Sunday, February 13, 2005

A game-theoretic model of providence

I. Setting the table

What are the leading objections to predestination and providence? They rob us of our freedom of choice. In so doing, they rob us of our responsibility. And they conduce to fatalism.

In considering these objections, I’d like us to consider a game-theoretic model of predestination and providence. Providence is the historical execution of the decree. Providence is, if you will, a temporal mirror-image of the timeless decree.

Although predestination and providence are not the same thing, yet objections to one are much the same as objections to the other. So, for purposes of this analysis, I won't differentiate between the two.

II. Scriptural precedent

A game-theoretic model of providence has a basis in Scripture. For the casting of lots was a common method of ascertaining God’s will. This was true, both with respect to the profane lottery, as well as the Urim and Thummin, which seem to have been a sacred lottery or the functional equivalent thereof. As such, it’s striking that this has not received a systematic treatment in the theological literature on providence.

One reason is a certain prejudice against the morality of gambling in general, as well as unspoken embarrassment over the presence of a lottery the in Scripture. Surely this reflects a superstitious outlook which we’ve learned to outgrow--or does it?

To begin with, if we take the inspiration of Scripture seriously, we cannot dismiss this phenomenon out of hand. There are at least three explanations, which may not be mutually exclusive.

i) Casting lots may have been a superstitious form of divination which God, in his forbearance, tolerated.

ii) Casting lots may have been an efficient, nonpartisan form of decision-making where the choices were either trivial or equiprobable--much as we use a coin toss today. This pragmatic use of lots would be without any inference or assumption of divine guidance.

iii) Casting lots may have been a superstitious form of divination which a longsuffering God not only tolerated, but co-opted--in his overruling providence. The providential use of the lottery is clearly articulated in Prov 16:33, and illustrated in such cases as Josh 7:14-18; 1 Sam 10:20-21; Acts 1:24-26.

III. Recipe

In trying to flesh out a game-theoretic model of providence, I’ll draw on some terms and distinctions from contemporary game-theory.

1. Epistemic uncertainty

A game of chance is so-called because the player is ignorant of the outcome. "Chance," in this context, is the measure of his ignorance.

In game-theory, a hypothetical player is said to enjoy "perfect" knowledge in the restricted sense of knowing the rules, the odds, the possible outcomes and personal consequences, as well as a knowledge of every move made up to the present move.

2. Metaphysical certainty

A game of chance may be uncertain at the epistemic level, but certain at the metaphysical level. A game of chance may be entirely deterministic, which why it is possible to calculate the odds with mathematical precision.

Metaphysical certainty can take different forms. In a game of cards, the sequence of cards can either be random or specified. A random order is the result of a randomized shuffling of the deck. A specified order is the result of a stacked deck.

In principle, a random order could be identical with a specified order. That is to say, the chances are that, sooner or later, a random process would eventually generate the same sequence as a stacked deck.

3. The player

Just as jurisprudence is predicated on the postulate a "reasonable person" in the jury box, game-theory posits a reasonable player, defined as a player who can imagine various outcomes, devise strategies which probilify various outcomes, and opt for a strategy best adapted to achieve his favored outcome.

This, in turn, presupposes that the player comes to the table with certain goal-oriented preferences ("utilities"). These values and incentives motivate the player to make the choices he does.

Some players are naturally risk-averse, while others are natural risk-takers. Some players have a good poker face, while others are a dead giveaway.

What ulterior factors predispose a player to be daring or cautious, to value some utilities over others, to be transparent or opaque, could be genetic, cultural, or some combination thereof. The ultimate origin of a player’s temperament or value-system is irrelevant to the necessary and sufficient conditions of a player in game-theory.

4. The play

Every player has two or more choices ("plays") available to him. Depending on the game, some choices are riskier than others.

In a zero-sum game, one player’s gain is another player’s loss. You win by making the other player lose. You take the jackpot.

Philosophically speaking, the stipulation of multiple choice raises both psychological questions (compatibilism/incompatibilism) as well as metaphysical questions (the grounding of counterfactuals). Game-theory is neutral on what necessary and sufficient conditions must be satisfied to constitute multiple-choice.

5. Backward induction

In chess and poker, each player will try to second-guess the oposing player. The success of this strategy depends, in part, on the degree of transparency or opacity of the players. A good poker player has a poker face. His body language does not betray his intentions.

Tim has a hunch that if he chooses A on the first move, then Jim will choose B on the second move, and so on. To some extent, then, a good player will impose a teleological order to the sequential order, reasoning in reverse from the last move to the first.

This is, of course, only a property of sequential games, i.e., games with sequential moves observable by other players. That’s a condition of "perfect" knowledge.

Of course, if Tim’s choice is logically contingent on Jim’s choice, and Jim’s choice is contingent on Tim’s choice, then this generates a logical dilemma.

How much a player is prepared to bet is affected by whether he has an opportunity, in a subsequent game, to recoup his losses. Backward induction must take in the possibility (or not) of subsequent games as well as the game at hand.

In game-theory, it is useful to postulate an ideal player. An ideal player is able to perform an infinite backward induction, i.e., compute an infinite series of moves and countermoves.

IV. Sitting down to dinner

Okay, let’s apply game-theory to providence.

In game-theory, a player is a free agent if three conditions are satisfied: (i) he enjoys "perfect" knowledge; (ii), he is reasonable, and (iii) he has two or more choices. Let’s take these in turn.

i) Perfect knowledge is not the same thing as exhaustive knowledge. Indeed, the appeal of a game like poker lies in the tension between epistemic uncertainty and metaphysical certainty.

At a metaphysical level, a game of chance has no element of chance. To begin with, there are fixed number of possible variables. In addition, once the deck is shuffled, the actual sequence is also fixed.

At an epistemic level, a game of chance becomes a game of skill because all the variables are not known in advance of the play.

No one says that a poker game is fatalistic. Since a player doesn’t know what card is coming next, his choice is not inwardly constrained by the next card. Whether he asks the dealer to hit him again, whether he chooses to place a bet, raise a bet, call, or fold, is based on the odds of what cards are already on the table, whether face up or face down, and remaining in the deck, in a probable order. There is, of course, the psychological game as well, which I’ll get to later.

So the player is under no external coercion. If he knew he had a losing hand, then he might feel compelled to fold; or if he knew he had a winning hand, he might feel compelled to bet all his chips--but his sphere of freedom lies in the dialectical tension between metaphysical certainty and epistemic uncertainty. And the same hold true for providence.

ii) As I say, game-theory posits certain conditions which must be satisfied for a player to be a reasonable player. It is, however, indifferent to the preconditions.

At the same time, it is easier to construe game-theory along compatibilist lines. A game of poker is not considered to be rigged just because a player’s childhood upbringing predisposed him to be sweaty or foolhardy.

Suppose a losing player demanded his money back because the game was unfair. And the reason he gives is that even though he was free to gamble however he chose, and free to refrain if he had wanted to refrain, he wasn’t free to want what he didn’t want and not to want what he did want. Is that even coherent?

But let’s take a fancier example. Suppose a poker player is so good that the casino is losing money. The casino arranges to have him kidnapped, and operated on without his knowledge. A neurosurgeon implants a microchip which subliminally directs the player not to make certain choices.

Suppose this device prevents the player from making a choice he would otherwise make, apart from surgery. I think we’d generally consider this as having robbed the player of his freedom of choice.

But suppose, instead, that this device prevents the player from making a choice he would not otherwise make, apart from surgery? He no longer has the freedom not to make that choice. But since he would not have made that choice even before he underwent surgery, does his post-op condition rob him of his freedom of choice? In a sense it does, but not in a morally relevant sense--that I can see.

Since he was never going to make that choice, preventing him from making that choice doesn’t look like a significant impediment on his sphere of freedom.

iii) Given (ii), what does multiple-choice really mean? If I can only make one choice at a time, and if I only want one out of the two or more choices, then what is the moral necessity in my having a wider range of abstract options from which to choose?

At most it might mean that the choice I don’t make has an influence on the choice I did make. By comparison and contrast, I see that one choice is better than the others.

Of course, if I didn’t have those other options in the first place, then, by definition, I’d end up making the same choice, sans the process of elimination. Perhaps the imaginative experience of toying with a host of hypotheticals gives me an added sense of satisfaction, but is that a necessary condition of freedom and responsibility?

In addition, as I’ve noted above, you have the same number of variables whether the deck is stacked or randomly shuffled. Indeed, the order may even be the same.

In any event, since a player doesn’t know the order of the cards in advance of the fact, a specified order would not affect his deliberations. He will, of course, play his hand differently depending on the hand he’s dealt, but that is true every time the deck is reshuffled.

Fatalism is a form of second-guessing. The victim of fate despairs of escaping his fate, for perhaps the very effort to escape is fate is the fated means of fulfilling his fate.

Poker is also a form of second-guessing. Yet, again, no one regards poker is fatalistic. The reason that paralysis of action does not ensue is because the abstract dilemma is dissolved by epistemic uncertainties. If you know either too much or too little, then that can be a disincentive to action. If you know too little about the odds and consequences, then any action will be reckless and precipitous. If you know you have a losing hand, any action, short of folding, will be doomed to failure.

But in poker there’s just enough ambiguity that you can make an educated guess and take a calculated risk. And the same holds true with respect to providence. By definition, belief in providence can have no countersuggestive influence on those who disbelieve in providence.

And even for those who do believe in it, their knowledge is abstract and general, rather than concrete and specific. They know "that" there is such a thing as providence, as well as having in mind a general outline of providence in its historical arc and broad promises, but of what future form it takes at any particular place and time--they’re in the dark.

Game-theory is an interdisciplinary field, tied into probability theory and Bayesian logic. The fact that game-theory resorts to the postulate of an ideal player illustrates the fact that uncertainty is parasitic on certainty.

The indeterminist can’t even quantify his epistemic indeterminism without the presupposition of something ontologically determinate (an actual infinite). And this, in turn, affords a striking parallel between the teleology of the decree and backward induction.

In a game-theoretic model of providence, God is the dealer, and the deck is stacked. We instinctively feel that a stacked deck is unfair. But our intuition overgeneralizes from a few irrelevant, paradigm cases.

Suppose, for example, that one of the players is using marked cards. So he is cheating the other players. Suppose the dealer stacks the deck in order to compensate the disadvantaged players. Is that still unfair? Or does such a process of equalizing the odds right the scales of justice? It takes one cardsharp to cancel out the damage done by another cardsharp.

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