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Examinations, and tax audits, are not always announced in advance. There are ‘surprise’ examinations. What is more, it seems that there might be surprise ones consistent with it being announced that they will occur in some speciﬁed period of time. There is, however, a line of argument to the conclusion that it is impossible to have a surprise examination when it has been announced ahead of time that it will occur in some speciﬁed period of time. This is very hard to believe, which is why the line of argument constitutes a paradox.
This research paper starts by stating the line of argument and various attempts to identify the fallacy. It concludes with a short discussion of some related issues concerning cooperative social arrangements, a matter often raised under the heading of the backwards induction paradox. This research paper does not discuss certain technical issues to do with self-reference sometimes raised in the context of the surprise examination paradox (for these see, e.g., Shaw 1958).
1. The Paradox
Suppose that on Monday morning the head of a school makes two announcements to the students.
(a) An examination will be held at 14:00 one afternoon this week.
(b) You will not know until shortly before 14:00 on the day which afternoon it is.
The examination will, that is, be a surprise one.
The argument to paradox runs as follows.
Suppose that the examination is held on Friday. In this case, it will be known in advance when the examination is to be held. For it will be known on Thursday after 14:00 that as the only day left to hold the examination is Friday, the examination has to be on Friday. It follows that the examination cannot be held on Friday consistent with the announced conditions (a) and (b). Now suppose that the examination is held on Thursday afternoon. After 14:00 on Wednesday, it will be known that the only afternoons left to hold the examination are Thursday and Friday. But Friday has just been ruled out as a possible day to hold a surprise examination, so that leaves Thursday as the only possible day. But then the students will know after 14:00 on Wednesday that the examination has to be on Thursday. But this is to say that Thursday as well as Friday can be ruled out as a possible day for a surprise examination. Suppose next that the examination is held on Wednesday. On Tuesday, after 14:00, it will be known that the only afternoons left for the examination are Wednesday, Thursday, and Friday. Thursday and Friday are not possible days for a surprise examination, by the reasoning just given, which leaves Wednesday as the only possible day. But then, if the examination is held on Wednesday, it will be known in advance that it is on Wednesday: Wednesday can, that is, be ruled out as a possible day for a surprise examination. Repeating the same reasoning rules out Monday and Tuesday. It follows that it is impossible to hold an examination under the announced conditions, that announcing that the examination will be held in a given period, and that it will be a surprise one makes it impossible for it to be a surprise one.
There is clearly a fallacy somewhere in this reasoning but it has proved hard to say exactly where it goes wrong. This is part of the reason the paradox has generated a large literature (see Kvanvig 1998).
2. Replies That Draw On The Possibility Of Doubt About (A) Or (B)
2.1 Uncertainty About (A)—The Examination Being Held At All
In most discussions of the paradox, it is granted that the examination cannot be held on the last day and still be a surprise. The focus of the discussion is on where the argument goes wrong after the step where Friday is excluded. However, as Quine (1953) points out, the matter is not that simple. If there is some doubt about whether or not the conditions set by the head will be satisﬁed, the argument that rules out the last day can fail. In particular, if there has been no examination by late Thursday afternoon and the students ﬁnd themselves, perhaps as a result of this fact or perhaps for some other reason, in doubt about whether condition (a) will be satisﬁed, about whether, that is, there will be an examination at all in the week in question, they do not know that there will be an examination on Friday. There maybe, for all they know, no examination at all. In this case, if the head proceeds to hold the examination on the Friday at 14:00, both conditions set down will be satisﬁed. The examination will have been held at 14:00 on one day in the week in question (which satisﬁes condition (a)), and the students will not have known in advance which day it was to be (which satisﬁes condition (b)).
However, as many commentators point out, the students might be in a position to be certain that the examination will be held in the week in question. Perhaps they know that holding the examination some time during that week is an unshakable prerequisite for government registration of the school. However, it seems that it would still be possible that the examination should be a surprise. There must, therefore, be something more to say about where the argument to the conclusion that a surprise examination is impossible goes wrong. The mistake cannot simply be the possibility of doubt about the examination being held in the relevant period, because the paradox survives the removal of that doubt.
2.2 Uncertainty About (B)—The Examination Being A Surprise
Suppose that it is certain that the examination will be held in the week in question, in which case the students can rule out Friday as a possible day for a surprise examination. This is consistent with it being a matter of doubt whether or not the examination will be a surprise one. Now suppose that there is doubt about this condition—(b)—being satisﬁed. Wright and Sudbury (1977) point out that this undermines the students’ reasoning on Thursday morning in the case where Monday through Wednesday prove to be examfree. The students cannot argue that Friday is excluded as a possible day for the examination to be held on the ground that then they would know in advance when the examination was to be held. For, if there is doubt about (b), they must allow that maybe the examination will not be a surprise. They must, therefore, allow that there are two possible days for the examination to be held: Friday, in which case (b) will be violated, or Thursday. In this case, the head can hold the examination on Thursday consistent with (a) and (b). This is because the students cannot, on Thursday morning, rule out Friday owing to doubt about the examination being a surprise. In eﬀect, the head can exploit doubt about the examination being a surprise—which leaves Friday as a possible day for the examination in the students’ minds—to ensure that it is a surprise by holding it on Thursday.
In sum, doubt about (a) would make it possible for the examination to be held on Friday consistent with (a) and (b). Doubt about (b), even in the absence of doubt about (a), would make it possible to hold the examination on Thursday consistent with (a) and (b).
The obvious question to ask at this point is what happens when there is no doubt about either (a) or (b). Could it not be the case that it is certain that the examination will be held on one afternoon of Monday through Friday, and certain that the students will not know in advance which afternoon it is? Isn’t there still a paradox to be addressed? In order to address this question, it is necessary to draw a distinction between whether there is doubt on some matter and whether there would be doubt on some matter were certain things to happen.
3. What Is Open To Doubt S. What Would Be Open To Doubt
It is not open to doubt given current evidence that Hitler died in the bunker, but were certain discoveries to be made, it would become open to doubt. Similarly, for most people who are here and now awake, it is not open to doubt that they are here and now awake, but were they to have the experience as of waking up from an extraordinarily life-like dream, it would be open to doubt. What is here and now open to doubt is distinct from what would be open to doubt were certain things to happen.
The view that the points made in the previous section do not dispose of the surprise examination paradox arises from the conviction that a surprise examination is possible when it is beyond doubt both that the examination will occur during the week and that the students will not know in advance. How then, runs the challenge, can we dispose of the paradox by pointing out that when there is doubt about (a) or (b), the student’s reasoning fails at some key point or other? However, on careful inspection, the failure of the students’ reasoning does not depend on there actually being doubt about (a) or (b). It depends on it being the case that there would be doubt were certain things to happen.
Provided it is the case that were Friday morning to come without the examination having so far taken place, there would then be doubt about the examination happening in the week in question, that is, about (a), the students’ cannot rule out the examination being held on Friday. If there would, should Friday morning arri e before the examination makes its appearance, be doubt on that Friday morning about (a), the head can hold the examination on Friday without violating either (a) or (b)—as was noted above. This does not require that there be, at any stage, actually any doubt. It requires simply that were Friday morning to arrive without an examination having been held, there would be doubt about (a).
A similar point applies to the role of doubt about the examination being a surprise, that is, of doubt about (b). There need not actually be any doubt that the examination will be a surprise. The students’ reasoning would be undermined simply by the fact that there would be doubt should enough days pass without the examination taking place. As was noted above, should Thursday morning arrive without there having been an examination and were this to mean that there would then be doubt about (b), the students cannot rule out Friday even if they are certain of (a). It is not required that there ever actually be any doubt about (b). All that is required is that should certain things come to pass, there would be doubt about (b).
It might be suggested that it could be built into the paradox that not only is there no doubt at any stage about (a) and (b), there would be no doubt about them no matter how many days pass without an examination. But this would be to build something absurd into the statement of the paradox. There would ha e to be doubts about one or both of (a) and (b) should enough days pass without the examination having taken place. For example, should Friday morning arrive without the examination having taken place, the students would know that one or both of (a) and (b) are open to doubt. It would be inconsistent of them to hold in that eventuality both that it is beyond doubt that the examination will take place on one of the designated days and that they cannot know in advance which it is. For example, if it would still be certain that the examination will take place on one of the designated days should Friday morning arrive without the examination having been held, they would know that should this happen, the examination would not be a surprise (for a longer development, see Jackson 1987).
4. The Backward Induction Paradox
4.1 The Rationality Of Cooperation
There are parallels between the style of argument in the surprise examination paradox and certain arguments concerning the rationality or otherwise of cooperative social arrangements. Many cooperative arrangements that have both costs and beneﬁts to the involved parties rest on the basis of expected returns in the future to all parties to the arrangement. The Robinsons work with the Smiths on something that advantages the Smiths in the expectation that in the future the Smiths will work with the Robinsons on something that advantages the Robinsons. A classic example is helping with building houses. It is much easier to build a house when there are a number of workers to lift heavy beams and the like. So the Robinsons give of their time to help the Smiths build the Smiths’ house, but do so in the expectation that the Smiths will later help the Robinsons build the Robinsons’ house. But why, in terms of self-interest, should the Smiths later help the Robinsons; they already have their house? The answer is that there will be other opportunities for cooperation, and the Smiths can expect to beneﬁt from some of them.
This intuitively appealing answer can be challenged by a ‘backwards induction.’ It will not be rational (in the sense of being advantageous) for the ‘giving’ party to cooperate on the last occasion for possible cooperation, because there is no prospect of their being the ‘receiving’ party in the future. But then, it seems, there is no reason for the giving party to cooperate on the second last occasion either, assuming rationality on the part of both parties. The only reason for the giving party to cooperate on the second last occasion is in the hope of being the receiving party on the last occasion, and that could only happen if the giving party on the last occasion is irrational. Similar reasoning applies to the third last, fourth last, etc. occasions, so yielding the conclusion that it is never rational to be a giving party in a cooperative arrangement.
It would be a mistake to reply to this argument by challenging the self-interest conception of rationality it employs. The puzzle is that it clearly is in parties’ self-interest sometimes to cooperate when they are the givers, in the expectation of being receivers in the future. The error in the argument is that it neglects the kind of epistemological points we saw to be crucial in diagnosing the error in the surprise examination paradox. Although it would be irrational for the giving party to cooperate on the known last occasion for cooperation, normally it is not known when the last occasion for cooperation is. Similarly, there may be doubt about all parties being rational.
4.2 Sequences Of Prisoner’s Dilemmas
A special version of the backwards induction paradox is sometimes deployed to argue that it is rational to defect throughout a sequence of prisoner’s dilemmas as well as in a one-oﬀ prisoner’s dilemma. A prisoner’s dilemma is deﬁned as follows. There are two parties, X and Y. If they both cooperate, they each do better than if they both defect. However, X and Y each do better if they defect, independently of whether or not the other cooperates or defects—defecting dominates cooperating, as it is put (see Resnick 1987).
Most game theorists accept that this means it is rational for each to defect despite the fact that if they both defect, they each end up worse oﬀ than if they both do the irrational thing of cooperating. What has proved harder to accept is the claim that, in a sequence of prisoner’s dilemmas of known length involving rational players, it would be rational to defect throughout. Surely it would be rational, sooner or later, for X and Y to start cooperating in order to set up an expectation of cooperation which means that, at least sometimes, they end up both cooperating, which is better for each than their both defecting?
A backwards induction argument has been used to resist this thought. Clearly, runs the resistance, it is not rational to cooperate at the last prisoner’s dilemma— engendering expectations of future cooperation has no pay oﬀ in that case. But cooperation at the second last prisoner’s dilemma is only rational if it leads to cooperation at the last prisoner’s dilemma, so this means that cooperation at the second last cannot be rational unless it leads to irrationality at the last prisoner’s dilemma. But that is to say that if X and Y are steadfastly rational, defection is rational at the second last prisoner’s dilemma as well as at the last. Repeating the reasoning delivers the result that it is rational to defect throughout the sequence of prisoner’s dilemmas.
A full discussion of this last backwards induction argument is beyond the scope of this research paper (but see Pettit and Sugden 1989 and Bicchieri 1998). But it should be noted, ﬁrst, that belief in rationality, or lack of it, is crucial to the argument, and, second, that if cooperating at early stages in a sequence of prisoner’s dilemmas sets up certain expectations, these early stages will not strictly speaking be prisoner’s dilemmas, for taking into account the longer term beneﬁts means that they will not be cases where defecting dominates cooperation.
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