Fair play: Viminitz's re-formulation of Newcomb's Problem

From Ben's Writing

A paper for Philosophy 3411 - Game Theory [Fall 04]

I'm a two-boxer: willing to forgo the million for the sake of rationality and maintaining my free will (or at the very least, the illusion of it), a defector in a Prisoners' Dilemma, opposed to the Symmetry argument and, as we shall see, to Viminitz's re-formulation of Newcomb's Problem.

Lets start with one version of the original Newcomb's Problem:

There are two closed boxes on the table. The first box contains $1,000. The second box contains either $1 million or no money at all. You have a choice between two actions: 1) taking what is in both boxes; or 2) taking just what is in the second box. Imagine a Being that can predict your choices with high accuracy. You can think of this Being as a genie, or a superior intelligence from another planet, or a supercomputer that can scan your mind, or [g]od [with a capital G]. He has correctly predicted your choices in the past, and you have enormous confidence in his predictive powers. Yesterday, the Being made a prediction as to which choice you are about to make, and it is this prediction that determines the contents of the second box. If the Being predicted that you will take what is in both boxes, he put nothing in the second box. If he predicted that you will take only what is in the second box, he put $1 million in it. You know these facts, he knows you know them, etc.

Beign

You

	predics 1-box	predics 2-box
1-box	$1,000,000	$0
2-box	$1,001,000	$1000

Payoff matrix for Newcomb's game

In his paper Artificial Prudence, Viminitz offers up a denial of Lewis' equivalence thesis, i.e. that a Prisoners' Dilemma is a Newcomb's Problem. He does this by re-formulation of Newcomb's Problem in such a way that no Prisoners' Dilemma can be reduced to it. The re-formulation is as follows:

Before me are two boxes, both transparent, one containing a million dollars, the other a thousand. The box containing the million, however, is bottomless, and it's placed firmly over a trap door, and this trap door, in turn, attached to the lid of the box, such that if I open the box to retrieve the million the trap door will open and the million will fall back down to God, from whence, ex hypothesi, it came in the first place. (Or, in what Lewis presents as the Prisoners' Dilemma version, the million will fall down into the lap of my co-player.) The aforementioned 'attachment', however, is in turn attached to the box containing the thousand, which is poised in perfect equilibrium on a fulcrum the other side of which is a steep chute which likewise leads back down to God from which it came in the first place. (Or, in what Lewis presents as the Prisoners' Dilemma version, the thousand will slide down into the lap of my co-player.) Thus by pushing the box containing the thousand down the chute and hence irretrievably out of reach - and only by doing so - can I snap the first attachment, thereby disabling the trap door, thereby enabling me to safely open the box containing the million, thereby enabling me to retrieve the million.

When I first read Viminitz's re-formulation Newcomb's Problem, it startled me. It seemed to me that he was claiming that Newcomb's Problem had different properties than it really did.

For two problems to be equivalent they must be reducible to each other. I don't think this is the case in Viminitz's re-formulation. In the original thought exercise the player's choice lies between taking the single one-million-dollar-touting box or both the million dollar one and the thousand dollar one. The choice of boxes, as is clearly stated, is most certainly not a mutually exclusive one. This property is, in fact, what makes the question interesting. Viminitz's game fails to offer us both these choices; we are limited to choosing either one or the other exclusively. I'll grant that in the original problem it is a possibility to end up with, figuratively speaking, one box or the other, as the million dollar box might be empty, but the possibility still remains that a player could conceivably walk away with the contents of both — a possibility not accounted for in Viminitz's.

Viminitz's so-called equivalence-preserving re-description only manages to re-describe Newcomb's Problem in the case where the all-seeing deity is completely accurate in its visions. To put it in the terms of the original problem: our choice between one- or two-boxing retro-causes the deposit — or withdraw, as is more likely the case — of the million dollars in the opaque box. This change only succeeds in destroying the thought exercise of the original, as it is no longer rational to be a two-boxer given the payoff structure. The net effect of his re-formulation is for the game to determine the player's decision^[1].

At this point it should be noted that one could be argued that this re-formulation might be modified to account for the discrepancy of the perfect predictor. This modification would involve fixing the so called 'attachment' on the boxes such that it no longer reliably secures the million from falling through the trap door to god if one forgoes the gain of the thousand. Actually, we need not restrict ourselves to one single malfunction we could extend the possibility of malfunctions to every system in the re-formulation. However, a modification of this sort would ruin the original intent of the re-formulation. That is, it would no longer be the case that the acquisition of the million dollars would remain causally independent of the acquisition of the thousand dollars. Thus I submit that the re-formulation still fails.

Finally, the re-formulation implies that players' actions are temporally commutable; that is, the original problem is isomorphic to one in which the player makes the first move, after which the deity is free to decide whether or not to populate the million dollar box. This means that the deity involved in this re-formulation must be a perfect predictor; otherwise the abovementioned case would be more preferable to the deity, as it has no prediction costs associated with it. However, if one reads the original problem, the deity is described only as having high accuracy and not the perfect one implied by this re-formulation. So, just as one's high attendance of a class does not necessarily imply one will always be there, the deity's high accuracy cannot imply its' predictions will always be correct. Thereby, since the deity in the original problem is not a perfect predictor, it cannot be the case that their equivalence holds.

[edit]

Crap Section: notes from the original writing

What if there was a certificate put in place of the placed in the could-possibly-contain-$1-million-dollars box. This marker could be coded in such a way that held information on the decision of the deity and of the authenticity of the marker itself. This marker could then later be used to verify the deities' decision

His re-formulation fails to satisfy the original form of Newcomb's problem, insofar as it does not grant its players the original options. If we are to play his game we only have 2 options. Either we take the $1 million or we take the $1000, exclusively. What is troubling about this is that it is no longer possible to walk with away with both boxes, even if they both contain the dough, which they obviously do.

In the original problem one might be able to ask: Why chose just the one box on the off chance that there may be a causal relationship between just doing so and there being a million dollars contained within it. But you re-formulation negates this, as it is now causal.

It also fails to satisfy Lewis' Prisoners' Dilemma insofar as it fails in reproducing the payoff matrix.

Lewis' Prisoners Dilemma version of Newcomb's Problem:

Ratting gets you $1000
Your opponent not ratting gets you $1 million
Escape from the long sentence (3 years) cost a million; escape from a short sentence (2 years) costs a thousand.

Leslies' deconstruction of Lewis' decision problem relies on the Symmetry argument for it to be equivalent to a PD. That is, the replica (or as the case may be, the other prisoner) choosing to cooperate will quasi-cause us to cooperate and visa-versa; similarly, if the replica chooses to defect it thereby quasi-causes us to defect as well. As a result of which, it is more to your favor to cooperate, as it'll more likely in that case that you will get you your million.

The assumptions / observations (Viminitz's view on Newcomb's Problem):

Box-filling-deity is 100% accurate
The god's decision to place the million influences your decision, or rather makes your decision, insofar as you will always then pick the choice god laid forth — you're fated to do so.

Conclusions?