1 Introduction
In a recent article, “The Meta-inductivist's Winning Strategy in the Prediction Game: A New Approach to Hume's Problem,” Gerhard Schurz (Reference Schurz2008) proposes the best-alternative justification as a new approach to the problem of induction. As acknowledged by Schurz, the original idea goes back to Hans Reichenbach (Reference Reichenbach, Kamlah and Reichenbach1935/1994, Reference Reichenbach, Kamlah and Reichenbach1938/1983). But Schurz furnishes this idea with a new technical approach relying on a certain class of prediction strategies that he calls “meta-inductivists.” Given that all attempts to solve the problem of induction have hitherto failed (Howson Reference Howson2000), what should induce us to reconsider the skeptical conclusion that there is no solution to the problem of induction? Prima facie, Schurz has a convincing answer to this question: Most of the proposed solutions to the problem of induction tried to prove the reliability of the inductive procedure. But Schurz, following Reichenbach, merely tries to show the optimality of a specific inductive strategy—namely, his “meta-inductivistic” strategy. Demonstrating the optimality of an inductive prediction strategy is a less ambitious task than is demonstrating its reliability, because for an inductive strategy to be reliable, one would have to prove that it works in any possible world. But an inductive strategy that is merely optimal is allowed to fail in some possible worlds, as long as in the worlds where it does not work all other possible prediction strategies are bound to fail too. Schurz does not raise the claim that he has solved the problem of induction literally, but the closing paragraph of his article suggests that he believes that he has at least provided a very good candidate for a solution to the problem of induction (Schurz Reference Schurz2008, 304).Footnote 1
Schurz discusses the problem of induction within the technical framework of prediction games, where a number of players have to predict the next event in a series of binary-valued (0 or 1) or real-valued (any real number from 0.0 to 1.0) world events. By proving two theorems regarding the optimality of the prediction strategies of the “weighted-average meta-inductivist” and the “collective weighted-average meta-inductivist,” Schurz is at least able to give a partial solution for the problem of induction that accounts for the case of finitely many prediction strategies. But, as shall be demonstrated in the following, Schurz's technical approach meets insurmountable limits once one tries to pass from a finite number to an infinite number of prediction strategies. This raises the philosophical question of whether an optimality result demonstrated for a finite number of prediction strategies might suffice to answer the problem of induction. If not, then providing a full solution to Hume's problem remains an open challenge.
In the following, I am going to briefly restate Schurz's central results and then demonstrate that the results for prediction games with finitely many strategies cannot be extended to prediction games with infinitely many strategies. Finally, there will be a brief discussion of open questions regarding Schurz's answer to Hume's problem.
2 Schurz's Central Results
The technical framework within which Schurz derives his results consists of a set of prediction strategies and a series of events that can either be 1 or 0 (binary prediction game) or take any real value in the closed interval from 0.0 to 1.0 (real-valued prediction game). In the binary prediction game, the prediction strategies have to correctly predict as many events as possible. In the real-valued prediction game, they have to predict the coming events as closely as possible.
Two kinds of prediction strategies occur in Schurz's framework: alternative predictors that predict the next event by some arbitrary algorithm without looking at any of the other predictors and meta-inductivists that may—although they do not have to—base their own predictions on any of the alternative predictors’ predictions. The alternative predictors’ predictions are considered to be at least output accessible; that is, the meta-inductivists get informed about the alternative predictors’ predictions before they place their own predictions. In order to choose between the alternative predictors, the meta-inductivists may take into account the predictors’ success rates as well as their previous predictions. For the further details of the technical framework, see Schurz (Reference Schurz2008).
Within this framework, is it possible to find a provably optimal meta-inductivistic strategy? Schurz believes it is, and he presents two important theorems regarding the optimality of certain types of meta-inductivists. But he also relates an argument (Schurz Reference Schurz2008, 298) by Cesa-Bianchi and Lugosi (Reference Cesa-Bianchi and Lugosi2006, 67) that points out certain limits. The argument is the following:
Impossibility Theorem 1. In the binary prediction game, a single meta-inductivist cannot be optimal in all possible worlds.
Proof. The proof of this theorem can informally be stated as follows. Assume a world with one meta-inductivist and two alternative predictors, one of which always predicts 1 and the other always 0. Whatever the sequence of world events is, the success rates of the two assumed alternative predictors will always add up to 1. It follows that in any round, the success rate of either one or the other of the two assumed alternative predictors is at least 50%. Now, if the sequence of world events is a demonic sequence that always delivers the event that was not predicted by the meta-inductivist, the meta-inductivist's success rate will always remain 0%. Thus, in any round, the meta-inductivist's success rate is significantly lower than that of the best player, which means that the meta-inductivist's strategy is not optimal.
Because of this, Schurz understands the different types of meta-inductivists, which he develops in sections 4–6 of his article (Reference Schurz2008, 285–96) as only partial solutions to the problem of induction. They are optimal in (large) classes of prediction games but not in all prediction games.
The “impossibility theorem” stated before does not cover the real-valued prediction game. And indeed it can be demonstrated for the real-valued prediction game that a meta-inductivist that averages over the alternative predictor's predictions weighted by the alternative predictor's relative success advantage over the meta-inductivist will quickly approximate the maximal success rate of the alternative predictors. This is Schurz's theorem 4 (Reference Schurz2008, 297), which shall be termed “optimality theorem 1” here:
Optimality Theorem 1. For a weighted-average meta-inductivist (wMI) in the real-valued prediction game, the following holds:
1. Long run: The success rate (
) of wMI approximates the alternative predictors’ maximal success rate (), that is, .2. Short run: In any round n, for the success rate of wMI the following holds:
, with m being the number of alternative predictors.
) of wMI approximates the alternative predictors’ maximal success rate (
), that is, 
.
, with m being the number of alternative predictors. Here P denotes the set of all alternative predictors, and 
denotes the maximal success rate of the alternative predictors in round n. For the precise definition of the strategy of the weighted-average meta-inductivist and the proof of the theorem, see Schurz (Reference Schurz2008, 296–99).
As can be seen, the long-run success of the meta-inductivists does not depend on how many alternative predictors are present. Only in the short run is the number of alternative predictors m important, insofar as the more alternative predictors are present, the more “distracted” the weighted-average meta-inductivist can become in the short run.
Schurz is able to prove a similar theorem for the binary-valued prediction game. The impossibility theorem stated earlier precludes that this will work for a single meta-inductivist in the binary prediction game. But, as Schurz is able to demonstrate, the mean success rate of a collective of meta-inductivists (cwMI) can approximate the maximal success of the alternative predictors—be they as demonic as they may—almost equally as well as the weighted-average meta-inductivist can in the real-valued prediction game. The more cwMI predictors are present, the better the approximation. Or, more precisely:
Optimality Theorem 2. For the mean success rate of a collective of k collective weighted-average meta-inductivists (cwMI) in the binary prediction game, the following holds:
1. Long run: The mean success rate (
) of the collective meta-inductivists 1/(2k)-approximates the alternative predictors’ maximal success rate (), that is, .2. Short run: In any round n, for the mean success rate of cwMI the following holds:
, with m being the number of alternative predictors.
) of the collective meta-inductivists 1/(2k)-approximates the alternative predictors’ maximal success rate (
), that is, 
.
, with m being the number of alternative predictors.For the proof of this theorem and for the precise algorithm of the collective weighted-average meta-inductivists (cwMI), see Schurz (Reference Schurz2008, 297–99). Again, the long-run success of the collective meta-inductivists (with regard to their mean success rates!) does not depend on the number of alternative predictors. This renders the finding nontrivial. For no matter how large the number of alternative predictors is, their maximal success can be approximated by a comparatively smaller collective of meta-inductivists, the precise number of which is determined only by what level of approximation is regarded as satisfactory. It is only in the short run that a large number of possibly demonic alternative predictors can effectively deceive the collective meta-inductivists.
3 Limitations of Schurz's Approach: Confinement to the Finite
3.1 Why Schurz's Approach Cannot Be Extended to the Infinite Case
Schurz explicitly restricts his investigation to prediction games with finitely many prediction strategies (Schurz Reference Schurz2008, 284). In this case, the number of meta-inductivists may even be much smaller than the number of alternative predictors. It will now be shown why a similar argument cannot be made in case the number of alternative predictors is infinite and why, therefore, Schurz's optimality argument is confined to the finite.
Impossibility Theorem 2. If there are an infinite number of alternative predictors, then even a collective of meta-inductivists cannot perform approximately optimally in all possible worlds in the binary prediction game.
Proof. Consider the following scenario. Let there be an arbitrary number of meta-inductivists. As there are only two possible events, namely 1 and 0, at least half of the meta-inductivists predict the same event. Obviously, in each round there exists an event that is not picked by a majority of meta-inductivists. Now, assume a demonic world where the world event is always an event that is not predicted by a majority of meta-inductivists. Then the average success rate of the meta-inductivists never exceeds 50%.
Now we only need to show that there can exist at least one (demonic) predictor that achieves a higher success rate. For this purpose, split the (infinite) set of alternative predictors into two infinite sets in the first round. The predictors in the first set predict 1, and the predictors in the other set predict 0. In the following rounds, take the (infinite) set of predictors that have always predicted true so far, split it into two infinite sets, and again let all predictors from the first set predict 1 and all predictors from the second set predict 0. In any round of the game, there are thus an infinite number of predictors left that have a success rate of 100%. Since the meta-inductivists’ average success is significantly lower (smaller than or equal to 50%), their strategy is not optimal.
And, as may be expected, there is a similar impossibility theorem for the real-valued prediction game:
Impossibility Theorem 3. If there are an infinite number of alternative predictors, then no meta-inductivist (xMI) can be approximately optimal in all possible worlds in the real-valued prediction game.
Proof. Assume a demonic world, where the world event is always 0 or 1, whichever of these two numbers is further away from the prediction that xMI makes. As to the infinite number of alternative predictors: In the first round, let half of them predict 1 and the other half 0. In the following rounds, let half of the alternative predictors that have always predicted correctly so far predict 1 and the other half 0. Then at any point in time n, there exist some predictors with complete success, while the average success of xMI does not exceed 50%.
Hence, the conclusion: neither in the binary nor in the real-valued prediction game exists an optimal meta-inductive strategy if the number of alternative predictors is infinite.
3.2 A Sidenote: Limitations of “One-Favorite” Meta-inductivists
Just how difficult it is to design a meta-inductivist that covers all possible or at least all desirable scenarios becomes apparent when considering a limitation of Schurz's avoidance meta-inductivist, which is the most universal type in a series of “one-favorite meta-inductivists” that Schurz develops in sections 4–6 of his article.
As Schurz proves mathematically (his theorem 3), the avoidance meta-inductivist (aMI) 
-approximates the maximal success of the nondeceiving alternative predictors. However, this proof does not cover all strategies that we might intuitively consider nondeceivers. For example, a strategy that starts as a deceiver and switches to a nondeceiving clairvoyant prediction algorithm only later in the game (after it has been classified as a deceiver by aMI) will remain classified as a deceiver by aMI.
Intuitively, though, we would probably not consider it a deceiver anymore after it has switched to a nondeceiving clairvoyant algorithm.
Further below, it will be demonstrated that this can even happen accidentally for a predictor that never deceives (in an intuitive sense).
This limitation is a consequence of the fact that Schurz's definition of “deception” is purely extensional. It is based on the predictor's overt behavior and not on the deceptive or nondeceptive algorithm the predictor uses: “A non-MI-player P (and the strategy played by P) is said to deceive (or to be a deceiver) at time n iff 
” (Schurz Reference Schurz2008, 293), where 
is the “deception threshold” and 
is P's conditional success rate when 
has P as a favorite. So, contrary to what we might intuitively think, it is not a necessary condition for being a deceiver to base the predictions on what favorites the meta-inductivists have.
As Schurz himself notices, “Even an object-strategy (such as [the object-inductivist] OI) may become a deceiver, namely, when a ‘demonic’ stream of events deceives the object-strategy” (Reference Schurz2008, 293). This is of course to be understood in terms of Schurz's previous definition of deception, because the algorithm that, say, OI uses is the same as in a nondemonic world and would intuitively not be considered deceptive.
But then there is a finite probability that an OI will be classified as a deceiver by aMI even though the stream of world events is not demonic in the sense that the events are computed from the predictions made by the predictors. For there is a finite probability that a random stream of world events accidentally mimics a demonic stream of world events up to round k, so that OI appears as a deceiver up to round k. If k is sufficiently large, then aMI classifies OI as a deceiver. And it will only reevaluate OI's status if OI lowers its unconditional success rate: “For a player P who is recorded as a deceiver will be ‘stigmatized’ by aMI as a deceiver as long as P does not decrease his unconditional success rate (since P's aMI-conditional success rate is frozen as long as aMI does not favor P)” (Schurz Reference Schurz2008, 295). As OI's success rate reflects the frequency of random world events in the binary prediction game, it is unlikely that it significantly lowers its success rate at a later stage in the game.
In such a situation, aMI would fail to 
-approximate the maximal success of OI even though no deception was ever intended and the conditions under which this situation can occur are completely natural (i.e., nondemonic world, no supernatural abilities like clairvoyance, etc.).
Thus, if aMI can fail to be optimal even with respect to OI under completely natural circumstances, the optimality result concerning the performance of aMI with regard to all nondeceivers may not quite deliver what we expect. For example, we cannot not say that aMI is optimal, save for demonic conditions or deception, if deception is understood in an intuitive sense as described above.
4 Open Questions
Schurz's new approach to Hume's problem is both a research program on prediction strategies and a proposed answer to the problem of induction. As a research program, it can be pursued more or less independently of its claim to offer a new approach to Hume's problem. Regarding this claim, the central question is, of course: can meta-inductivists solve the problem of induction? This question can be broken down into three separate questions.
1. How is Schurz's model related to the problem of induction, if the latter is understood as the problem of justifying scientific inference?
2. Does it matter that no single meta-inductivist but only a collective of meta-inductivists constitutes an optimal strategy in the (binary) prediction game?
3. Is it acceptable that the best alternative justification works only when the number of predictors is finite?
In order to better understand these questions, each of them shall briefly be discussed.
1. Justification of scientific inference.—Schurz answers the problem of induction within a technical framework that consists of a highly stylized world that produces a sequence of (binary or real-valued) events at discrete time intervals. Further interpretative work might be necessary to show that the problem of induction that has been solved within Schurz's technical framework matches the problem(s) of induction that occur in the real world. In particular, the kind of induction in Schurz's model is the induction from previous events to the next event—in contradistinction to the inductive or abductive inference from a number of single instances to a general rule. For the justification of scientific inference, the latter type of induction seems to be even more important. After all, we would like to know whether we can rely on a law of nature if it has been confirmed in a finite number of instances and has never been disconfirmed. Thus, the question would be whether and how Schurz's answer to Hume's problem can also be transferred to this case.
2. Admissibility of collectives of meta-inductivists.—In the binary-valued game, a collective of meta-inductivists is required to assure optimality in the prediction game. Now, if we are looking for a reliable method of induction to the next event, then we are faced with the somewhat puzzling fact that we do not get a single proposal but a multitude of proposals instead. As follows from the impossibility theorem for single meta-inductivists (impossibility theorem 1 above), it is impossible to melt down the different predictions of the collective of meta-inductivists to a single prediction without becoming vulnerable to deception by a demonic world. There is no grave problem involved if the prediction game is considered in a decision-theoretical framework (Schurz Reference Schurz2008, 301ff.). For then the primary goal would not be to get the next prediction right, but rather to derive as much utility as possible from a number of predictions. However, this implies a shift of emphasis from the original problem of induction to a closely related decision-theoretical problem.
3. Confinement to the finite.—If only a finite number of prediction strategies are taken into account, then we exclude the overwhelming majority of possible prediction strategies from the game right from the beginning. For, given that the sequence of events is infinite, there exist an uncountably large number of possible event sequences. And even if we take into consideration only those prediction strategies that can be described by an algorithm, there still remains a countable infinity of possible alternative prediction strategies. Unfortunately, neither a single meta-inductivist nor a collective of meta-inductivists can perform optimally in all possible worlds if the number of alternative predictors is infinite (see sec. 3.1).
Schurz deliberately restricts his investigation to prediction games with finitely many players, because he makes “the realistic assumption that xMI has finite computational means” (Schurz Reference Schurz2008, 284). But in order to justify this restriction, one would need to show that an infinite number of alternative players is impossible rather than argue that xMI cannot deal with an infinite number of alternative players. Otherwise, the notice “that xMI has finite computational means” merely amounts to admitting that under this “realistic assumption” xMI simply cannot always perform optimally.
As there is no logical contradiction involved in the assumption of an infinite number of alternative players, the only grounds on which it could be defended are empirical. But then it is hard to see how the general impossibility of an infinite number of alternative players can be demonstrated without silently or explicitly making use of inductive inference.
Very simply put, there seems to be no good reason why a theoretical framework for answering Hume's problem that allows for “clairvoyants,” “deceivers,” and “demonic” streams of world events should not also admit infinite sets of alternative predictors. Surely, that an xMI that has only finite computational means does not work under this condition is not a sufficient reason.
5 Conclusion
Summing it up, what can be achieved with prediction games, with respect to inductive inferences, is a nontrivial result that shows that if only a finite number of prediction strategies are involved, there exists with the “(collective) weighted-average meta-inductivist” a strategy that is an approximately best strategy in all possible worlds. However, it can also be demonstrated that no approximately best strategy exists if an infinite number of alternative prediction strategies are involved. And there is some reason to believe that for a full solution to the problem of induction, the restriction to a finite number of alternative prediction methods is insufficient. If this is true, then the best alternative justification cannot offer a full solution to the problem of induction.
