareto-Optimum, Pareto-Superiorität, Verschobene Pareto-Verteilung, Paretodiagramm kannst Du Dich nicht nur bei den Damen noch mehr profilieren. Wenn Du Dich dann auch noch outest, dass Du weißt, dass Vilfredo Pareto diese Wahrscheinlichkeitsverteilung entwickelt hat …

  4. rd (Thursday November 11th, 2010)

    @Chris:
    zu Deinem Kommentar:

    … Game theory explains various aspects of behaviour of humans and other living things. (Evolution can be regarded as a big game with winners and losers) …

    Genau das bezweifle ich, besonders den Satz in Klammern. Um das zu sagen, muss man schon Gott ähnlich sein. Vielleicht sind wirs ja, aber sichern nicht auf dieser rationalen Ebene.

    ich wüsste schon gar nicht, wie ich im Kontext mit Evolution – die ich als zuerst Mal als total zweckfrei ansehen würde – Gewinner und Verlierer definieren sollte.

    :-; Oder meinst Du, dass die Dinosaurier die Verlierer der Evolution und die Menschen die Gewinner (nur weil wir in der vierten Dimension an einem anderen Punkt eingezeichnet sind? 😉

    Und selbst wenn ich annehme, dass die Evolution doch einen Zweck hat und dieser ist, Leben und Varianz des Lebens zu mehren (eine kühne Annahme!), finde ich keine sinnvolle Möglichkeit, die beiden Begriffe Gewinnen und Verlieren für die entstehenden und verschwindenden Elemente der Evolution zu definieren.

    Also wieder die Gretchenfrage:
    Ist Evolution ein p-Spiel oder ein s-Spiel?

  5. Chris Wood (Friday November 12th, 2010)

    @Roland; I used “winners and losers” informally. I realise that all are eventually losers. We shall all die, and life in our universe will end in the big crunch or in an entropy desert. Nevertheless all beings that propagate can be regarded as winners, especially ones like the sponges that became the ancestors of all animals. Some genes will probably continue to reproduce as long as DNA-based life exists on Earth.
    Your use of the word “Zweck”, (purpose), implies that games can be played only consciously. This is an unhelpful restriction. My computer plays chess. Animals play, although it is a matter of dispute (i.e. word definition), whether they are conscious. Fate plays games with us all. The workings of evolution produce results predictable by game theory.
    On a more serious note, I believe I am not God-like. Firstly God probably does not exist, whereas currently I do not doubt my own existence. Secondly, if God exists, She is probably a beautiful woman. Surely no man would design a universe on the basis of two contradictory principles, (quanta and relativity). Only a beautiful woman could convince millions of men, contrary to all the evidence, that She loves them and that they will spend eternity in heaven with Her.
    I cannot find p-spiel or s-spiel in wikipedia. Do you think Gretchen can answer your question?

  6. rd (Friday November 12th, 2010)

    @Chris zu s-Spiel und p-Spiel:

    Bei S geht es ums “Siegen”, beim P ums “Partnern” (Optimierung des gemeinsamen Erfolges).

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