Game Theory and the Kingdom of God (A Quirky Series Installment), Part 3: "The Prisoner’s Dilemma"

Posted on 5.24.2006


Of all the non-zerosum "games" the one that has received the most attention is a "game" called the Prisoner's Dilemma (henceforth "PD"). What I love about the PD is what a great lens it gives us to look at life and social interactions. I hope after these next few posts you feel the same.

The classic formulation of the PD is via a story:

Imagine you are criminal with co-conspirator. You both are captured by the police and are being interrogated in different rooms. You cannot communicate with each other. As you are being interrogated you are asked to implicate your partner. If you cooperate with the police and rat out your partner ("It was all his idea! I had nothing to do with it!") you are told that you could go free. If you keep quiet you'll do some time in jail.

What do you do?

The problem is that your partner, in the other room, is also being given the same deal. He's being asked to implicate you to save his own neck. What the police are hoping is that you both crack, revealing vital evidence, as you implicate each other. This way they can send both of you to jail.

That is the classic scenario. Game theorists then specify the following "payoff" structure:

If you both rat each other out, both you and your partner go to jail for 3 years.

If you both stay quiet, both you and your partner go to jail for just 1 year.

If you rat out your partner and he stays quiet, you go free and your partner goes to jail for 5 years.

If you stay quiet while your partner rats you out, your partner goes free and you go to jail for 5 years.

(Ignore the legal feasibility or reality of such a payoff structure, the payoffs are taken as given for point of illustration.)

As we have learned, when we can specify choices ("moves") and players and can quantify outcomes, we can use the idea of a game to analyze the decision making process. How then should we "play" the PD? Should I stay quiet and cooperate with my partner against the police? Or, should I defect on my partner and rat him out?

Ratting out my partner, the "defection" move, looks good. If I rat and my partner say quiet, then I get to go free. Further, if he's ratting me out right now I better rat him out. Because if I say quiet while being ratted out I could go to jail for 5 years. By ratting him out at the same time he's ratting me out we will both go to jail for 3 years, but that is better than going to jail for 5 years.

The point: No matter what my partner does MY BEST MOVE IS TO DEFECT (to rat out my partner).

So far so good. But the problem comes in the symmetry of the situation. Your partner is going through the exact same decision making process. In short, rationality is leading both of you, inexorably, to mutual defection and a shared 3 year jail term. This outcome has perplexed social scientists and game theorists in that we expect rationality to lead to optimal outcomes. The PD is a "dilemma" in that rational play leads to mutual defection and a sub-optimal outcome. That is, both players could have cooperated with each other by staying quiet. In that event, the jail term is only a year. But reason draws the players away from mutual cooperation to mutual defection leading to a worse outcome. And the logic of the players is impeccable. You cannot fault their thinking or strategizing. For just this reason, some game theorists have called the PD the "failure of rationality."

As a psychologist, what is intriguing about the PD is how it illustrates, tragically, failures of trust in our world. PD-type encounters are all around us. We play the PD everyday, with all kinds of people. And, tragically, people "defect" on us and we are left holding the bag or paying the price. Many of us have deep emotional scars because of the way someone treated us in a PD-type interaction: Relational interactions where we trusted a person (or a group of people) or financial interactions where we trusted employers, insurance companies, or people handling our money.

Much more remains to be said about the PD. Much more about trust, multiple player PDs, iterated PDs, free riders, etc. Today I just want to make a point about vulnernability. Trust means you are vulnerable. And it is our vulnerability in the PD that makes us choose the "protective," "cautious," and "paranoid" move: Defection. To trust, to cooperate, just leaves us too vulnerable in this life.

So, how does that fit with the Christian life? Too often, I think I play the game of life cautiously. That is, I don't risk my life serving others, fearing that they will take advantage of me. And here's the thing: I'm totally rational for thinking this. I can feel justified that self-protection is the "best" choice.

But we see God, particularly with Jesus, display this crazy willingness to trust humanity. To trust you, to trust me. And He takes risks when he trusts us. He hasn't played the game conservatively.

And we are called to do the same. To do the crazy thing, to allow rationality to fail. We don't keep a record of wrongs (I Cor. 13). We turn the other cheek. We forgive seventy times seven.

That is a difficult thing to do in this world, to trust. But, like God, I trust. I risk.

So, you can play with me. I won't rat you out.

But you've got to trust me...

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6 thoughts on “Game Theory and the Kingdom of God (A Quirky Series Installment), Part 3: "The Prisoner’s Dilemma"”

  1. Brilliant insight. Many of us have been taught that the world is basically bad and out to get us. Basically a gnostic view of the world falling out of a misreading of Augustine. But when we start to see through this kind of lens the basic goodness of the world, especially the basic goodness of God towards Creation. This vantage really opens us up towards a life of generosity. My experience is that people are attracted to that expression and although there have been a few who really took advantage of our generous nature - even when that happened (worse was a con artist that took us for a grand) it really evokes pity, not anger. Pity that they didn't trust the basic goodness of life and felt they had to take what we joyfully gave anyway.

    But isn't that what the Kingdom of God is about, turning things on their heads. Great post.

  2. Jesus is great and all, but not part of rational game theory. Jesus cannot be a solution since rational games are based on a person optimizing his utility. Jesus would simply be a variable that changes the outcomes to utility levels with more favorable circumstances, but then it isn't the prisoner's dilemma, it's a game you made up.

    Secondly, people do not behave according to rational game theory, rational game theory works to an extent on groups, firms, nations, but rarely on the individual. You would best look into Behavioral Game theory for that.

    Thirdly, you and your pal obviously have never actually studied game theory and simply heard the prisoner's dilemma referenced in some other blog.

    I say this because the prisoner's dilemma is famous not just because of one-time rounds, but repeated games. Your argument only works on a finite game. If the game has three rounds, then you can know in the third round, your partner will choose "defect" and therefore you should choose defect. In the round before that, the partner will "defect" so you should then defect. In the first round....

    This is somewhat trivial. The repeated games are what is studied. By "repeated" I mean non-finite games. There is no last "round" in which to reference, therefore, you are forced to create a "strategy."

    A strategy consists of a first move, coop or defect, followed by an "algorithm" such as "always defect" or "defect every seventh time."

    In a sort of "contest," R. Axlerod had hundreds of programmers send in a "strategy" to compete against other strategies in the prisoners dilemma. Whoever, after playing all the others for a non-finite round (note, not the same as "infinite" mathematically), the utility accrued by each strategy is weighed out. Remember that you acrue utility every time you "defect...

    It just occured to me that you never actually explained to everyone how the Prisoner's Dilemma works. You just gave a simplification that allows for this sort of misinterpretation of game theory.

    Alright, consider this. Each move combined with your opponents move gives you a certain amount of "utility points." If you both cooperate, both of you get 8 points (8 for me, 8 for partner). If I cooperate and my partner defects, he gets 10 points and I get 4 (4,10). If he cooperates and I defect I get 10 (10,4). If both of us defect we get (6,6).

    Now, with that in mind, the contest is to see what each strategy gains eventually after competing with each other.

    Imagine a strategy was like a trait in an organism. The contest is between many of these organisms are competing with each other. If you dont get enough points natural selection weeds you out.

    The answer was that, for that contest and many others, the strategy is "tit-for-tat." I begin by cooperating, and from then on mirror every move of yours. If you defect, then next time I will defect. If you cooperate, then next time I will cooperate. Among most strategies, this creates a "reciprocal altruism" that accrues many more points than pure defect and eventually wins.

  4. So, to clarify, your article is taking an example of rational game theory, applying it to human behavior and trust. The fact is that, oddly, we do sometimes trust people. That's why defection-based strategies can work in real life; because you aren't necessarily trying to optimize your utility, initial reactions and reactions following your partners first defection don't have to create the "cycle of defection" mentioned.

    To reference behavior game theory, cooperation in single PDs is common when the two players share some bond. Even knowing the other player is a human, as opposed to a computer, may make that player more likely to cooperate.

    Arguing that Christ was "irrational" by not following the logic of the prisoners dilemma is not surprising. Almost no one acts according to rational game theory.

  6. Very late comment, I realize. But to address Ross's point about Jesus not being part of game theory, I've got 2 counter-points.

    1. Part of what Jesus offers followers is eternity. In this sense, Jesus is creating the non-finite repetition of the cooperate/not-cooperate (PD) game required for 'cooperate' to become the long-term equilibrium strategy. If there is a 'final' instance of the game, the system collapses and 'not-cooperate' becomes the equilibrium.

    2. Jesus also teaches forgiveness. The tit-for-tat strategy is essentially a demonstration of forgiveness (and 'eye-for-an-eye' at the same time). For the tit-for-tat strategy to be successful, you cannot harbor ill will. You respond in kind, but do not cut off the promise of forgiveness if the opponent comes back to cooperation.

    Indefinite time horizons and forgiveness are required for success in the repeated play of the PD. These are both central concepts of Christianity.

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