What Are Analytic Narratives?

The recently born expression "analytic narratives" refers to studies that have appeared at the boundaries of history, political science, and economics. These studies purport to explain specific historical events by combining the usual narrative way of historians with the analytic tools that economists and political scientists find in rational choice theory. Game theory is prominent among these tools. The paper explains what analytic narratives are by sampling from the eponymous book Analytic Narratives by Bates, Greif, Levi, Rosenthal, and Weingast (1998) and covering one outside study by Mongin (2008). It first evaluates the explanatory performance of the new genre, using some philosophy of historical explanation and then checks its discursive consistency, using some narratology. The paper concludes that analytic narratives can usefully complement standard narratives in historical explanation, provided they specialize in the gaps that these narratives reveal and that they are discursively consistent, despite the tension that combining a formal model with a narration creates. Two expository modes, called alternation and local supplementation, emerge from the discussion as the most appropriate ones to resolve this tension


Analytic narratives and the problems they raise
Narratives report on their subject matter -stories, or how human actions proceed from and result in events -by obeying definite rules that are both formal and substantial. Competing accounts of these rules exist, and they are all controversial, but they share a common feature, which is to endow narratives with various functions besides the basic one of organizing actions and events in a temporal order (e.g., they causally explain the actions and events, make teleological sense of them, turn them into objects of entertainment, and so on). In essence, narratives are synthetic objects, and this feature makes surprising and even paradoxical a recent expression of social sciences that is slowly gaining ground, "analytic narratives".
The studies this term refers to emerged at the boundaries of history, political science, and economics. They purport to explain specific historical events and claim to do that by combining the usual narrative way of historians with the analytic tools that economists and political scientists find in rational choice theory. The paradox of a narrative that is also analytic is compounded by the fact that game theory, a formalized discipline, often provides the analytic tools, whereas narratives are of necessity limited to natural languages. If specialists in narratology had been aware of the new genre (they are not), they would no doubt have expressed worries about the discursive contradiction it may involve. However, unaware of this objection, the promoters of analytic narratives have strongly argued that, by collaborating on the same explanatory problems, narration and formal analysis contribute to better explanations of historical events than either could ever do in isolation. This argument on explanation is at the core of the book, Analytic Narratives (1998), by Bates, Greif, Levi, Rosenthal, and Weingast, which has created the tag and provided the project that this tag expresses with a manifesto and five illustrative case studies. 1 The present paper explains what analytic narratives are by relying on two studies from the eponymous book, as well as one outside study by this author. At the same time, it explores the two claims just mentioned, i.e., that analytic narratives might involve a discursive contradiction and that they contribute to renewing historical explanation. The former claim is more foundational than the latter, but to discuss the explanatory possibilities of analytic narratives turns out to be an easy way of introducing them, so we reverse the theoretical order of examination. This will also permit discussing the thornier issues in concreto.
The paper is organized as follows. Section 2 briefly summarizes Greif's and Levi's case studies in Bates et al. (1998) and uses them to evaluate the explanatory record of analytic narratives. Section 3 extends the sample with Mongin's (2008) study and carries the discussion on explanation to an end. At this stage, we will have selected two schemes of analytical narratives, alternation and local complementation, as being the most relevant.
Section 4 moves from explanation to exposition, raises the discursive problem again, and sketches an argument from narratology to resolve it.

Sampling from Analytic Narratives and discussing explanation
It would be better to cover Analytic Narratives as a whole, but the studies by Greif and Levi are good representatives of both the manifesto and its implementation.

Case 1 (Greif).
In the Middle Ages, the city-state of Genoa was first governed by elected consuls (1096-1194, period I) and then by an appointed magistrate, the podestà, who was chosen from outside the city (1194-1334, period II). Under consulate, civil peace prevailed from 1096 to 1194 (I.1), and then civil war lasted from 1164 to 1194 (I.2). Under podesteria, civil peace prevailed all along. Genoa's main economic activity was long-distance trade in the Mediterranean, and this activity was prosperous concomitantly with civil peace, i.e., for subperiod I.1 and period II, but with a noticeable peak at the end of the former. The main actors of economic and political life were the clans, which appear to have kept their identity and relative influence fixed for much of the period under study. In view of this fact, the time sequence becomes problematic. Why did the clans first cooperate and then fight under the politically unchanged conditions of consulate? Why did they cooperate most efficiently at the end of the first civil peace? How did the institutional move to podesteria contribute to reestablishing the civil peace that prevailed henceforth and why did it occur when it did?
These are Greif's main explanatory questions. He notes that the historians' work fails to answer them satisfactorily and even to raise them in full clarity.
Greif responds by constructing two extensive-form games of perfect information involving the clans as strategic players. The first is devised for the consulate regime and is played between two representative clans; it explores their trade-off between maintaining mutual deterrence and collaborating on maritime undertakings. This game explains the changes within period I by using the German Emperor's external threat as a variable parameter.
Depending on whether the threat is absent or present, Greif retains a different mutual deterrence equilibrium (MDE) for his game. The presence of the external threat pushes the clans towards mutually advantageous MDE by the following mechanism. In general, clans compete to gain control over the consulate, which would guarantee them a higher share of trade benefits, and this competition stabilizes peacefully only because they spend on deterrence resources they could more profitably spend on joint trade; this is what MDE formally captures. Now, the controlling clan also incurs the burden of external wars when they happen, so that the external threat changes the clans' ex ante costs and benefits of conquering the consulate; this is why MDE with fewer resources spent on deterrence, and more on joint trade, arise when there is such a threat. The second game, again of the perfectinformation, extensive form, is intended for the next political regime and has thus the podestà as a third player. An MDE in this game explains the stabilizing effect of podesteria. It relies on the more elementary mechanism that, if one clan challenges the other, the podestà can join forces with the challenged clan to defeat the challenging one, and he is motivated to do so by his rewarding scheme, given the military means at his disposal. Each time, the subgame perfection concept serves to define what an equilibrium is. 2 Case 2 (Levi). In the 19th century, several Western states changed their regulations of military service, moving from conscription with provisions for buying out of the duty to more or less universal conscription. Historians have usually emphasized democratization and military efficiency as being the two likely reasons. However, the latter is not clear technically (a professional army would have dominated all other arrangements), and the former is objectionable in view of the timing of reforms (they often took place either before or after universal suffrage prevailed). Starting from these objections, Levi compares the changes in France, the USA, and Prussia, paying attention not only to the chronological pattern, but also to the spatially and temporally variable pattern of buying out. (There are three distinctive forms, i.e., substitution, replacement, commutation.) She does not mean to displace the previous explanations entirely, but rather to subsume them under her own.
To do so, Levi uses an informal model in the spirit of formal political economy, whereby three main actors contribute to shape national decisions on the conscription regime. They are the army, which wants only military efficiency, the government, which wants it but also pays attention to social and economic considerations, such as employing the population efficiently, and the legislator, who aligns itself on the coalition that emerges among three social groups (traditional elites, middle class, workers). With this informal model at hand, the pattern of reform in each country can be explained by hypothesizing changes in the (informal) parameters. Levi proposes two such changes, i.e., the increased demand for troops from army and government, and the legislator's evolving preferences, both of which push in the direction of universal conscription. She relates the latter change to a reshuffle of the politically influential coalition (the pivotal middle class turning away from the traditional elites and becoming allied with the workers), as well as to an increased taste for equality within the social groups themselves. The two main hypotheses from the historical literature appear again, though included within a systematic explanation scheme. The study includes carefully collected evidence that clarifies and supports this scheme.
How analytic narratives explain. There are both significant similarities and dissimilarities between the two cases. They start by pointing out where the historical literature failed in its attempted explanations and define their own explanatory problems accordingly. The gaps to be filled typically concern fine temporal patterns (e.g., the succession of civil peace and war under apparently unchanged conditions) or fine-grained variations (e.g., between the forms of buying out) that historians did not properly account for. The explanatory questions following the initial assessment are answered in terms of a model, and there are two basic distinctions to be made in this respect.
Case 1 includes a formal model and case 2 an informal one, and the model of case 1 draws on game theory alone, whereas that of case 2 exploits more resources. This diversity raises a definitional issue, and we respond to it by taking the view on analytic narratives: (i) they require formal models and (ii) they can borrow them from any formal branch of rational choice theory. With restriction (i), we avoid taking the edge off the new methodology. If it just means applying rational choice hypotheses to historical events, this was practiced many times before any talk of analytic narratives, and even by traditional historians themselves. 3 Levi (2002), who does not make formalism a condition for analytic narratives, quite consistently claims that they are not a major innovation. The shortcoming of her position is that it deflates the tensions between the modeling and ordinary modes, which make the cases thought-provoking over and beyond their substantial contribution to history. For the sake of the discussion, it seems fruitful to dramatize these tensions and thus take the analytic element also to be a formal element. At the same time, (ii) enlarges the toolbox beyond game theory alone, and here we join forces with Levi, whose study draws inspiration from political economy models that do not technically reduce to games. Analytic narratives should be able to accommodate not only more complex formal models, as hers would be if it were mathematized, but also less complex ones, such as those of individual decision theory. These are meant to capture the decision makers' uncertainty when they are faced with an imperfectly known phenomenon there is no strategic interaction with. As a basic example, think of the kind of cost-benefit analysis that might be offered to explain the foundation of Pompeii and Herculaneum, on such prosperous but also dangerous farmland as the slopes of Vesuvius.
Mongin (2009) cites a more controversial case in Clausewitz, suggesting that a prima facie interactive situation can be represented, to a degree of approximation, by a standard expected utility (EU) model of individual decision-making.
From the angle of the philosophy of explanation, the three case studies have much in common: they can be likened to the so-called hypothetico-deductive scheme of scientific explanation. The assumptions of the models play the role of theoretical hypotheses, and the conclusions drawn from them play the role of the empirical consequences logically deduced from these hypotheses. According to this familiar scheme, if empirical consequences match the explanandum, then theoretical hypotheses can serve as explanans, provided however that some other deductions from the same hypotheses match some other facts, so as to discard ad hoc explanations. The three cases testify to an effort to avoid adhocness by carrying out loose forms of independent testing. Thus, Greif supports his account of podesteria in Genoa by discussing other Italian cities, and Levi actually states her study as a comparison between France and the USA, with Prussia being more like a control case. The effort to secure independent testing would be more successful if the explanans contained proper regularities, in which case not simply the hypothetico-deductive, but also the richer deductive-nomological scheme of scientific explanation would apply. The authors clearly believe to have uncovered theoretical patterns that can be transferred elsewhere. 4 However, by distancing themselves explicitly from this scheme, they implicitly recognize that they have not laid their hands on proper regularities. 5 This negative observation is more embarrassing to political scientists and economists than it is to historians. We will return to it after extending our collection of analytic narratives.

Sampling from military history and concluding about explanation
This section extends the sample from the history of institutions to that of war events. While revisiting a famous military campaign, Mongin (2008Mongin ( , 2016 offers his study as an analytic narrative and expands on the methodological principles of the new genre. To make progress with this explanatory question, Mongin proposes a zero-sum game in normal form with two players, Napoleon and Blücher, allowing for uncertainty in several ways. First, Napoleon is uncertain whether Blücher was crushed or only weakened after his Ligny victory. Second, both Napoleon and Blücher are uncertain which battles will result from their independent decisions. 6 Third, external circumstances matter besides the players' decisions, and both are uncertain of the issue of each given battle. Taken together, the first two forms of uncertainly liken the game to those of incomplete information, whereas the third form is nonstrategic; it is treated as if it were objective and amenable to EU calculations.
Given suitable parameter restrictions, von Neumann and Morgenstern's solution concept for a 2-person, zero-sum game delivers a unique equilibrium, which involves pure strategies. This arguably delivers not only an equilibrium, but also rational choice recommendations. From his three pure strategies, keeping the army together, sending out Grouchy for a mere pursuit, and sending him out to prevent Blücher from joining Wellington, Napoleon should choose the last; and from his two strategies, retreating to Germany or joining forces with Wellington, Blücher should choose the latter. That Napoleon effectively chose the interposition strategy can only be conjectured, because his orders were not fully reported, but the game reinforces this hypothesis. The ex post failure is not an objection since it could result from unfavorable external circumstances and from Grouchy misapprehending the plan -some historical evidence points in these two directions. Overall, the study exemplifies how an analytic narrative can be both formal and hermeneutic, since assumptions and conclusions are assessed in terms of evidential reports that are always incomplete, equivocal, and, given the high stakes, unavoidably biased. The conclusions adjudicate among existing positions, indeed by reinforcing classic pro-Napoleonic arguments against equally classic anti-Napoleonic ones.
More on how analytic narratives explain. Like cases 1 and 2, case 3 starts from the extant historical literature, defines explanatory questions negatively from this corpus, and reserves the modeling treatment to the major gap found in it. Mongin (2008Mongin ( , 2016 goes so far as to argue that analytic narratives can be justified only if they meet these three conditions; otherwise, there would be reason for breaking with the traditional narrative mode of historical explanation. 7 Unlike case 1, case 2 represents the game in normal form (in which players move once, simultaneously and independently) instead of representing it in extensive form (in which players make successive moves and dependencies result from this succession). Bates et al. (1998, p. 9 and elsewhere) make a plea for the latter form, as if this were the only one adapted to historical work, but case 3 is sufficient evidence that the former is also workable. If the extensive form is so acclaimed, this is presumably because its sequential structure seems capable of paralleling concrete sequences of actions, as made available by historical narratives, but we doubt whether this harmonious parallelism can really take place. Greif's games involves abstract moves, such as "to challenge", "to give in", and "to retaliate" in the podesteria game, which capture the actions the Genoan clans once made only very broadly.
Similar reservations apply to the other extensive forms in Analytic Narratives. If the moves are idealizations, their sequential ordering cannot represent the passing of historical time, and the alleged superiority of extensive forms over normal forms vanishes. 8 Another defense of the latter is more technical. They allow more easily than the former for an elaborate treatment of uncertainty, as the above discussion illustrates. Bates and al. (2000) honestly recognize that their work remains short of the target here. 9 Perhaps even more strikingly, case 3 differs from the first two by the nature of its explanandum and proposed explanans. It is directly concerned with interactive decisions, and these are made there by designated individuals. By contrast, cases 1 and 2 are directly concerned with events (states of affairs), like civil war or the prevalence of a conscription type, and only indirectly with interactive decisions, so that an interpretive reduction step is needed; moreover, the hypothesized decision makers, such as clans or the government, are not individuals, let alone designated individuals, but collectives. This is not all there is to the comparison. The explanandum of cases 3 is narrowly circumscribed in time and space, 7 This claim presupposes another, i.e., that the traditional narrative mode can be explanatory. Many philosophers of history agree, but some doubt or even disagree; see, e.g., White (1984). 8 Note that we do not discuss the quality of the idealizations made to define the moves, which is variable across analytic narratives. We only stress a consequence of the fact that such idealizations are (and presumably need to be) made. 9 Zagare's (2011) more sophisticated analysis of the July 1914 crisis takes a step forward by resorting to games of extensive form with incomplete information. The equilibrium concepts for this case are substantially complex.
whereas those of cases 1 and 2 extend in time or even time and space. To some extent, we recover here a topos of 20 th -century historical methodology, i.e., the contrast between traditional narrative history, which centers on the deeds of historical characters, and the more recent trends that deemphasize decisions and designated actors, while also enlarging the spatio-temporal scope of inquiry. Famously, the French Annales school has dramatized this contrast; for a classic methodological statement, see Braudel (1969).
Work along the older line is more clearly amenable to explanations given in terms individual actions, hence to analytic narratives, than work according to the Annales trends. Ironically, Braudel missed this point when he made game theory part of the mathematical toolkit he promoted to capture longue durée. 10 Whatever their persuasion, professional historians are more likely to concede case 3 than case 1, and case 1 than case 2, assuming equal scholarly merit across these three studies. By a similar reaction, Mongin (2008Mongin ( , 2016 recommends exploring more cases in military strategy on the grounds that the assumptions underlying the technical treatment have a better empirical warrant there than elsewhere. At the same time, one must acknowledge that those cases are in some sense too easy, and it is much more challenging to apply game-and decision-theoretic tools to cases in which the individual decision structure needs hypothesizing and that cover broader time lags. A balance must be struck between theoretical audacity and empirical reasonableness. Case 1 is perhaps the best candidate for a right balance, because the large scope is made acceptable by the stability of the proposed decision structure in terms of clans, and because clans, despite being collectives, can be given the attributes of strategic players with some plausibility.
From the vantage point of the philosophy of explanation, the last case obeys the hypotheticodeductive scheme of scientific explanation no less than the previous ones, but leaves even less room for independent testing than these do. This connects with the choice of not only a more specific explanandum, but also of a more specific explanans. The Waterloo game is usable only on the explanandum for which it was devised, whereas the podesteria game, having a more abstract structure, can in principle be used elsewhere. However, Greif contents himself with informally comparing Genoa with other towns, and the likely reason for this is that he does not have sufficient data outside Genoa. One should expect practical failures of independent testing to be very common with analytic narratives. It follows that the difference in the way explanations proceed is much less significant than the difference in the generality of explanantia would entail.
We can reinforce this point as follows. While it is often impossible to test a game as a whole on something else than the explanandum it is devised for, this may be possible for some of its individual assumptions. The podesteria game shares assumptions with the mutual deterrence games that have been explored fruitfully in international relations, 11 and the Waterloo game shares its two-person, zero-sum assumptions with at least one predecessor. 12 It is conceivable to test these structural assumptions in a variety of contexts, whether historical or not, and even experimentally. That independent testing can proceed in this way further erodes the difference between specific and general games.
We can conclude generally that the cases covered here do not exhibit critical variations in their explanatory functioning, despite the preferential disciplinary connections, which may suggest otherwise (with political science and economics for the first two, and history for the third). To use a familiar contrast, all analytic narratives are to be located on the "ideographic" rather than the "nomothetic" side of explanation. 13 They are in this sense closer to the hermeneutic way of traditional history than to any construal of scientific explanation, be it the hypothetico-deductive one or any other. However, this first conclusion must be weighed against the opening point of the paper, i.e., that modeling and narration are discrepant expository modes. We now move from the issues of explanation to those of exposition.

Analytic narratives as a mode of exposition
Consider how Greif presents case 1. He organizes it chronologically, covering first civil peace under consulate, then civil war under consulate, and finally civil peace under podesteria. He subjects each period or sub-period (more particularly the first two) to a ternary exposition, i.e., he first provides a standard narrative that states the facts and introduces the explananda, 11 See, e.g., Brams (1975Brams ( , 2011 and Zagare (2009Zagare ( , 2011. 12 Haywood (1954) used this apparatus to revisit the historical account of two campaigns in World War II. Zero-sum games are prima facie more relevant to war studies than to international relations studies, but this picture must be refined by taking the war objectives into account, as Clausewitz famously recommends. 13 The classic contrast goes back to Windelband (1904). Bates and al. (1998, p. 12) prefer to see their work as constituting a middle ground between the "nomothetic" and the "ideographic". second states a game-theoretic model based on relevant explanatory hypotheses, and third provides another narrative that finalizes the explanations. This revised narrative differs from the initial one, both because it borrows theoretical terms from the modeling part (e.g., "mutual deterrence equilibrium") and because it brings out additional historical evidence. This is, however, a narrative all right, so we have an alternation scheme. 14 This exposition does not appear in an earlier piece by Greif (1993), which uses a game-theoretic model to explain how 11 th -century Mediterranean traders entered in binding agreements, despite the unavailability of formal sanction mechanisms. The exposition there proceeds through the previous stages one and two, but differs concerning the third, because it ends up, very classically, with a nonnarrative discussion of the explanatory achievements of the model. 15 Consider now how Levi develops case 2. She (i) states the historical problems of conscription, (ii) introduces the theoretical hypotheses of the political economy model, (iii) narrates in detail how conscription laws changed, and (iv) compares relevant facts from this narrative with the previously stated hypotheses. Ignoring (i), one would have a classic hypothesistesting scheme, i.e., a statement of hypotheses, followed by a statement of empirical evidence, followed by a conclusive assessment. If hypothesis testing is the aim, the evidence should remain as much as possible independent of the hypotheses, and it becomes consistent to avoid the theoretical language of (ii) when proceeding to the narrative part (iii). Levi's writing precisely conforms to this requisite. However, her study also clearly obeys an explanatory purpose, for one can fuse parts (i) and (iii), considering that both state the explananda, and that the explanatory hypotheses come in (ii) before being evaluated in (iv). Viewed in this way, case 2 is not far off Greif's (1993) piece. Both allow narration to enter only the explanandum, not the explanans, and they are best described as analyzed narratives.
Case 3 exemplifies still another form of exposition. It begins with an ordinary narrative of the Waterloo campaign written like (and to some extent parodying) extant histories; this serves to introduce both the main facts and the (here unique) explanandum. Then the study introduces the game-theoretic model that contains the (also unique) explanatory hypothesis, and a reflective stage assesses this hypothesis while bringing out more historical evidence. A distinctive feature of this study is that it considers the initial narrative as being essentially satisfactory, except for an explanatory gap found at a critical juncture. The model and its assessment are like a parenthesis in the initial narrative, which can be resumed once the gap is filled. This scheme differs from that of analyzed narratives, because the narrative plays the role of an explanans, not simply of an explanandum, and it also differs from alternation, because it is the initial narrative, not a revised one, that has the final explanatory word. This distinctive scheme can be referred to as local supplementation. Consistently with a relatively modest purpose, it is more likely to arbitrate between existing sketches of explanations than to provide entirely new explanations.
Conceptually, an analytic narrative is a narrative that is somehow made analytic, not an analyzed narrative, and we will henceforth restrict attention to the alternation and local supplementation schemes. This is not to say, of course, that the analyzed narrative scheme is unimportant for historical explanation; quite the contrary, it is basic to most of the work done in economic history and historical political science. We only mean to say that it does not capture what is new and challenging with analytic narratives.
We now have to check whether the two schemes of interest are not subject to discursive inconsistencies. Here we need some basic groundwork from narratology, and to begin with, the distinction between a narrative text and the narrative form of discourse. 16 As such, a narrative text can very well include parts that are not narrative in the sense of the form of discourse, but rather descriptive, informative, explanatory, argumentative. All these received forms are prima facie acceptable within a narrative text, especially when it is factual (nonfictional), as those of history must be. 17 By contrast, it appears that a narrative text cannot accommodate some other received forms, such as the injunctive or demonstrative ones.
These observations suggest a simple argument to conclude that analytic narratives are consistent when they conform to the local complementation scheme. This consists merely in saying that such analytic narratives are narrative texts, which include, among their parts in non-narrative form, the statements of formal models and their consequences. These parts are clearly not descriptive, but explanatory, and although this may be less obvious, they are also informative and argumentative. To see that, consider how the author of case 3 informs the reader of basic mathematical facts about zero-sum, two-person games and argues for the relevance of his model despite the shortcomings he also concedes. Here information and argument together serve to approximate a demonstration that has to be provided somewhere, but certainly not within the narrative text, which cannot accommodate this form of discourse.
When it comes to analytic narratives, mathematical demonstrations can appear only in appendices or in separate publications.
To distinguish between prima facie acceptable and unacceptable discourse forms is only one step in the argument. Narrative texts put implicit constraints on what they can effectively absorb from the former. It is often suggested that quantity matters -non-narrative parts should be brief so that the author does not lose the thread of the narration. But this view is usually held concerning fictional narratives, whereas factual narratives, especially historical ones, appear to be more tolerant of lengthy interludes. What seems to matter most is not the absolute length of the interlude, but rather how it connects with the narrative parts. An example from military history will bring this point home. Clausewitz's (1835) monographic study of the Waterloo campaign is based on a nearly perfect alternation of narrative and non-narrative chapters. The former state the background facts and the protagonists' strategic moves with extreme temporal precision and refrain from passing judgments; the latter critically evaluate the adequacy of the strategic moves. Despite the high proportion and trenchantly different style of non-narrative exposition, the book remains a narrative text, because Clausewitz submits his examinations to an implicit rule of temporal consistency. That is, they never extend beyond what has been narrated thus far and they principally relate to what was narrated last. Explanations and arguments can thus be dated on the same time scale as the events and actions they discuss. In a sense, they are part of the story told by the book, and this is why it remains a narrative despite widely using other forms of discourse.
Clausewitz sets an illuminating precedent for the local supplementation scheme. The intrusions into the narrative flow can be as lengthy and complex as the topic requires, but they should respect the rule of temporal consistency and in particular avoid irrelevant anticipations, generalizations, and comparisons. Within the constraint imposed by this rule, it is possible, as Clausewitz does, to adopt the viewpoint of a best-informed narrator (who knows more than the protagonists and also knows what their knowledge consists of). This is one possibility, but there is another, which consists in idealizing the protagonists' subjective deliberations without supposing a summary viewpoint. This further division of the analytic narrative genre connects with the narratologists' now received distinction between the standpoints of the narrator and the characters. 18 Thus far, we have been discussing how a narrative text can feasibly accommodate nonnarrative forms of discourse. This may be sufficient for local complementation, but clearly not for alternation. There, the revised narrative aims at combining forms of discourseprincipally, the narrative with the explanatory form. We are led back to a more fundamental problem, which we already mentioned and did not pursue: is the narrative form at all capable of conveying explanations? If it is not, it seems impossible to rewrite a traditional narrative to make it explanatory; but if it is, the new question arises whether explanations that originate in theoretical models can be expressed by narrative means without losing their epistemic virtues.
Whatever the answer here, a more pressing point remains to be addressed. We have said nothing yet on the tension that the use of a formalism creates with the narrator's use of natural language. Plainly, to establish the discursive consistency of analytic narratives, it is not enough to check it against informal ancestor forms such as Clausewitz's. In our opinion, the tension primarily comes from the semantics of the artificial language, not from this language itself. For one thing, the recourse to mathematical symbols can be minimized and it is effectively modest in the way our two examples, cases 1 and 3, are written. For another, charts of numerical data, geographical maps, diagrams, and other figures, which are collections of symbols of a different kind, have long been used in historiography, and they are by now considered acceptable (and even commendable, according to Annales writers). The substantial question seems to be what meanings the chosen symbols (or the expressions used to designate them) are invested with. In most historical texts, charts and figures relate to concepts that do not call for specialized understanding, but this is not the case with such expressions as "zero-sum game" or "mutual deterrence equilibrium". 19 In our view, the solution to this difficulty could be found in the genesis of the formal theories of rational choice. They evolved from informal conceptions of human action that belong to an immemorial source -what cognitive scientists call folk psychology. 20 The guiding idea of these conceptions is to explain an observable action by what can be hypothesized from the agent's desires, on the one hand, and his or her beliefs, on the other. In individual decision theory, this duality has become utility and probability. Game theory, which historically emerged later, borrows from a more complex background of folk ideas, but its organizing concepts, and even those of equilibrium, are not esoteric either. With more effort, those of political economy could also be traced back to refinements of common usage. The fact that analytic narratives do not draw upon formal theories indiscriminately, but on this threefold group exclusively, means that there is a common ground between the specialized semantics they need and the intuitions that come to the mind of a cultivated, but non-specialized reader (perhaps the typical historian?). To substantiate this claim, one should revisit cases 1 and 3 in semantic detail.
To supersede the semantic objection is important for the local supplementation scheme, but it is even crucial for the alternation scheme. In this scheme, the revised narrative departs from the initial one by using some technical terms from the intervening modeling stage (think of "mutual deterrence equilibrium" again). If this material brings with it semantic ideas that conflict with those accompanying the use of ordinary words, the revised narrative will not be properly understandable, and the scheme as a whole will be incoherent discursively.

Conclusion
We have explained what analytic narratives are by starting from what exists and gradually becoming more normative. We have restricted them severally, first emphasizing that they raise interesting new questions only when they use rational choice theory in formal versions, and eventually retaining only two operational types, local supplementation and alternation.
We have gone some way toward giving these two types discursive foundations, although more thoroughly for the former than the latter. A fuller argument would need not only a more sustained appeal to narratology, but also a detour through the semantics of formal rational choice theories, and this also exceeded the limits of the paper. A perhaps easier task, we have shown that analytic narratives can improve historical explanations when they conceive of their contributions differentially, i.e., as ways of filling explanatory gaps in standard narratives. Meanwhile, we have argued that a hermeneutic element was inevitable in these contributions and that this made them less distant from standard narratives than appeared at first glance.