`determining rationality. The dominance principle argues that performing an action is rational if
`performing the action leads to the best possible outcome for yourself irrespective of whatever
`else may happen (in the case here, regardless of prisoner B's actions it is best for me to confess)
`and that in at least one eventuality performing the action leads to the best possible of all
`outcomes for me (i.e. in this case, I confess whilst prisoner B remains silent and hence I go free).
`The principle of maximum expected utility differs from the dominance principle in that the strategy here is to maximise our expected gains (in this case the maximum utility to us is the minimisation of our expected sentences). This strategy is concerned with the probability of an outcome multiplied by it's utility to ourselves. In the case of the prisoner's dilemma we have four possible outcomes, two in which prisoner B and I perform alike and two in which we
`perform differently. Given that we know from our original parameters that we are both rational
`agents with the same goals then it is fair to assume that we will act similarly, thus giving high
`probabilities to the cases in which either we both confess or we both remain silent and low
`probabilities to the situations in which we act differently. Applying the maximum expected utility principle to the outcomes we have here, we can see in which ones the 'correct' strategy is not to confess. This can be shown statistically if we assign a probability value to the two sets of situations - acting alike or acting differently. Assuming that we are sufficiently similar we can assign a high probability for our acting similarly - let us conservatively set this to 0.9. Thus the probability of us acting differently is 0.1. To look now at the outcomes for each of the cases, we see that in the case of confession our expected utility (remembering that we are looking in this case for the smallest no. of years for us to serve) is probability * outcome, or 0.9 * 5 + 0.1 * 0 = 4.5 yrs. If we look at the other case we can see that the utility is 0.9 * 1 + 0.1 * 10 = 1.9 yrs.
`Thus it is seen that if the two of us are sufficiently similar then not confessing will lead to a smaller sentence. (This is not the case if the probabilities are lower, for example if set at 0.7 for us performing the same then we would indeed do best to confess, however for the purposes of this case we can assume that the probabilities are sufficiently high).
`An interesting point is made by Sainsbury with regards to the case for remaining silent.
`He argues that "the way of silence is unstable", by virtue of the fact that remaining silent leaves
`me open to punishment - i.e. it gives the other prisoner the chance to win freedom by indicting
`me. His argument posits that knowing that my counterpart will reach the same conclusions as I
`do and hence will act the same, leads to my knowledge that he will not confess, for the reasons
`given above. However, Sainsbury argues, knowing this means that I can reach the optimal
`situation in which I receive no sentence at all by simply confessing and thus escaping
`punishment, whilst sentencing the other prisoner to ten years. Sainsbury then correctly points out that, of course, the other prisoner is also in posession of this knowledge since he reasons in the same way as I, and he too, will therefore confess, Q.E.D. - silence is unstable. Whilst this
`argument seems intuitively sound, and does indeed suggest that remaining silent is an unstable
`situation since either one of us has the opportunity to win our freedom by implicating the other,
`a problem seems inherent by virtue of the circularity of the argument. If it is indeed true that the two protagonists reason in the same fashion and hence arrive at the situation that Sainsbury
`describes, we would then find ourselves in a position in which we both serve four needless years - had we both kept our silence we would be free in a year. Thus knowing, we arrive back in the same situation as encountered in the argument for keeping silent - we will both act in the same fashion and thus it is rational to remain silent to minimize our sentences. But, knowing that the other prisoner will do this, I can simply confess... and so the argument continues ad infinitum.
`Whilst this is evidence for instability, it does not, as Sainsbury states, refute only the case
`for silence. Rather, it seems to refute either case, since at alternate points in the circle we can
`see an argument at first for one rationale, then for the next. If this argument does indeed provide grounds for not acting in one fashion then it must also provide grounds for not acting in the opposite fashion.
`Given that the essential problem underlying the Prisoner's Dilemma is one of conflicting
`appeals to rationality by two opposing principles of choice it has been observed by several
`authors that this problem is similar to other choice problems, most notably Newcomb's Problem. Indeed, David Lewis (1978) goes so far as to suggest that they are in fact one and the same problem. In view of this, it would seem rational (!) for us to now consider Newcomb's Problem and examine the arguments for it's similarity to the Prisoner's Dilemma to see if any further light can be shed upon the problem in hand.
`First thought of in 1960 by William Newcomb, apparently whilst cogitating on the
`Prisoner's Dilemma, Newcomb's problem is a choice problem of a slightly different kind. The
`basic premises are:
`þ You are offered two sealed boxes, A and B. You have the option of taking either the
`contents of just box A, or taking the contents of boxes A and B.
`þ You know that box B definitely contains œ1,000.
`þ Box A either contains nothing, or it contains œ1,000,000.
`þ The contents of box A are determined by some "higher being" who has predicted which
`of the two options you will choose. If he predicts that you will take only box A, he has
`put œ1,000,000 in it. If he predicts that you will take both boxes then he has put nothing
`in box A.
`þ The being is extremely accurate in his predictions.
To look at the competing strategies in this problem, we first see a very clear argument
`for choosing both boxes, obtained via the dominance principle. Reminding ourselves that the
`dominance principle requires the best scenario to be reached via one of the possible outcomes
`and for the other outcomes to lead to a situation which is no worse than obtainable by taking any of the other courses of action open to us, clearly we must choose to open the two boxes. If box A contains the œ1,000,000 then we make œ1,001,000, whereas taking only box A leaves us with œ1,000,000. If, on the other hand, box A was empty, taking both boxes at least leaves us with œ1,000, whereas taking just box A would have left us penniless. In either situation then, we are better off by œ1,000 for taking both boxes. However, things are never quite this simple, and it seems that an equally convincing argument can be made for taking only the one box. Since we know that the predictor is extremely accurate, and we have seen from experience that one- boxers make more money, there is a high probability that he will have correctly predicted what we will do. Thus, if we take the two boxes there will be nothing in box A. If however, we choose to only take box A, again the probability is high that he will have predicted this, and hence will have placed œ1,000,000 there. Given this knowledge, would we not be extremely foolish to take both boxes? Surely the rational thing to do is to only take one box?
`It is important to carefully consider the role of causality when considering these problems.
`In the case above, it may be true that there is a causal relationship between what it is predicted
`that we will do and the contents of box A. However, since no actual causal link exists between
`what decision we make once the boxes are in front of us and the contents of the boxes, the
`arguments for one-boxing must be refuted. Whatever you do once the boxes are presented can
`have no effect on their contents, therefore the only rational thing to do is to open both boxes. But how can this be a rational thing to do when one knows that everyone else who reasoned thus made only œ1,000 whilst the one- boxers made œ1,000,000? Supporters of the one-box school of thought such as Bar-Hillel and Margalit would argue that although there is no actual backward causality operating, you must resign yourself to the fact that your best strategy is to act as if there was, and therefore choose only box A. The logic of this strategy has it's roots in the maximum expected utility principle. Assuming, as we did with the Prisoner's Dilemma, a
`conservative estimate of the predictive power, 0.9, say, we find a utility for one-boxing of
`œ900,000, whilst two-boxing returns œ101,000. Clearly the advantage is with the one- boxers. But is this argument really as sound as it seems?
`Consider again the problem of the prisoner's dilemma. We saw that MEU gave support
`to the condition in which maintaining silence was best. In that situation it was discussed that if
`the probability levels were altered then we could arrive at a different conclusion, although for
`the purposes of the example the probabilities could be assumed to be sufficiently high. The same is then true of Newcomb's Problem, although in it's present state the probability would have to be close to chance for MEU to recommend two-boxing, since the relative pay-offs are so different. But let us consider just this point. If we made the relative sums more equal, would this change our actions? Changing the probabilities changes a fundamental part of either of these two problems; in the case of the Prisoner's Dilemma it would imply that the two agents did not reason in a similar fashion, whilst in the case of Newcomb's Problem it would imply less predictive power than was specified in the problem. But is it also the case that altering the sums involved changes the fundamental nature of the problem? In my opinion it does not. Let us then consider what happens by doing this.
`Suppose we changed the problem so that box B contained not œ1,000, but œ500,000.
`(Alternatively we could lower the potential amount of money in box A; it matters not. The aim
`here is simply to lower the potential loss in choosing both boxes when A is empty.) Keeping the
`same probability for the predictor as before, we then see that the utility of one-boxing is still
`œ900,000, whilst two-boxing has risen to œ600,000. Carrying this further, at a probability level of 0.9, setting B to a value greater than 8/11 of A will lead to MEU supporting the strategy of two-boxing. Let us then suppose that we set B not to greater than 8/11 of A, but to a value slightly less than this. Would one- boxers still be one-boxers? Knowing that taking both boxes would at worst give them slightly less money than one-boxing and at best give them nearly double, would they still be ardent supporters of the one-box strategy or would they then consider the gamble worthwhile? In short, would the lowering of their expected losses and the elevation of their potential gains not sway them towards two-boxing? Whilst in the original problem the differences in these amounts are substantial and therefore the risk may not seem a reasonable one, in our new scenario it may be considered to be a worthwhile risk. If this is so (and it seems to this author, not intuitively unreasonable) is there then a point at which that risk becomes an unreasonable one and can cause a switch in strategy?
`It is my belief that this is the case. Further, it would seem likely that if such a threshold
`existed it would be a subjective one and would thus invite other subjective factors to enter the
`scenario. For example, it must be considered what the guaranteed amount of money in box B
`'meant' to the subject. If it were considered to be a small amount then less concern might be
`placed upon gambling it by two-boxing. If however it were considered to be a significantly large amount of money, then the gamble may be considered unwise, however high the probability of success. The very fact that we are dealing with probabilities and not certainties means that at some point the predictor may fail. It is equally likely that this may occur on the next turn as it is that it will not occur for the next ten thousand turns, therefore if the guaranteed sum is considered ample, subjectively it may seem wise to take it. It would seem then, that Newcomb's Problem only presents conflicts when in specific contrived conditions but breaks down if these are manipulated.
`Since, as was stated earlier, it has been observed that the Prisoner's Dilemma and
`Newcomb's Problem are very similar if not one and the same, the same arguments can be
`extended to the former. Presumably then, if the difference in sentence times were manipulated
`a similar effect would be witnessed for supporters of remaining silent (if we believe Sainsbury's
`view that it is only logically consistent to be a one-boxer and a believer in not confessing). Thus, by the same arguments, not confessing is an unstable and inconsistent choice, dependent on subjectively experienced factors.
`In first publishing details of Newcomb's problem, Nozick seems to be undecided as to
`what is the best strategy. As he points out, it would on the one hand be perfectly rational for an
`observer to place a bet that opening both boxes will lead to A being empty. However, if the
`boxes were transparent on the observer's side, he would then want you to take both boxes
`regardless of the state of box A. Given that two equally persuasive arguments seem to exist in
`this and other examples of the genre, and that either of these seem able to be used to sway
`proponents of the other, it is perhaps the role of the philosopher to play the devil's advocate in
`paradigms of this type. It is my opinion however, that in this case the arguments for one-boxing
`(or not confessing) seem too subjective and are thus unstable and logically inconsistent.
`
`approx 3,000 words
Bibliography
`Bar-Hillel, M. & Margalit, A. (1972) Newcomb's Paradox Revisited. British Journal for the
`Philosophy of Science, vol. 23 pp. 295-304.
`Gardner, M. (19__) Mathematical Games.
`Lewis, D. (1979) Prisoner's Dilemma is a Newcomb Problem. Philosophy & Public Affairs, vol.
`8 pp. 235-240.
`Nozick, R. Newcomb's Problem and Two Principles of choice. in "Essays in honour of Carl G
`Hempel". Reidel (1969).
`Sainsbury, _. (19__) Paradoxes. __
