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A game-theoretic analysis of the Waterloo campaign and some comments on the analytic narrative project

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Abstract

The paper has a twofold aim. On the one hand, it provides what appears to be the first game-theoretic modeling of Napoléon’s last campaign, which ended dramatically on June 18, 1815, at Waterloo. It is specifically concerned with the decision Napoléon made on June 17, 1815, to detach part of his army and send it against the Prussians, whom he had defeated, though not destroyed, on June 16 at Ligny. Military strategists and historians agree that this decision was crucial but disagree about whether it was rational. Hypothesizing a zero-sum game between Napoléon and Blücher, and computing its solution, we show that dividing his army could have been a cautious strategy on Napoléon’s part, a conclusion which runs counter to the charges of misjudgment commonly heard since Clausewitz. On the other hand, the paper addresses some methodological issues relative to “analytic narratives”. Some political scientists and economists who are both formally and historically minded have proposed to explain historical events in terms of properly mathematical game-theoretic models. We liken the present study to this “analytic narrative” methodology, which we defend against some of objections that it has aroused. Generalizing beyond the Waterloo case, we argue that military campaigns provide an especially good opportunity for testing this new methodology.

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Notes

  1. The 1914 crisis has been explored in game-theoretic detail by Zagare (2011). Regarding the Cuban crisis, the classic study by Allison (1971) was quickly followed by more advanced game-theoretic studies (one of the first being by Brams 1975, ch. 1). Of lesser historical relevance are the many game-theoretic pieces written on deterrence in general, as in Schelling (1960) and followers. See O’Neill (1994) for early references along both lines.

  2. Bates et al. (1998) push analytic narratives in two directions at once, i.e., historical explanation and the neo-institutionalist analysis of institutional change (along the same lines as North 1990). Here we interpret them only in the former sense, which the authors’ own division of the issues permits (see their methodological introduction). Zagare (2011, 2015) and Brams (2011) understand the expression “analytic narratives” in the same way.

  3. Besides the 1998 introduction, see Bates et al. (2000) and the further elucidations in Levi (2002, 2004).

  4. A rare counterexample is the review published in History and Theory by Downing (2000).

  5. Von Neumann and Morgenstern took their basic examples from parlor games; see Leonard (2010).

  6. See, e.g., Erikson et al. (2013).

  7. The two decisions analyzed by Haywood belong to the US Pacific campaign and the US Normandy campaign, respectively. Despite Brams’s efforts (1975, ch. 1), which breathed new life into Haywood’s work, this interesting precursor has remained virtually unnoticed.

  8. This objection was put to us by Margaret Levi; we try to answer it here and later in the text.

  9. Las Cases (1823) includes “Relation de la campagne de Waterloo, dictée par Napoléon” in Mémorial de Sainte-Hélène under the date of August 26, 1816. The other reports are Gourgaud’s (1818) La campagne de 1815 and Bertrand’s (1816–1821) Cahiers de Sainte-Hélène. The last work was published long after its author’s death and played no role in the Waterloo controversy, unlike the first two.

  10. Der Feldzug von 1815 in Frankreich. Posthumous like the others, this work appeared in 1835 in the Hinterlassene Werke edited by Marie von Clausewitz; it was written in 1827. Clausewitz’s commentators do not spend much time on his campaign narratives. Aron (1976), for example, hardly mentions them, while Paret (1992, ch. 9) is somewhat derogatory about them.

  11. Davidson (1980) is famous for emphasizing this connection.

  12. La campagne de France en 1815, tr. Niessel, 1973, pp. 37–43. From now on, all page references to Clausewitz are to his monograph and this French version, from which we translated the quotations.

  13. Hofschröer (1998–1999) stresses that Wellington had weakened himself in order to prepare for an attack from the west, which there was little reason to expect. The Duke had already faced the charge in his reaction to Clausewitz; see Bassford (1994, pp. 42–45, and 2001 for a transcript of Wellington’s comments).

  14. Mémorial de Sainte-Hélène, Garnier reprint, p. 237 (all page references to this edition).

  15. Cf. Weber (1922b, pp. 435–439). The distinction between objective and subjective rationality has since become established; see, e.g., Popper’s (1967) classic restatement.

  16. Although Thielemann finally had to surrender Wavre, he had fulfilled his role by holding back the enemy for half a day. Clausewitz, then a colonel, served as his chief of staff.

  17. Compiled and published by his descendants long after his death, Grouchy’s (1873-1874) Mémoires discuss these instructions at length, but the effort at exculpation is so blatant that it is impossible to take them seriously.

  18. Even the careful study by Hofschröer (1998–1999) falls far from making Clausewitz’s case compelling.

  19. De Mauduit (1847) eloquently illustrates the beginning of this line of interpretation, the first of many to blame the weakness of Soult and the staff in general.

  20. Here Fuller joins forces with Houssaye (1905–1906), a classic of the French rehabilitation literature.

  21. Cited by de Mauduit (1847–2006, p. 142) and subsequent authors, Bertrand’s letter is missing from Clausewitz, which weakens his chapter XXXVII.

  22. We use de Mauduit’s (1847–2006, pp. 160–161) version of this letter. Fuller (pp. 285–286) summarizes it accordingly, while Grouchy’s Mémoires (LV, pp. 58–59) distance themselves significantly from the text.

  23. The letter from Soult appears in Clausewitz (p. 141), as do all the subsequent dispatches.

  24. Houssaye (1905–1961, pp. 294–295) explains the desirable path. Grouchy would leave Gembloux to the west, marching to Mousty and Ottignies, where he would cross the Dyle and follow the river’s left bank.

  25. It would be more accurate to call the battle after Mont-Saint-Jean, where it took place, than after the neighboring village of Waterloo, but Wellington wanted that name to be chosen. The Germans—Clausewitz among them—have long preferred to call the battle after the farm of Belle-Alliance, where Blücher and Wellington met on the evening of June 18.

  26. Roberts (2005) puts the best moment for withdrawal even earlier.

  27. See Clausewitz, p. 157. This is a brilliant insight for a time when the concept of risk attitude was not yet separated from those of risk or uncertainty; see Mongin (2009).

  28. The tension between these two goals of war can be seen throughout On War, and Aron’s (1976, ch. III) commentary brings it even more clearly to light.

  29. Herbert-Rothe (2005) also compares the two battles of Waterloo and Borodino.

  30. This idea comes from Roberts (2005, p. 95).

  31. Following the previous analysis, at Borodino Napoléon did not aim at Kutuzov’s total annihilation. Unless the payoffs are redefined, a zero-sum game would therefore not correctly represent the strategic interaction of the two adversaries. Haywood (1954) also underlines that not every battle is appropriately modeled as a zero-sum game.

  32. B 2 B 1 minimizes the payoff of S 1 if and only if (a 1 + θa 2 − b 1 − b 2)/(a 1 + a 2 − b 1 − b 2) > l, l′. As θ grows, the central expression increases toward 1, thus binding l, l′ less and less. This is not a very constraining assumption.

  33. We derive V 1 < V 2 from m > k (1 − ld) + ld, putting d = (a 1 − b 1 + a 2 − b 2 )/(a 1 − b 1). This is a substantial and constraining assumption, which is simplified as m > k when l = 0. To ensure that the right-hand side is between 0 and 1, we also impose that ld < 1—another bound on l—and that m > 0, k < 1.

  34. A sufficient condition for V 3 < V 2 is that m  > k(1 − θ)a 2/(1 − k)(a 1 − b 1). The right-hand side is less than 1 if a 1 − b 1 > a 2 and either k < ½ or θ > k. The first assumption is fully justified in the context of June 17. Both of the latter two enter the historical explanation merely as conjectures. There are other sufficient conditions available.

  35. See Nash (1950) and Luce and Raiffa (1957, appendix 2).

  36. Due to von Neumann (1928), this theorem owes its fame to von Neumann and Morgenstern (1944–1947).

  37. Observe that dominance reasoning does not suffice to eliminate S 3. Recall that one strategy is dominated by another if it returns a smaller payoff for all the opponents’ responses, in all states of the world. Our game does not give dominated strategies to Napoléon, but it does to Blücher; see the “Appendix”.

  38. Inconclusive as they also are, two already discussed staff documents suggest a low k. On June, Bertrand warns Grouchy about Blücher’s remaining possible maneuvers, and Soult’s dispatch of June 18 confirms that Napoléon was concerned about an offensive return of the Prussians.

  39. We pursue this hermeneutical line at greater length in Mongin (2009).

  40. We focus on this early debate because it puts the issues sharply, perhaps at the risk of oversimplification. For complementary viewpoints, see the collection of articles in Social Science History (2000), with another response by Bates et al., and the symposium in Sociologica (2007); see also the introductory comments in Zagare (2011).

  41. This answers a technical question raised by Steve Brams and Françoise Forges.

  42. However, some of Zagare’s (2011) work uses perfect Bayesian equilibrium and thus takes the step of integrating strategic uncertainty with the extensive form.

  43. This is essentially the distinction made by On War, I, II.

  44. When a single decision is at issue, as in Haywood’s (1954) Pacific example, the distinctions between strategy and tactic, or campaign and battle, of course vanish.

  45. Arguing from On War (VIII), Aron (1976) concludes that Clausewitz promoted a novel conception of victory. But Paret (1992, p. 106) makes it clear that this is not entirely the case.

  46. Even some Napoleonic events are not easy to classify. With Borodino, Eylau is the classic example of an ambiguous victory, and Tolstoy in War and Peace goes as far as to claim these two battles for the Russian camp.

  47. Davidson (2004, p. 26) claims that the explanatory asymmetry between desires and beliefs is structural, but others see it only as a contingent property of the given explanations.

  48. On these changes, see, among many others, Earle’s (1943) collection or Aron’s (1976) comments on Clausewitz’s heritage.

  49. Compare the campaigns investigated by Fuller (1954–1956) on a very broad time range. Some are evidently more amenable to rational choice modeling, and there is no such obvious time dependency as may seem at first glance.

  50. Largeaud (2006) provides a thorough account of how interpretations of Waterloo have succeeded—and to an extent generated—each other, on the French scene. One would welcome similar reviews for the British and German scenes.

  51. Some comments along the present lines can already be found in Mongin (2010, 2016). A troubling suggestion we make there is that analytic narratives tend to work well when traditional narratives of the same events already work well.

  52. Even though some cliometricians and analytic narrators endow their respective fields with an inductive potential. Thus, Diebolt and Haupert (2016) and Haupert (2016) connect cliometrics with the second German historical school, which promoted inductive economics, and Bates et al. (1998) express the hope that their models would be applicable across similar historical cases, thus revealing an inductive potential.

  53. See again Diebolt and Haupert (2016) and Haupert (2016) on this.

  54. This is a classic assessment; see, e.g., Heffer (1977) and McCloskey (1978).

  55. Some game-theoretic applications that border on cliometrics are mentioned in Greif (2002).

  56. Diebolt (2016) is already willing to include analytic narratives in cliometric works.

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Correspondence to Philippe Mongin.

Appendix

Appendix

As stated in the text, the zero-sum game between Napoleon and Blücher has a unique equilibrium (S 2, B 2 B 1) if some optional conditions hold, beyond those which are automatically ensured by the definitions of final payoffs, probabilities and the abatement coefficient θ. The present appendix gives some details on the computation of this equilibrium.

Let us label Napoleon’s payoffs in the following way:

$$\begin{array}{*{20}l} {} \hfill & {} \hfill & {B_{1} B_{1} } \hfill & {B_{1} B_{2} } \hfill & {B_{2} B_{1} } \hfill & {B_{2} B_{2} } \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {S_{1} } \hfill & {} \hfill & {V_{11} } \hfill & {V_{12} } \hfill & {V_{13} } \hfill & {V_{14} } \hfill \\ {S_{2} } \hfill & {} \hfill & {V_{21} } \hfill & {V_{22} } \hfill & {V_{23} } \hfill & {V_{24} } \hfill \\ {S_{3} } \hfill & {} \hfill & {V_{31} } \hfill & {V_{32} } \hfill & {V_{33} } \hfill & {V_{34} } \hfill \\ \end{array}$$

Each V ij is obtained by an expected utility calculation:

$$V_{11} = k\left[ {l^{\prime } (a_{1} + a_{2} ) + (1 - l^{\prime } )(b_{1} + b_{2} )} \right] + (1 - k)\left[ {l(a_{1} + a_{2} ) + (1 - l)(b_{1} + b_{2} )} \right]$$
$$V_{12} = k\left[ {l'(a_{1} + a_{2} ) + (1 - l')(b_{1} + b_{2} )} \right] + (1 - k)\left[ {a_{1} + \theta a_{2} )} \right]$$
$$V_{13} = k(a_{1} + \theta a_{2} ) + (1 - k)\left[ {l(a_{1} + a_{2} ) + (1 - l)(b_{1} + b_{2} )} \right]$$
$$V_{14} = a_{1} + \theta a_{2}$$
$$V_{21} = k\left[ {ma_{1} + (1 - m)b_{1} + a_{2} } \right] + (1 - k)\left[ {ma_{1} + (1 - m)b_{1} + b_{2} } \right]$$
$$V_{22} = k\left[ {ma_{1} + (1 - m)b_{1} + a_{2} } \right] + (1 - k)\left[ {ma_{1} + (1 - m)b_{1} + \theta a_{2} } \right]$$
$$V_{23} = k\left[ {ma_{1} + (1 - m)b_{1} + \theta a_{2} } \right] + (1 - k)\left[ {ma_{1} + (1 - m)b_{1} + b_{2} } \right]$$
$$V_{24} = ma_{1} + (1 - m)b_{1} + \theta a_{2}$$
$$V_{31} = k\left[ {l^{\prime \prime } (a_{1} + a_{2} ) + (1 - l^{\prime \prime } )(b_{1} + b_{2} )} \right] + (1 - k)(b_{1} + b_{2} )$$
$$V_{32} = k\left[ {l^{\prime \prime } (a_{1} + a_{2} ) + (1 - l^{\prime \prime } )(b_{1} + b_{2} )} \right] + (1 - k)\left[ {ma_{1} + (1 - m)b_{1} + b_{2} } \right]$$
$$V_{33} = k\left[ {ma_{1} + (1 - m)b_{1} + a_{2} )} \right] + (1 - k)(b_{1} + b_{2} )$$
$$V_{34} = k\left[ {ma_{1} + (1 - m)b_{1} + a_{2} )} \right] + (1 - k)\left[ {ma_{1} + (1 - m)b_{1} + b_{2} )} \right]$$

The definitions of \(k,\;m,\;l,\;l^{\prime } ,\;l^{\prime \prime } ,\;\theta\) and the sign restrictions on \(a_{1} ,\;b_{1} ,\;a_{2} ,\;b_{2}\) imply a number of inequalities between the \(V_{ij}\):

$$\begin{aligned} V_{11} > V_{31} ,\;V_{21} > V_{31} ,\;V_{12} > V_{32} ,\;V_{22} > V_{32} ,\;V_{14} > V_{24} , \hfill \\ V_{22} > V_{21} ,\;V_{21} > V_{23} ,\;V_{22} > V_{23} ,\;V_{24} > V_{23} , \hfill \\ V_{32} > V_{31} ,\;V_{33} > V_{31} ,\;V_{34} > V_{33} . \hfill \\ \end{aligned}$$

The strategic analysis on Napoleon’s side proceeds as follows. It is the case that:

$$V_{{14}} > V_{{13}} \;{\text{and}}\;V_{{12}} > V_{{11}} \;{\text{iff}}\;(^{*} )\;\frac{{a_{1} + \theta a_{2} - b_{1} - b_{2} }}{{a_{1} + a_{2} - b_{1} - b_{2} }} > l,$$

and:

$$V_{{13}} > V_{{11}} \;{\text{iff}}\;\left( {^{{*^{\prime } }} } \right)\;\frac{{a_{1} + \theta a_{2} - b_{1} - b_{2} }}{{a_{1} + a_{2} - b_{1} - b_{2} }} > l^{\prime } .$$

We assume both (*) and (*′) to hold, thus ensuring that V 11 = V 1 is the security payoff of S 1 (cf. fn. 32). By inspecting the definitional inequalities, we observe that V 23 = V 2 is the security payoff of S 2 and that V 31 = V 3 is the security payoff of S 3.

Now to compare the three values V 1, V 2, V 3. In view of (*′), the inequality V 2 > V 1 can be obtained from V 2 > V 13, which is equivalent to:

$$(m - k)(a_{1} - b_{1} ) > (l - kl)(a_{1} - b_{1} + a_{2} - b_{2} ),$$

or:

$$\left( {^{{**}} } \right)\;m > k(1 - ld) + ld\;,{\text{with}}\;d = \frac{{a_{1} - b_{1} + a_{2} - b_{2} }}{{a_{1} - b_{1} }}.$$

This inequality makes sense only if the right-hand side is between 0 and 1, i.e., only if m > 0, k < 1, and

$$\left( {^{{**^{\prime } }} } \right)\;l < 1/d,$$

which implies that l < 1. We impose these conditions (cf. fn. 33). Notice that (**) implies that m > k, a condition to which we return below.

Since V 33 > V 3 holds, the inequality V 2 > V 3 can be obtained from V 2 > V 33, or equivalently:

$$\left( {^{{***}} } \right)\;m > \left( {\frac{k}{{1 - k}}} \right)(1 - \theta )\left( {\frac{{a_{2} }}{{a_{1} - b_{1} }}} \right),$$

which requires that k < 1. The right-hand side is less than 1 under one of the two conditions:

$$\left( {^{{***^{\prime } }} } \right)\;k < \frac{1}{2},\;a_{2} {\text{ }} < a_{1} - b_{1} ,$$

or:

$$\left( {^{{***^{{\prime \prime }} }} } \right)\;\theta > k,\;a_{2} < a_{1} - b_{1}.$$

We assume (***) to hold, as well as either (***′) or (***″) (cf. fn. 33). Hence, V 2 is Napoleon’s maxmin.

Here are the computations on Blücher’s side. From what has just been shown in the last paragraphs, −V 2 is the security payoff of the conditional strategy B 2 B 1. We will show that it is also the maxmin by checking that no other strategy can deliver a higher security payoff.

Concerning B 1 B 1: from definitional inequalities, the security payoff is either −V 11 or −V 21. It cannot be −V 11 because V 11 > V 21 would imply a cycle, given that V 21 > V 2 > V 13 > V 11; so it is −V 21, which cannot be the maxmin, given the first inequality in this sequence.

Concerning B 1 B 2: definitional inequalities entail −V 12 or −V 22 being the security payoff, but neither can be the maxmin because V 22 > V 2 holds (if −V 22 is the security payoff, it falls below −V 2, and the same if it is −V 12, since this implies V 12 > V 22).

Concerning B 2 B 2: again from definitional inequalities, either −V 14 or −V 24 is the security payoff, and by a similar argument, V 24 > V 2 precludes either value from being the maxmin.

Thus, we conclude that the equilibrium of the game, in the von Neumann–Morgenstern sense, is (S 2, B 2 B 1).

A Nash equilibrium calculation would have reached the same conclusion somewhat differently and more quickly. It would have used the fact that Blücher’s strategies B 2 B 2 and B 1 B 2 are dominated, respectively, by B 2 B 1 and B 1 B 1, once condition (*) is granted. So they are discarded from consideration for Napoleon too, and his strategy S 3 becomes dominated by S 2 from (***), (***′) or (***″), and definitional inequalities. The game is now 2 × 2, and the remaining conditions, i.e., (*′), (**), (**′), ensure that (S 2, B 2 B 1) is an equilibrium in the Nash sense and that it is unique.

As mentioned in the text, the assumption that l = 0 simplifies the analysis. Then, (*), (*′), (**′) are trivially satisfied. The binding conditions are (***), (***′) or (***″), and (**), which reduces to the straightforward inequality m > k. Thus, in this limiting case, the necessary condition becomes sufficient.

Not all the probabilities can take extreme values. The conditions make it necessary that m > 0, l < 1 and k < 1, with k being further bounded from above and θ unrestrained between 0 and 1, or alternatively k and θ being mutually related. Notice that l″ is the least constrained parameter, being only subjected to the definitional inequalities l′, m > l″.

It is trivial to find non-extreme values that satisfy all the conditions. For example, take as utility parameters:

$$a_{1} = 1,\;b_{1} = - 1,\;a_{2} = 1/2,\;b_{2} = - 1/2,\;\theta = 1/2,$$

and as probability parameters:

$$l = 0.1,\;l^{\prime } = 1/2,\;l^{\prime \prime } = 1/3,\;k = 1/3,\;m = 2/3.$$

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Mongin, P. A game-theoretic analysis of the Waterloo campaign and some comments on the analytic narrative project. Cliometrica 12, 451–480 (2018). https://doi.org/10.1007/s11698-017-0162-0

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