God and Game Theory

March 18, 2004 | 23 comments
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Ars Disputandi, which is a journal on the philosophy of religion, has a review of what looks like a very interesting book using game theory to analyze stories in the Old Testament. Game theory is part of the rational-actor branch of social science. It attempts to understand social interactions by creating mathematical models of different “games” and then deriving the optimal strategy for pursuing those games. The most famous example is the so-called prisoner’s dilemma. (The optimal strategy in a single round game is to rat; in a multi-round game it is to co-operate and punish non-cooperators). So here is an exmple of applying this kind of thing to the Bible.

Consider the story of the Fall. God is playing a “game” with Adam and Eve. (In game theory lingo a “game” is not something you do for fun or amusment, etc. It simply means a situation in which people make strategic choices.) God has two choices: He can give them a commandment or not give them a commandment. Adam and Eve have two choices: They can follow God or not follow God. Combining these choices we have four possible outcomes, which can be represented by this box:

FallGame.jpg

We now rank the different potential outcomes for each team. 4 is the best outcome and 1 is the worst outcome. If we assume that God prefers unthreatened obedience to everything and would prefer anything to consequenceless disobedience by Adam and Eve, then you get the rankings represented by the first set of numbers in the boxes. If you make the assumption that Adam and Eve would prefer disobedience to obedience and no punishment to punishment, then you get the rankings represented by the second set of numbers in the boxes. Still with me?

Now imagine that God is trying to figure out which “strategy” to adopt. Should he issue a commandment or not? If he doesn’t issue a commandment, then there is the possiblity that Adam and Eve will follow him without any threatening, which would be his most favored outcome. However, knowing Adam and Eve’s preferences, God will expect them to abandon him without any consequences, which is their most preferred outcome and his least preferred outcome. On the other hand, if he imposes a commandment, he loses out on the possiblity of completely voluntary obedience (his most preferred option), but he also forecloses the possiblity of consequenceless disobedience, his least prefered option. Faced with a choice between obedience and disobedience in the presence of a commandment, Adam and Eve will still choose disobedience, but God will still prefer the outcome to the one that he could have gotten without imposing a commandment. (Look at the boxes, God gets his 2 rather than his 1). Thus the story in Genesis is explained. It represents what is called a Nash Equilibrium (named after the John Nash of A Beautiful Mind fame), namely niether party has any incentive to change their strategy since they cannot get a more desired outcome by doing so. This also explains why you get the same outcome if Adam and Eve “move” first, ie if they choose whether or not to follow God and God, after the fact, chooses whether to impose a commandment or not.*

Of course the whole “game” depends on rather arbitrarily assigning preferences to God and Adam and Eve, so there is something rather ad hoc to say the least about the way that game theory “explains” this story. Still it is fun.

* Here is how it works: If Adam and Eve choose “Disobedience” they know that God will choose “Impose.” If they choose “Obedience” they know that God will choose “Not-Impose.” In a choice between “Disobedience/Impose” and “Obedience/Not-Impose,” Adam and Eve prefer the latter (ranking 2) to the former (ranking 1). Hence they will always choose to disobey.

23 Responses to God and Game Theory

  1. Adam Greenwood on March 18, 2004 at 12:40 pm

    This is fun! We could do it with the Canaanites:

    Choice 1: don’t resist Israel and get wiped out.
    Choice 2: do resist Israel and get wiped out.
    Choice 3: be a harlot . . .

  2. Nate Oman on March 18, 2004 at 12:45 pm

    Apparently the book that got reviewed (I don’t have the book) does this for a number of stories, showing how the outcomes represent various game theory equalibria.

  3. brayden on March 18, 2004 at 12:48 pm

    Nate – I like the story here, but I’m a little confused. How can Adam and Eve be disobedient if there is no commandment imposed by God? Isn’t the presence of obedience dependent on the imposition of some commandment?

    Are you saying that Adam and Eve could act in a way that is not pleasing to God even if he did not previously impose some kind of restriction on them?

  4. Nate Oman on March 18, 2004 at 12:48 pm

    Another problem with the analysis presented above, is that it assumes that God is playing a single round game with humanity. If God thought he was playing a multi-round game, he would offer no commandment in the first round, punish Adam and Eve in the second round by offering a commandment, then returing in round three to offering no commandment, etc.

    Perhaps their is a game theoretic account of dispensationalims in here some place?

  5. Nate Oman on March 18, 2004 at 12:51 pm

    Brayden: I think that you are right. I didn’t make the box, so I am stuck with the terminology. Perhaps, a better way of think about it is the question of whether God should impose consequences for the violation of the commandment, ie the game is being played after God has said “Don’t eat the fruit” and we are trying to figure out whether or not God will impose a punishment. (This would require me to rephrase the story above in terms of God issuing threats, or perhaps it would simply be more intuitive to walk through the game with Adam and Eve as the first movers, as I do in the footnote.)

  6. Dave on March 18, 2004 at 1:48 pm

    Nate,

    That reminds me of a scripture I read once: “And this is my work and my glory, to generate optimal payoffs in one-shot, two-stage perfect information games.”

  7. Clark Goble on March 18, 2004 at 3:41 pm

    The link to the article doesn’t work. It should be:

    http://www.arsdisputandi.org/publish/articles/000097/index.html

  8. Clark Goble on March 18, 2004 at 3:48 pm

    I’m not sure I agree with the way the values are decided. As you say it is ad hoc, but it seems also to reflect a view slightly at odds with Mormonism. I’ve tried to think of a way to reconcile this to 2 Nephi’s conception but can’t really manage it.

  9. Nate Oman on March 18, 2004 at 3:57 pm

    “And this is my work and my glory, to generate optimal payoffs in one-shot, two-stage perfect information games.”

    I like it. It goes with the passage in the Book of Mormon stating, “Adam fell that he might avoid a sub-optimal, non-equalibrium strategy, and man is that he might realize Pareto optimal distributions of resources.”

  10. lyle on March 18, 2004 at 5:29 pm

    Nate:

    along those lines, perhaps on this thread we could work out a prisoner’s dilemna for Book of Mormon stories? Possible suggestions:

    1. Alma, Amulek & asking to save the righteous from the fire?
    2. Nephi on whether to kill Laban or not?
    3. Laman & Lemuel on whether to repent after each individual episode when they are cowed?

  11. Kevin on March 18, 2004 at 7:11 pm

    The question of how well game theory applies to reality, as ultimately described by religious doctrine, is an interesting one. In a multi-round prisoner’s dilemma game, the tit-for-tat strategy is superior to a strategy in which the player always cooperates. Can we apply this to real life and conclude that an eye-for-an-eye approach is superior to turning the other cheek?

    Christianity says no (although it amounts to little more than lip service for most Christians). Why does the prisoner’s dilemma model not apply? Is it because people can act for reasons other than self-interest?

  12. clark goble on March 18, 2004 at 7:17 pm

    An other way to look at it Kevin, is to question the very categories a game theory analysis rests on. i.e. does viewing others as the self fundamentally change how game theory applies? Consider a family. Are the other members of my family really others?

    This is not a point to neglect as I think it leads to conflicting models which lead to a fundamental instability. (I’d make a disparaging point about economics, but I’ll leave that alone for the moment (grin))

    With regards to 2 Nephi, I tend to think God intentionally set up an inherently paradoxical situation. I’d like to analyze it with game theory, but I think there really are some difficulties that I can’t wrap my brain around.

  13. Nate Oman on March 18, 2004 at 10:04 pm

    Clark: There are actually some economists who have built models that assume one regards others as ones self. The basic idea is that you predict behavior by assuming actors take the welfare of another as a maximand. I believe the term of art is “interpersonal utility function.”

  14. Clark Goble on March 18, 2004 at 10:32 pm

    I think the problem is that we don’t consider others ourselves in a stable way.

  15. Wendell on April 8, 2004 at 1:25 pm

    Very interesting theory. However, I was left in great apprehension, mainly because I anticipated your inclusion of the other player of the game. Therefore,an account of external forces that influenced the final out come of this game should have been included, Making your theory even more complete in actuality.

  16. Jack on May 5, 2004 at 9:36 am

    Taking a single round instance of the prisoners dilemma the outcome has alawys struck as being contrary to that which would be desired. However if we introduce the christian concept of charity to the equation, allowing both prisoners to act for the good of the other i.e. not co-operate with the authorities, then the outcome is much happier for both. A good arguement for charity as a teaching I think. In the event of only one prisoner acting in a christian manner, at least he has the comfort of his faith to see him through the sentence.

  17. Scott on July 7, 2004 at 7:47 pm

    I have a comment concerning the initial structure of the game. Although the description involves God deciding to either give a commandment or not, the real choice He had was to give us free will or not. If you substitute “free will” for “commandment” the game seems to make more sense to me. God wanted a companion and had to decide if the companion (mankind) would have the option to reject him.

  18. Tutissima Cassis on March 19, 2004 at 3:35 pm

    Game Theory and Biblical Hermeneutics
    Here is an strange and interesting application of game theory and Nash equilibria….

  19. Sergio on October 5, 2004 at 12:14 am

    My question:
    Is there any group that apply game theory and other branches of mathematical sciences to try to explain the nature of our creators and why they are observing and change the parameters of this lab called planet earth?

  20. Preston Regehr on November 18, 2004 at 2:59 am

    This is an interesting topic. However, 2 Nephi 2:11 reminds us that opposition is a requirement in all things. God, in establishing the game, knew opposition was necessary. And the devil was part of the mix in the Garden of Eden. So, we have to factor his expected moves into God’s decision to command/not command, and anticipated responses of Adam/Eve.

  21. Frank McIntyre on November 18, 2004 at 11:48 am

    Nate and Clark,

    The interpersonal utility function is fine in principle in economics. It just can create equilibrium problems in general (so that there may be no solution or there may be lots of solutions). But other games have this feature. As for “considering others ourselves”, perhaps you mean considering their happiness as joint with our own. This can be done quite readily, and I think we do consider others’ utility to matter to us in a stable way. Even if it wasn’t stable, one could then just model it as having a random component (although this semmes needlessly complicated).

    Kevin,

    “In a multi-round prisoner’s dilemma game, the tit-for-tat strategy is superior to a strategy in which the player always cooperates.”

    This isn’t true. In a repeated prisoner’s dilemma it is an equilibrium strategy to play the outcome that helps both of you. This behavior falls apart if the game is known to end at a certain date. But even then, your factoring of Christianity as being opposed to game theory is also wrong. Christianity claims that the payoffs are not as originally expressed. Once you write down a utility function based on a Christian ethic, you will get agents that behave in ways that reflect the gospel. Since economics does not restrict how the utility function is specified, you are free to write models entirely consisten with Christianity.

  22. Nate Oman on November 18, 2004 at 12:04 pm

    Frank: To a certain extent it seems that the widely precieved tension between Christianity and economics comes from the equivicol way in which economists use the word “utility.” If I say that utility is simply the maximand of some function that is supposed to describe an agent’s behavior, then there is nothing per se about “utility maximization” that is opposed to Christianity (unless there is something in the Sermon on the Mount that would lead me to believe that Christians must have intransitive preferences and the like.)

    On the other hand, welfare economics also uses utlity as a normative criteria by which to judge the desirablity of particular states of the world. The problem here is that the use the concept of “utility” as a maximand of a function describing behavior is completely agnostic as to the content of that maximand as it were. This concept gets taken over by welfare economists and we are left in a situation in which we should maximize something that Christianity teaches to be evil merely because people have a strong expressed preference in its favor.

  23. Frank McIntyre on November 18, 2004 at 2:41 pm

    Nate,

    That’s true. I am often dismayed by how some economists use utillity theory. It is almost as bad as how non-economists use it.

    In any case, the normative goals in economis are all predicated upon the “if” statement, if one wishes to maximize realized preferences, do this. “If” one has some other maximand, then one will need to tweak the model. As you may have guessed, I have no problem with tweaking the model. I have a problem with people who don’t like a particular model and then think that that one model is the only way to do it so if that is wrong then economics is useless.

    That’s like rejecting English literature because you take an English class from a loony professor who sees everything through a Marxist or feminist lens.

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