Newcomb’s Paradox, Particular Redemption, and Sincere Offers

Newcomb’s paradox is a famous puzzle in decision theory that has provoked much discussion. It has been formulated in different ways, but a standard formulation runs as follows.

The Predictor is a person who is able to make a prediction about a future choice of yours with a very high degree of certainty. (In some versions, the Predictor is infallible — a point to which we will return.) The Predictor invites you to play a game involving two boxes: A and B. Box A is transparent and you can see that it contains $1,000. Box B is opaque. You’re now given a straight binary choice: you may pick either both boxes or only box B. But before you choose, the Predictor informs you that he has already predicted which choice you will make and has arranged the contents of box B accordingly. If he predicted that you will pick only box B then he placed $1,000,000 in that box; but if he predicted that you will pick both boxes then he left box B empty.

The million-dollar question is this: What choice should you make? (The thought experiment assumes, of course, that you want to maximize your winnings!)

Some argue that the rational decision is to pick both boxes. They reason thus: the prediction has already been made and nothing can change that now. The contents of the boxes are settled. At this point, either box B contains $1,000,000 or it does not. But either way, given that box A contains $1,000, the decision to pick both boxes must result in greater overall winnings than picking only box B.

Others argue that the rational decision is to pick box B. After all, the Predictor is extremely good at his predictions, which means that if you were to pick only box B then you’d be extremely likely to go home with $1,000,000, whereas if you were to pick both boxes the chances are very high that you’d go home empty-handed.

On the face of it, both lines of reasoning seem cogent — and therein lies the paradox.

My purpose in this post is not to try to resolve Newcomb’s paradox or to defend one particular answer to Newcomb’s question. (To those who ask, I’d say that how one should answer the question depends crucially on what assumptions one makes about foreknowledge and free choice; the paradox arises because commonplace intuitions give support to conflicting assumptions.) Instead I want to ask a very different question in order to develop a theological analogy. The question is this:

Does the Predictor make a sincere offer?

In other words, is the Predictor somehow deceptive or dishonest in the way he presents the offer? Is he as good as his word?

For the sake of simplicity (and for other reasons that will become clear shortly) let’s assume that the Predictor is infallible. What this means is that his predicting choice C logically entails that you make choice C. (Note that this entailment is a purely logical relationship and implies nothing about causation, explanation, metaphysical dependence, or any other ‘thicker’ relationship.)

Given this assumption there are only two scenarios to consider, one for each possible choice:

Scenario 1: The Predictor predicts that you will pick both boxes; he therefore leaves box B empty; he explains the game to you; you decide to pick both boxes; you take home $1,000.

Scenario 2: The Predictor predicts that you will pick only box B; he therefore puts $1,000,000 in box B; he explains the game to you; you decide to pick only box B; you take home $1,000,000.

Now consider what your reasonable response should be after playing the game and receiving your winnings. Start with Scenario 2. In this case you would hardly have cause for complaint. What the Predictor told you implied that if you picked only box B then there would be $1,000,000 in that box — and that’s precisely what happened. Only an ingrate would grumble about not also receiving the $1,000. The Predictor was true to his word and you’re one million bucks the richer for it.

But what about Scenario 1? It’s tempting to argue that here you could legitimately raise the following complaint. “Sure, I was free to choose only box B. But the fact is that there never was $1,000,000 in that box. You implied that I had the chance to win a million bucks, but given that fact about the past, I never really had that chance. So your offer wasn’t sincere.”

It shouldn’t take much reflection to see that this complaint is quite misguided, for two reasons. First, it’s true that there never was $1,000,000 in box B; but it’s equally true (given the way things were set up) that had you chosen only box B then that box would have contained $1,000,000. The reason you didn’t get that money is straightforward: you chose, in effect, not to get it.

Secondly, the complaint is misguided because the veracity of the Predictor’s claims and his willingness to fulfill his promises in Scenario 1 are no different than they are in Scenario 2. He didn’t say that there would be $1,000,000 in box B and he didn’t promise that you would take home more than $1,000. Rather, what he told you was that things were arranged such that you would get a million dollars if and only if you were to choose box B alone.

So the complaint that the Predictor’s offer was insincere is quite unfounded. Whatever the outcome, you made your choice based on what was offered to you, an offer that involved neither dishonesty nor any promise that would not be fulfilled. If you end up disappointed with the outcome of your choice, you have no one to blame but yourself.

Now what does this have to do with theology?

I propose that if the argument above is sound then it serves to refute one popular objection to the Calvinist doctrine of particular redemption (or “limited atonement” as it is more commonly, and infelicitously, labeled). According this doctrine, Christ died in an atoning sense only for the elect, that is, for those who put their faith in Christ and receive eternal life. (This is not to say that Christ’s death doesn’t bring any benefits to the non-elect, but only that it was not intended to make atonement for their sins.)

It has often been argued that if the atonement really was particular rather than universal then the gospel offer can’t be sincere, because salvation is offered indiscriminately to everyone yet provision for salvation has been made only for the elect. So forgiveness is offered to some people whose sins have not been paid for by Christ. If this is the case, so the argument goes, not only can the evangelist be fairly charged with insincerity — perhaps even duplicity — but so can God, from whom the evangelist receives his commission.

But this objection is as misguided as the complaint against the Predictor, for the two ‘offers’ are parallel in every relevant respect. (Note: in every relevant respect.) For on the Calvinist view, the gospel offer doesn’t amount to “Jesus died to atone for all your sins, but you must believe in him in order to receive the benefits of that atonement” (where ‘you’ could be just anyone). That would be misleading if not outright dishonest. (It’s worth noting, incidentally, that none of the evangelistic appeals recorded in Acts sound like that.)

A far better rendering of the gospel offer would be this:

Jesus came to save his people from their sins (Matt. 1:21), to give his life as a ransom for many (Mark 10:45), to lay down his life for his friends (John 15:13). If you will only repent of your sins, put your trust in him, and confess him as Lord and Savior, then all your sins will be forgiven, you will be reconciled to God, and you will inherit eternal life.

It should be clear that this wonderful free offer of salvation is neither dishonest nor duplicitous. It makes no false claims. It makes no promises that would not be fulfilled without reservation.

On what basis then could the unbeliever on the Day of Judgment have cause for complaint? “Sure, I had the opportunity to repent and trust in Christ. But the fact is that Christ never actually made atonement for my sins. So the gospel offer wasn’t sincere.” He would be quite correct about the atonement, given the doctrine of particular redemption. Nevertheless, his complaint is without foundation precisely because of this pivotal truth: had he repented and trusted in Christ then atonement would have been made for his sins.

The reason he didn’t receive forgiveness and eternal life is straightforward: he chose not to receive them. So if he ends up disappointed with the final outcome, he has no one to blame but himself. The offer of salvation was absolutely sincere, the particularity of the atonement notwithstanding.

It should be noted, by the way, that the argument above doesn’t beg any questions by taking for granted a compatibilist view of human freedom, or unconditional election, or any other distinctively Calvinist doctrines. Even if it did, however, that wouldn’t be problematic, since the objection in question concerns the internal consistency of Calvinism.

None of this provides any positive support for the doctrine of particular redemption; that must come from elsewhere. But it does successfully defuse, I think, one popular objection to it.

22 thoughts on “Newcomb’s Paradox, Particular Redemption, and Sincere Offers”

  1. Prof. Anderson,

    That’s an interesting analogy, and I largely agree with your point. But it might also be helpful to consider how the disclosure of information will affect our intuitions about what is fair and what is unfair.

    Suppose that in addition to the Predictor’s game, another game is run by a dice Thrower. In this game, the Thrower asks a player to lay down $10. The Thrower promises that the next step depends on the throw of two six-sided dice. If the dice show 2-7, then the Thrower wins, and the player is out $10. If the dice show 8-12, then the player wins, and the Thrower rewards him a whopping $100 (a net gain of $90). The only catch, the Thrower warns the players, is that he insists on tossing the dice in private, by himself.

    Suppose now that John and Mary hear about the Thrower’s game. They are impressed by the almost-even odds and the potential winnings. And although they don’t know for sure, they assume, perhaps naively, that the Thrower’s game is honest. So John and Mary approach the Thrower, and, one after another, hand him $10, agreeing to play. Without hesitation, and without tossing any dice, the Thrower announces that John has won and that Mary has lost. He hands John $100 accordingly.

    But Mary is upset. “You couldn’t have thrown any dice,” she complains, “because you never left my sight.” The Thrower informs her that he threw the dice before they ever agreed to play the game. “The first throw was a win, and the second a loss,” he tells her. “Since you played the second game, that means it was your loss.” This does not assuage Mary’s anger, however, and she declares, “Your game is dishonest!” Even the winner John is disturbed by this spectacle.

    But why should Mary be upset? Why should John be uneasy? The Thrower delivered exactly what was promised, and is out $80 as a result. Not only that, given the odds of the game, he would have been better off having left it to chance than to have fixed the results.

    In this way, people seem to be especially sensitive to the advance fixing of results, even when it preserves the odds of the game. Maybe John and Mary shouldn’t care about that particular aspect of it, but, for better or worse, they do care about it. So too with libertarians: Maybe they should not care about God’s advance-fixing of who goes where after death, but the fact remains that they DO care about it. Moreover, their concern is intuitive, and resistant to dispassionate reflection.

    It’s easy, I think, for the Calvinist to rationally defend the consistency and sensibility of his view on determinism. However, he still needs to extend sympathy to the libertarian for his strong intuition that advance-fixing is somehow unfair. The Calvinist needs to sufficiently motivate the libertarian to fight off that intuition. This is no easy task, and it is perhaps the most difficult roadblock to reaching the libertarian.

    Just my 2c.

    –Ben Wallis

    1. Interesting, Ben!

      Two quick comments:

      1. I think the players are upset in your scenario because there’s a natural expectation that the dice have yet to be thrown. If that’s the case, John and Mary have some cause for complaint; not because the Thrower lied to them or because the pay-out was unfair (and they ought to recognize that on reflection) but because they were arguably misled on that particular point. I don’t see anything equivalent to that in my scenarios.

      2. Molinists are libertarians but also hold that, in some strong sense, God fixes in advance who goes where after death. So if there’s a genuine objection here, it’s one that some libertarians have to face too.

      1. 1. That’s certainly a possibility. In fact, that’s the very observation I had in mind when I began to write the post! Hence my unfulfilled promise to talk about “the disclosure of information.” In order to secure the sense of fairness, the Thrower must disclose that he has already thrown the dice. But by the time I had typed out some of the particulars of the analogy, I realized that even such disclosure may not help a great deal. I think you are correct that Mary thought the Thrower was DISHONEST because he permitted her to remain ignorant. But, her sense of FAIRNESS was violated, I think, by the fixedness, so to speak, of the game. Imagine, for example, that the Thrower discloses that information before they agree to play. Would that not disconcert the would-be players? I think it would, even if the Thrower was thought to be otherwise even-handed and trustworthy.

        2. I don’t know much of anything about Molinism, I’m sorry to say, so I can’t really comment on that. However, I should perhaps clarify that my post was not meant as an objection to Calvinism, election, limited atonement, etc. Rather, I was trying to make a point about the PSYCHOLOGY of libertarians, and indeed of people in general. I think we should accept that people really do have these intuitions, and though they may be false, they need to be dealt with because they have a very real effect on us, and inspire a sometimes acute bias.

        Sadly, these idle thoughts-o-mine are not at all relevant to your original post, except insofar as it elicited them in me. So, my apologies for the tangent.

  2. James,

    for the two ‘offers’ are parallel in every relevant respect.

    In the case of the Predictor, the subject has the power to choose between the two options, and his [i.e. the subject’s] choice determines whether he gets $1G or $1M. The infallible Predictor does not efficaciously determine which of the two options the subject picks. But the God of Calvinism pre-determines in the divine decrees, not on the basis of foreseen free choice, to which subset of persons to give saving grace, and to which to leave dead in sins eternally. Given Calvinism, all unregenerate persons can’t but reject God unless He gives them saving grace; without saving grace, it is not in their power to “trust Christ.” So if you want to make the Predictor scenario parallel in every respect to Calvinism, you need to make the Predictor install a device in the brain of all subjects that continually prevents the choice of opaque boxes, except when the Predictor turns off the device. Then, the “authentic offer” question is whether the Predictor’s offer is authentic when, in a subset of cases, he presents the offer to subjects without turning off the device.

    In the peace of Christ,

    – Bryan

    1. Bryan,

      I think you must have missed this sentence:

      It should be noted, by the way, that the argument above doesn’t beg any questions by taking for granted a compatibilist view of human freedom, or unconditional election, or any other distinctively Calvinist doctrines.

      The parallel I drew was not between Newcomb’s paradox and the whole Calvinist package, so to speak, but between Newcomb’s paradox and the doctrine of particular redemption. The argument is neutral with respect to one’s theory of free will.

      If your point is that the gospel offer is insincere because (on Calvinism) human choices are foreordained by God, that’s a completely different objection and therefore irrelevant to my argument. (It also begs the question against compatibilism, but that’s another story.)

      1. James,

        What makes the offer sincere in the Predictor case is not only that two conditionals are true [if you do x, y will occur, and if you w, z will occur], but also that the Predictor’s putting the $1M in the box, or not putting it in, is causally responsive to the subject’s choice. If his putting (or not putting) the $1M in the box is not causally responsive to the subject’s choice, then it is not an offer, but just information, such as you might get from a consultant, adviser or strategist. But in the Calvinist system, the individual’s being included in (or excluded from) the atonement is not causally responsive to his choice for or against Christ. Rather, his choice for or against Christ is causally determined by the same divine decree that determined (not on the basis of his foreseen free choice) whether he is included in (or excluded from) the atonement. So for this reason salvation [in the Calvinist system] cannot be offered sincerely, because it cannot be offered at all; it can only be presented as a theoretical space on a logical map composed of a set of conditionals.

        But presenting such logical maps is pointless because (given Calvinism) there is nothing the speaker or listener can do to move anyone from one logical space to another. If the person is non-elect, there is nothing he or anyone can do to save him. And if he is elect, there is nothing he or anyone can do to prevent him from being saved, because all his sins have already been punished on the cross. So the Calvinist gospel cannot be offered, and neither is there any point to presenting it.

        In the peace of Christ,

        – Bryan

      2. Bryan,

        But in the Calvinist system, the individual’s being included in (or excluded from) the atonement is not causally responsive to his choice for or against Christ.

        Once again, I have to point out that your remarks are not germane to the argument. I’m not offering a defense of “the Calvinist system” but a rebuttal of a specific objection to particular redemption. The issue here is whether the the offer is insincere because the box is empty in one of the cases. Your comments do nothing to address that issue.

  3. James A Gibson

    It has been too long for me to remember what is said, but Tom Crisp has made the connection between Newcomb and foreknowledge here: “On Divine Foreknowledge and Newcomb’s Paradox”, Philosophia Christi 1 (1999): 33-42.

    I thought you might be interested in taking a look since it is a short piece.

    1. Bill Craig made the connection too (see here). But Craig focuses his attention on quite a different issue than I did. He argues that Newcomb’s paradox illustrates the compatibility of divine foreknowledge and human free will.

  4. That’s a really interesting argument.

    I am sceptical of this part: “Note that this entailment is a purely logical relationship and implies nothing about causation, explanation, metaphysical dependence, or any other ‘thicker’ relationship.” I don’t think there are unexplained necessary coincidences between contingent facts.

    Another thought. I can’t deliberate about events that aren’t my choices or at least explained by my choices. But given a perfect predictor, I clearly can deliberate about whether the second box was to have contained a million dollars. I know how to make sure that it did and I know how to make sure that it did not. Therefore, whether the second box contains a million dollars is explained by my choice. (Notice that the deliberative principle here does not depend on incompatibilism.)

    So in perfect predictor cases, the prediction is explained by one’s choice. But then the Calvinist analogy doesn’t seem to work.

    A final remark. Sometimes we phrase offers like this: “There’s ten bucks in the top drawer if you want it.” Normally that’s a biscuit conditional–it’s there even if you don’t want it. But on the Calvinist story, there is a sense in which God’s offer is “Here’s salvation if you want it”, but it’s not a biscuit conditional–it’s not there if you don’t want it. This isn’t any kind of objection to Calvinism. It might even be an alternate line of defense of the sincerity claim. “There’s ten bucks in the top drawer if you want it” seems to be a sincere offer even if it’s not a biscuit conditional.

    1. Thanks for the comments, Alex.

      On the first point: I think I agree with you and I should have spoken more precisely. My point was that the entailment doesn’t imply any particular direction of causation or explanation. The point is that NP doesn’t assume any particular model of foreknowledge, e.g., passive foreknowledge versus causative foreknowledge. Of course, Calvinists are committed to a particular model (or subset of models) but I was trying to emphasize that the parallel isn’t affected by that point.

      On the second point: I need to think some more about this, but my first thought is that you’re raising a different objection. The doctrine of particular redemption as such doesn’t entail either that the extent of the atonement explains the responses to the gospel offer or that the reverse is true. The issue I was addressing was whether the atonement had to be universal in extent in order for the offer to be sincere; that is, whether the “emptiness of the box” in the case of rejection undermines the sincerity of the offer. And I think your final remark helps to reinforce my conclusion.

  5. “But it does successfully defuse, I think, one popular objection to it.”

    It would be good to see staunch Lutherans recognize and acknowledge the success of your argument against their objection to “L”.

  6. I suppose the $1,000 is common grace? Not that it makes any difference to the argument, but it’s worth noting that those who don’t experience salvation as a result of the atonement still receive free, undeserved, unearned, unmerited gifts from God.

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  8. James,

    Two questions:

    1.) “…had he repented and trusted in Christ then atonement would have been made for his sins.”

    How does this sentence not entail retrocausality?

    2.) Do you think most objections to particular redemption (including the “sincere gospel offer” objection) rest on logically prior (even if unstated) objections to unconditional election?

    1. Good questions!

      1. There are ‘thin’ and ‘thick’ answers to this question. The ‘thin’ answer is that, in general, propositions of the form If S had done X at t2 then Y would have been the case at t1 don’t entail that S’s doing X at t2 would have caused Y’s being the case at t1. Anyone who accepts the possibility of infallible foreknowledge ought to grant this point. For example: If Judas had not betrayed Jesus then Jesus would not have predicted Judas’s betrayal (John 13:21-26). However, certain models of foreknowledge may imply retrocausality.

      The ‘thick’ answer is that on the Calvinist view, it isn’t the case that the responses to the gospel explain the extent of the atonement. Rather, God’s decree of election explains both the extent of the atonement and the responses to the gospel.

      2. Yes, I think that’s probably right. Most objections to PR reduce to objections to UE. That’s why I think so-called 4-point Calvinism is an inherently unstable position.

  9. The connection to particular redemption is eluding me.

    What does God have to offer someone whose sins have not been atoned for except His wrath?

  10. James, I need to think about this some more, but my tendency is to agree with previous commenters who’ve pointed out the prima facie disparity between how foreknowledge works in your analogy and how it works in particular atonement; viz:

    1. In your analogy, the existence of the reward (call it R) in Box B is decided by the choice (C) of the agent (S). The Predictor is a passive party to S’s choice, and merely honors that choice (in advance).

    2. But in Calvinism, the reward (namely salvation through atonement) in Box B (Jesus?) is decided by the choice of the Predictor. Agent S is a passive party to the Predictor’s choice, and only chooses Box B if the reward has already been placed there for him.

    Ie, for any agent S:

    1. SCB @ t2 > PR @ t1
    2. PR @ t1 > SCB @ t2

    Notice you can still construct the same conditional statement for both:

    1′ . If SCB @ t2, then PR @ t1
    2.′ If SCB @ t2, then PR @ t1

    …but of course, the conditional is obscuring an ambiguity by not taking into account the reasons for each clause:

    1″. Because SCB @ t2, PR @ t1
    2″. Because PR @ t1, SCB @ t2

    You’ve tried to deflect this by saying that it isn’t an objection to particular redemption per se, but rather to unconditional election. I don’t think that flies for three reasons, though:

    Firstly, it’s not clear that the reasons for S and P’s actions—ie, the causal relationship between them—are ancillary to the paradox’s success. In fact, it seems to me that they’re integral to it. The paradox plays out from P’s acting to give the reward (placing it in the box), based on what he foreknows about S’s choosing it. But then your analogy should play out from God’s acting to give the reward (having Christ atone S’s sin) based on what he foreknows about S’s choosing it (which doesn’t reflect Calvinism).

    It’s easy to lose sight of this fact in the conditional statements you formulated, because they conceal the reasons relationship between P’s and S’s actions. But these reasons seem important to the issue of sincerity. It’s one thing for S to lament, after choosing A, “If only I’d chosen B, then I’d have gotten R”. But it’s quite another for him to complain, “If only P had made R available, then I could have chosen B”! The latter complaint seems perfectly valid in this situation. But it’s not valid in the original paradox.

    Secondly, although it’s tempting to try to treat particular redemption in isolation, I think (if you’ll excuse me) it’s rather naive. TULIP isn’t a handy acronym for five important but unrelated doctrines. Rather, perseverance is true because we are totally depraved and would otherwise fall away, obviating our unconditional election. Irresistible calling is true because we are totally depraved and would otherwise not choose to be saved at all, obviating his unconditional election. Unconditional election is true because we are totally depraved and cannot help ourselves, so God has to elect us without consideration for our own inclinations. We can’t separate the various doctrines from each other without undermining the reasons for holding them.

    Thirdly, and in line with my second reason, the “sincere offer objection” is an objection against the internal consistency of Calvinism. If you want to defuse it, then, you need to have a solution that stands up when Calvinism is consistently applied to it. I don’t think it’s right that you can formulate a conditional and say (arguendo), “Here, if we ignore the rest of Calvinism, this conditional shows that particular redemption doesn’t entail an insincere offer” when, if you bring in the rest of Calvinism, the conditional seems to fall apart. Particular redemption can’t be isolated from Calvinism in general. If you can only defend it against the sincere offer objection by ignoring other Calvinist doctrines, then it doesn’t seem you’ve successfully defended it at all.

    Look forward to your own comments. As I say, I need to think about this some more (:

    Kind regards,

  11. Hey James,

    It strikes me that the problem is that in the analogy, no actual thing is offered. Its not an offer, in the normal sense.

    At most, what are you are only offering are possible counter-factual realities.

    If the man chose B, then it would have turned out that 1mill was in the box.

    If a man accepts Christ, then it turns out that Christ would have atoned for his sins.

    But in reality such statements or propositions are simply not offers.

    In actuality a satisfaction made for X and only X, simply cannot be offered to Y. Right?

    The other thing is that the issues have become a little clouded, as I would say limited expiation is incompatible with the offer, but election is compatible.

    What am I missing?

  12. Hey James,

    Something else too, given what you’ve said (re: causal relationships), wouldn’t it follow that the sincere offer is grounded in divine middle knowledge?

    Still thinking,

  13. hey James,

    If I may, something else you said has caught my attention.

    You say: “Yes, I think that’s probably right. Most objections to PR reduce to objections to UE. That’s why I think so-called 4-point Calvinism is an inherently unstable position.”

    David: Leaving aside the 4-point descriptor as vague and fairly inaccurate term that it is, 2 things come to mind:

    1) We’ve found that its the strict TULIPers who generally are the ones insisting on the indispensable connectivity between unconditional election and limited expiation: such that it is alleged that to deny the latter is to undermine the former. Some of us think that there are some really good reasons why this is not so.

    2) The alleged idea of instability. You know that hypercalvinists object to well-meant offer Calvinism because in their minds, their position is more internally robust and tight, while well-meant offer Calvinism is unstable, tending to Arminianism. Clarkians think the same thing regarding Van Tillianism. Internal stability can be in the eye of the ‘systematician,’ and not actually in the heart of Scripture. We all need to be careful about such lines of thoughts even as soft “objections.” And the answer of the well-meant offer Calvinist to the hypercalvinist, would be the same in principle a moderate Calvinist would give to you, I am sure.


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