Mathematical modeling of strategic behavior: the Battle of Midway,1942

Abstract

In this paper, an innovative discrete dynamical model is presented, which is used to predict the kind of strategic behavior the participants should adopt to win a battle. For study purposes, a computer model is developed to reveal the most critical factors that strategically affect combat and the relationship of dependence between the warring parties. Besides, it can predict the outcome of a battle under specific scenarios. Furthermore, the proposed dynamical system is applied in Midway’s air–naval battle, which was one of the most decisive battles of World War II (WWII). It was a significant turning point in the history of naval warfare in the Pacific Ocean since the victory of the United States marked an end to Japanese expansionist policy, and these are the reason this battle was chosen. The numerical results of the analysis were presented, and the key factors (e.g., persons, decisions, and weather conditions represented by the critical values of model parameters) were highlighted, defining the outcome of the conflict.

Keywords

Modeling and simulation a combat discrete dynamical systems Battle of Midway

1. Introduction

Modeling and simulation play an essential role in the military domain.¹ Many researchers have developed models to simulate combats and predict the outcome of a significant battle under different scenarios. Moreover, they help military planners to prepare for major military engagements. The most notable models are as follows: Richardson’s,² Lanchester’s,³ Intriligator and Brito’s^4,5 Strategic Deterrence-Attack, Saperstein’s,⁶ and Gross-Germeier’s “attack-defence.”⁷

A recent paper that adopts a similar approach to ours is that of Coram and Noakes.⁸ They extend Richardson’s arms race model to highlight how a strong participant attempts to influence the behavior of a weaker participant. This idea is considered a mixed system of symmetric and asymmetric interactions between the two players, examining specific parameters that affect a battle simulation. They present a general pattern and an analytical framework to prove some important policy implications. In a similar vein, Reuveny et al.⁹ developed a repeated, stochastic conflict model, entitled the “winner-take-all model.” The main features are the FOW (fog of war) and the POP (paradox of power). The FOW is associated with randomness by making the outcome of the conflict dependent on both opponents’ decisions. At the same time, the POP phenomenon is included as a part of the game structure. They argue that the number of runs in the model exhibits a critical role. A conflict over time generates more significant differences in players’ behavior than in a single run, leading to lasting differences in the system dynamics over long-time periods.

Ishida¹⁰ highlights how historical changes play a crucial role in interactions between influential leaders’ choices and public sentiment—not only in terms of arms races and war but also as a social phenomenon. The following researched strategies are proposed in this work: (a) a type of micro–macro linkage for social change that complements the type of micro–macro link described by Coleman’s boat and (b) an “initial condition game” of Richardson’s arms race model. It is an interdependent rational choice situation for influential players based on public sentiment. The model’s implications are proportional to various conditions of an arms race, armed conflict, or other analogous social phenomena. Finally, You et al.¹¹ present a new methodology entitled capability-based GMCR (graph model for conflict resolution). They applied it in the arms race between the United States and the Union of Soviet Socialist Republics USSR during the Cold War. One of the most highlighted points refers to the stability analysis results being consistent with the actual outcome evolution of the Cold War. The main contribution of their research is the provision of a novel investigation path to expand the application fields for GMCR, particularly in asymmetrical conflicts.

To the best of our knowledge and regarding the previous studies, it can be argued that the aspect of the optimum strategic behavior is neglected or underestimated. We adopt a well-known paradigm from the game theory “Hawk-Dove game” and contribute to the literature by developing a new discrete dynamical model. It considers the strategic behavior (aggressive or defensive) of actors in a battle and serves as a decision-making mathematical tool to study and better understand the crucial factors of combat separately. For instance, the battlefield environment is simulated mathematically, but also the two parties’ blunders and bad decisions are highlighted through alternative scenarios.

In the following sections, we explain the discrete dynamical system (section 2) and its application (section 3) in the Battle of Midway. In particular, section 3 presents a brief overview of the combat, and a few details of the military forces are mentioned. Afterward, the analysis process and the numerical results are presented. In section 4, the concluded remarks are summarized, and some further research ideas are mentioned.

2. Model presentation

Over the years, numerous simulations have been developed using a wide variety of modeling methodologies. Simulation modeling is an effective tool for evaluating different investigating scenarios and alternative hypotheses.

This section presents an extended version of the model developed by Founta and Zachilas.¹² In that paper, their model simulates the strategic behavior of two opponents under the hypothesis that one is weaker than the other in terms of military strength. In the extended version as presented in Equation (1), we added the following parts:

We divided the technological capabilities of each force into three parts (Navy, Air Force, and Intelligence).

$2 \cdot x_{t} \cdot (1 - x_{t})$ (and $2 \cdot y_{t} \cdot (1 - y_{t})$ ) express the next move depends not only on how its opponent acts but also the participant itself at the time t. The constant in front of each logistic equation is defined in [0, 4], and we chose a suitable value to achieve a smooth behavior of the model.

The term $G \cdot x_{t}^{2} \cdot y_{t}$ (in the first equation of Equation (1)) is added for the following reasons: (a) increases the aggressiveness of player x at the time t, (b) affects the opponent’s strategic behavior at the time t, and (c) they multiplied by the parameter G, because both player’s aggressiveness is affected by the difficulty of the battlefield by developing new changes that can occur during a battle. For the same reasons, we add, respectively, the term $(1 - G) \cdot y_{t}^{2} \cdot x_{t}$ in the second equation of Equation (1).

Colonel Lykke¹³ mentioned that the “military strategy is the combination of ends, ways, and means.” According to Lykke,¹³ strategy equals Ends (objectives toward which one strives) plus Ways (courses of action) plus Means (instruments by which some end can be achieved). This general concept can be used as a basis for the formulation of any type of strategy—military, political, economic, etc.^14–17

Based on this argument, we developed a novel nonlinear discrete dynamical system to express mathematically the participants’ strategic planning and how they act during the battle. In particular, we consider the “means” as the sum of technological capabilities (TN, TA, TI) for each participant, i.e., how they used their military technology to win a battle. We conceive that the parameter G expresses the “ways,” namely, how they were influenced by the geographical terrain of the battle for each favor (positively or negatively). The final part, the “ends,” is associated with the parameter (D). Both sides aim to cause considerable damages to each other, and thus win the conflict. The parameter (P) shows each participant’s military strength (substance), and the parameter (E) expresses the military expenditures that each warring part has (Table 1).

Table 1.

Lykke model and definitions.

Lykke model	Parameters	Definitions
Means	TN, TA, TI	Technological capabilities
Ways	G	Geographical terrain/battlefield
Ends	D	Damage
Extra parameters	P	Military strength (substance)
Extra parameters	E	Military expenditures

The variables and the parameters are defined in the closed interval [0, 1] because of the logistic equation at the end of each equation of the system. The logistic equation comes from studying biological populations reproduced in discrete time.¹⁸ It is the evolution of Malthus’s¹⁹ population model and shows that exponential growth cannot tend to infinity, but there is a critical point. In other words, it is not possible for someone to win and the other to continuously lose.

We aim to study significant battles (such as the Battle of Midway) through the mathematical model and show its accuracy. Moreover, we mathematically prove the historical facts of the combat, and some alternative scenarios are presented to highlight the mistakes of the admirals and the possible ways to overturn the outcome of the conflict.

The external model, which is applied in short-term combats and describes the strategic behavior during the battle, is given by the discrete system (Equation (1)):

\begin{matrix} {\begin{matrix} x_{t + 1} = 2 \cdot x_{t} \cdot (1 - x_{t}) \cdot (P_{x} + T N_{x} + T A_{x} + T I_{x}) \\ - G \cdot x_{t}^{2} \cdot y_{t} - G \cdot (D_{y} + E_{x}) \cdot 2 \cdot y_{t} \cdot (1 - y_{t}) \\ y_{t + 1} = 2 \cdot y_{t} \cdot (1 - y_{t}) \cdot (P_{y} + T N_{y} + T A_{y} + T I_{y}) \\ - (1 - G) \cdot y_{t}^{2} \cdot x_{t} - (1 - G) \cdot (D_{x} + E_{y}) \cdot 2 \cdot x_{t} \cdot (1 - x_{t}) \end{matrix} \end{matrix}

(1)

where $x_{t}$ is the strategic behavior of country x at the time t. $y_{t}$ is the strategic behavior of country y at the time t. $x_{t + 1}$ is the optimal strategic behavior of country x at the time t + 1. $y_{t + 1}$ is the optimal strategic behavior of country y at the time t + 1.

We consider $x_{t}, y_{t}, x_{t + 1}, y_{t + 1} [0, 1]$ , because the logistic equation is defined in [0, 1]. If the value of $x_{t + 1}$ (or $y_{t + 1}$ , respectively) equals 0, representing the entirely defensive strategic behavior of participant x (or y, respectively). Concurrently, if it equals 1, then it means the downright aggressive behavior of participant x (or y, respectively).¹² Also, the interval of values is defined as $x_{t}, y_{t}, x_{t + 1}, y_{t + 1} [0, 1]$ .

We consider the battle as a war game, and it is possible to predict the outcome of the battle. The game “Hawk-Dove” refers to the strategic behavior the players should adopt. Each one can behave aggressively (Hawk) or defensively (Dove). The winner of the game is the one who behaves as a Hawk (aggressively) and the other player who has more defensive behavior losses. Thus, we assume that if the value of the fixed point is close to 1, the participant has aggressive behavior (Hawk), while if the value is close to 0, then the participant has defensive behavior (Dove).

2.1. Explanation of parameters

$P_{x}$ represents the military strength of x and $P_{y}$ is the military strength of y, respectively. These two parameters indicate the military substance (either as a combination of quantity and quality or separately) of each country compared to the other.

$T N_{x}$ and $T N_{y}$ represent the technological naval capability of country x and y, respectively. These two parameters are also defined in comparison to the technological capability of the opponent.

$T A_{x}$ and $T A_{y}$ represent the technological aerial capability of x and y, respectively. These two parameters are also defined in comparison to the technological capability of the other participant.

$T I_{x}$ and $T I_{y}$ indicate the ability to decode the opponent’s messages (Intelligence) for x and y, respectively. Gilad et al.²⁰ explain that military intelligence aims to protect national security from interstate threats. Moreover, the intelligence process is focused on time relevancy and policy-making decisions. All military intelligence organizations collect and analyze data to generate valuable information for the relevant decision-makers.

The parameter G represents the geographical location (geophysical terrain) of the area where the battle occurs. The closer to 1 the parameter’s value is, the more complex the geophysical terrain of the area will be considered.

$D_{x}$ is the damage caused by x to y and $D_{y}$ is the damage y brings to x, respectively. The damages which are referred to may be economical, territorial, military, etc., or even deception, thus damaging the psychological part of the opponent.

$E_{x}$ represents the military expenditures of participant x and $E_{y}$ represents the military expenditures of participant y, respectively. In other words, these parameters denote the preparation costs of each participant for a battle compared to each other.

All the parameters that have been analyzed above should belong in the interval [0, 1], highlighting the criticality of the changes in the initial conditions of the parameters.

In other words, the proposed methodology and the analysis process provide the modeling of participants’ optimum strategic behavior during the battle. The model tries to simulate the historical facts of the Midway battle, by choosing realistic parameters and variables taken by historical books,^21–23 to study the interactions of two opponents that affect the outcome of the clash.

In the next section, the dynamical analysis along with the numerical results from the application in the naval Battle of Midway is presented.

3. Application to the Battle of Midway

3.1. A brief overview of the battle

The Battle of Midway was an air–naval battle that occurred between 4 and 7 June 1942, in the Pacific Ocean during WWII. It was one of the decisive battles of WWII. Specifically, it took place just 6 months after the Japanese attack at Pearl Harbor. The two warring parties were the United States Navy (USN) and the Imperial Japanese Navy (IJN). The leader of the USN side was Admiral Nimitz, while Admiral Yamamoto led the IJN side. More detailed descriptions of the battle can be found in the book of Stille and Howard.²¹.

During WWII, one of Japan’s main strategic aims was to reduce the USN power in the Pacific Ocean and gain ground in the Pacific Ocean islands. Japan hoped to defeat the US Pacific Ocean fleet and use the Midway’s airbase to gain dominance over the region, and thus force them into a negotiated peace.²⁴

Table 2 shows the military forces of the two sides. It overviews the two naval forces’ principal and most important military assets. Specifically, it presents the most important warships of the two navies as these are comprehensively presented and analyzed by Stille and Howard.²¹

Table 2.

IJN and USN military forces.

IJN	USN
Four heavy aircraft carriers (248 aircraft)	Three heavy aircraft carriers (233 aircraft)
Nine battleships	Eight cruisers (heavy, light)
Twelve cruisers (heavy, light)	Twenty-one destroyers
Forty-four destroyers	Twelve other warships
Ten other warships	Midway forces: 112 aircraft, 3652 soldiers, and 14 support ships
Total number of available aircraft: 208

Source: Stille and Howard.²¹

IJN: Imperial Japanese Navy; USN: United States Navy.

Yamamoto’s plan was the same as Pearl Harbor: a surprise attack. He considered that if he reapplied this successful strategy, the victory would be assured. His plan was complex, and it consisted of three parts. In brief, the first part referred to an attack in the Aleutian Islands for diversion, the second part to bombing the Midway Islands, and the final part was to capture the airbase by land. The Japanese strategic plan was already known to USN because they broke the IJN communication code many days before the impending battle. Thus, the USN’s purpose was to prevent IJN’s surprising attack.²¹

At the end of the battle, the Japanese abandoned the expansion into the Pacific Ocean and remained in defense for the rest of WWII. The US victory showed how important the power of correct information on the opponent’s plans was, and they felt that the Japanese could be defeated as they were the only ones who were able to attack them.

Although the United States won this battle, both sides—particularly Japan—suffered heavy losses. On one hand, Japan lost all 4 carriers and its 248 aircraft. In addition, 1 destroyer and 1 cruiser were sunk, while 3057 soldiers, sailors, and pilots died. On the other hand, the USN lost only 1 carrier (i.e., Yorktown), 144 aircraft were destroyed, and 1 destroyer sank. As for the soldiers, sailors, and pilots, 362 lost their lives during this battle.²⁴

In 1942, the Japanese military expenses were US$38.58 billion, while the United States spent US$227.24 billion. The two sizes diverge significantly because the United States devoted significant military spending for equipment renewal (damage repairs, purchase of new weapons, etc.) after its overwhelming defeat at Pearl Harbor. It reflects the difference in gross domestic product (GDP) production capacity since the GDP of Japan was less than the GDP of the United States or the United Kingdom or Germany or France.^25,26

3.2. Simulation and analysis results

3.2.1. Initial approach

At the beginning of the analysis, initial conditions were defined in variables, and numerical values were given to the parameters, representing the battle’s historical facts.

Specifically:

(a) We assume that the USN is the participant (x), while the IJN is the participant (y).

(b) We have set the military strength of the US fleet equal to $P_{x} = 0.3$ , and the military strength of the Japanese fleet equal to $P_{y} = 0.6$ . These values are selected according to Table 2, where we observe that the IJN forces are greater in quantity than USN.

(c) The technological naval capability of the USN was chosen as $T N_{x} = 0.3$ , while the IJN technological naval capability $T N_{y} = 0.5$ . The values of these two parameters are slightly different because Japan had included battleships in their fleet, but USN did not.

(d) The aerial technological capability of the United States is equal to $T A_{x} = 0.4$ and the Japanese is equal to $T A_{y} = 0.5 .$ These values describe the slight difference in the number of aircraft involved in the battle (quantity element). Furthermore, the United States had not improved air capabilities, compared to the Japanese pilots, because they were not well trained in the air–naval battles, but in air combats (quality element). We assume that these values combine the quantity and quality of participants’ aerial technological capability.

(e) The ability to decode enemy messages was chosen as $T I_{x} = 0.8$ and $T I_{y} = 0.4$ , respectively. According to the book of Stille and Howard²¹, the USN intelligence was better (hence, the parameter value is close to 1).

(f) The parameter that describes the battle’s geographical terrain, including the weather phenomena during the battle, was set as G = 0.4. The physical phenomena we refer to are the sunny or cloudy weather during combat. In addition, these phenomena are used in favor of each opponent. Specifically, the US pilots flew through the clouds when their first reconnaissance flight was 4 June. In this way, they investigated the enemy’s position undetected. The value of this parameter G = 0.4, which is close to 0, expresses this weather condition’s intelligent “exploitation.”

Regarding the Japanese side, we have set 1 − G = 0.6. Namely, when the aircraft were bombing Midway, the weather was sunny, and the Midway air forces quickly located the enemy aircraft. It is the reason why there was an air battle before the bombing of the airbase.

(g) The damages caused by the USN to the Japanese side was more considerable compared to the deterioration of the USN, so we have set $D_{x} = 0.6$ and $D_{y} = 0.4$ , respectively. Specifically, as mentioned above, Japan lost 4 carriers, 248 aircraft, 1 destroyer, 1 cruiser, and 3057 soldiers, sailors, and pilots died. USN lost only 1 carrier, 144 aircraft, 1 destroyer, and 362 soldiers, sailors, and pilots.

(h) The preparation costs of the United States were higher than the Japanese military expenditures, so we have set $E_{x} = 0.6$ and $E_{y} = 0.3$ , respectively. Particularly, the military expenditures of Japan were US$38.58 billion, while the United States spent US$227.24 billion.

Based on the above initial values, Equation (1) was solved, and we have got seven real equilibrium points. The complex fixed points were rejected (in all scenarios) simply due to not falling into reality.

The equilibrium points are presented below, while the kind of stability and instability is also mentioned.

(x^* = −6.3874, y^* = 28.5558) → Unstable

(x^* = −5.8733, y^* = −6.9249) → Saddle point

(x^* = 0.0694, y^* = 0.7182) → Saddle point (called E₁)

(x^* = 0.5926, y^* = 0.5867) → Stable (node) (called E₂)

(x^* = 0.6893, y^* = 0.0886) → Saddle point (called E₃)

(x^* = 50.2425, y^* = −8.8022) → Unstable

(x^* = 0, y^* = 0)

While the final fixed point (x^* = 0, y^* = 0) is trivial (and thus with no interest), among the rest six solutions, we only accept the fixed points which belong in the interval [0, 1], i.e., E₁: (x^* = 0.0694, y^* = 0.7182), E₂: (x^* = 0.5926, y^* = 0.5867), and E₃: (x^* = 0.6893, y^* =0.0886). All the other solutions have been rejected. The mathematical proof of stability for E₁, E₂, and E₃ is analytically presented below.

The Jacobian matrix is

\begin{matrix} J = \\ (\begin{matrix} - 0.8 xy - 3.6 x + 3.6 (1 - x) & 0.8 y - 0.8 (1 - y) - 0.4 x^{2} \\ - 0.6 y^{2} + 1.08 x - 1.08 (1 - x) & - 1.2 xy - 4 y + 4 (1 - y) \end{matrix}) \end{matrix}

(2)

The Jacobian matrix (Equation (2)) at the equilibrium point E₁ is

J^{*} = (\begin{matrix} 3.0597 & 0.3471 \\ - 1.2393 & - 1.8055 \end{matrix})

(3)

The determinant of $J^{*}$ is det $(J^{*}) = - 5.0940 < 0$ . The trace of $J^{*}$ is tr $(J^{*}) = 1.2541 > 0$ . The eigenvalues of $J^{*}$ are ( $2.9696, - 1.7154$ ).

The discriminant is $Δ = [tr (J^{*})]^{2} - 4 \det (J^{*}) = 21.9493 > 0$ . Therefore, the equilibrium point E₁ is a saddle point.

Studying the second equilibrium point E₂, the Jacobian matrix is

J^{*} = (\begin{matrix} - 0.9454 & - 0.0017 \\ - 0.0064 & - 1.1112 \end{matrix})

(4)

The determinant of $J^{*}$ is det $(J^{*}) = 1.0506 > 0$ . The trace of $J^{*}$ is tr $(J^{*}) = - 2.0566 < 0$ . The eigenvalues of $J^{*}$ are ( $- 0.9453, - 1.1113$ ).

The discriminant is $Δ = [tr (J^{*})]^{2} - 4 det (J^{*}) = 0.0275 > 0$ . Therefore, the equilibrium point is E₂ stable (node).

Studying the third equilibrium point E₃, the Jacobian matrix is

J^{*} = (\begin{matrix} - 1.4124 & - 0.8482 \\ 0.4043 & 3.2174 \end{matrix})

(5)

The determinant of $J^{*}$ is det $(J^{*}) = - 4.2016 < 0$ . The trace of $J^{*}$ is tr $(J^{*}) = 1.8049 > 0$ . The eigenvalues of $J^{*}$ is ( $- 1.3371, 3.1421$ ).

The discriminant is $Δ = [tr (J^{*})]^{2} - 4 det (J^{*}) = 20.0645 > 0$ . Therefore, the equilibrium point E₃ is a saddle point.

The fixed point E₂ describes (more realistically than E₁ and E₃) the strategic behavior of these two warring parties. The values of x^* and y^* are very close to each other, meaning that both sides had similar strategic behavior. Both opponents tried to astonish each other, making multiple and simultaneous attacks. In other words, both tried to plan surprise attacks or both were decoding the opponents’ messages, and so on. The USN victory was achieved through key moves, so this is the reason why the value of x^* is slightly higher than the value of y^*.

Figure 1 presents the strategic behavior of the United States (blue) and Japan (red) as the parameter $T A_{x}$ is varied.

We can observe (lower plot) an already bifurcated line (in red color) with two stable solutions, which describes the Japanese’s two possible choices. This phenomenon signifies that the Japanese side, especially Admiral Nagumo, was facing a dilemma regarding the decisions that needed to be taken. Indeed, Nagumo’s dilemma is historically well known. When he was informed of enemy ships’ location, he had to decide either to send airplanes to attack them immediately or wait for the Tomonaga’s team to return, refuel, and continue the strategic plan initially set by Yamamoto. Nagumo chose the second version since the loading of torpedoes instead of bombs would be a waste of critical time. Moreover, as the value of the parameter increases, the situation becomes even more complicated, and values higher than 0.65 leads to complete chaos. It means that if the US’s aerial technological capabilities were even more advanced, they would spot them quickly (via reconnaissance flights), and the Japanese would be unable to react in time.

Regarding the United States (upper plot, in blue), we observe two stable fixed points, which are very close to each other. On one hand, it proves that the USN side had to make crucial decisions, but, on the other hand, since the values of these two solutions are somehow similar, this fact is not a dramatic change in the outcome of the battle. Moving forward, as the value of the parameter $T A_{x}$ increases, there is an upward trend, which means that the United States had a successful strategic plan against Japan. When the value of the parameter is $T A_{x} \approx 0.4$ (initial value in the model), there is a bifurcation describing the US pilot McClusky’s two options. Specifically, McClusky had flown to search the IJN exact location. Unfortunately, he did not see them, and thus had to decide the following: if he would come back to Enterprise with his team to refuel and fly again, or whether he would move northwest, using the remaining fuels, lest they would find IJN there. McClusky ultimately chose the second option, which was proved a crucial move, and significant damages were caused to the Japanese carriers. If he had chosen the first option, the battle’s outcome would have been entirely different.

Figure 2 represents the strategic behavior of the USN (blue line) and IJN (red line), as the parameter G is varied. The purpose of this parameter is to highlight the weather conditions that existed at that time and affected the battle either positively or negatively. Particularly in both figures, there is chaos when the parameter value is close to G = 0. As the value of the parameter increases (up to G ≈ 0.09), there are two period-doubling bifurcations. From G ≈ 0.09 up to $G \approx 0.35$ approximately, we have just a pair of period-doubling bifurcations (i.e., two stable solutions), while for G≈ 0.35, there is a small interval where chaos appears. As $G = 0.4$ (initial value), we observe in both figures a two-period bifurcation, which corresponds to the two possible aspects of weather phenomena (cloudy or sunny). Each side uses these for each favor. For instance, when the first Japanese (y) airwave was launched, the weather was clear and sunny; thus, they bombed the Midway airbase, and as a result, the first part of the Japanese mission was accomplished. Hours later, USN (x) made several reconnaissance flights to locate the Japanese carriers, covered up within the clouds, so they were unnoticed.

3.2.1.1. Studying stability index

Each equilibrium point which satisfies the requirements: $\det (J^{*}) > 0$ , $trace (J^{*}) > 0$ , $Δ > 0$ , and identical negative eigenvalues, the stability index²¹ is calculated by $b = trace (J^{*}) / 2$ . When the stability index b belongs to the interval [−1, 1], the fixed point is stable, while if the value of b does not belong to the interval [−1, 1], the fixed point is unstable.²⁴

In Figure 3, we can observe that as the value of parameter G increases, the stability index b has an upward trend, while the values of b do not belong in the interval [−1, 1]. It means that the battle situation was unstable due to the weather conditions. As a result of this, it is visible that a weather phenomenon is crucial, affecting the outcome of the battle and causing instability. In the interval [0.33, 0.35], there is a gap in the line because the fixed points do not satisfy the requirements $\det (J^{*}) > 0$ , $trace (J^{*}) > 0$ , $Δ > 0$ , and identical negative eigenvalues.

3.2.2. Alternative scenarios of the battle

In this section, several hypothetical scenarios are presented, investigating: (a) first, what could happen during the battle if the opponents’ Intelligence would be different and (b) second, if the damage that IJN would bring to USN would be even more significant than the actual damages.

3.2.2.1. Changing the intelligence

Case 1: a reduction of American Intelligence

In this scenario, we have set the parameter $T I_{x} = 0.5$ , which means that United States had similar skills as the Japanese in decoding messages. All the other initial values are the same. We have repeated the analysis process by solving Equation (1). The real equilibrium points are as follows:

(x* = −6.3720, y* = 27.4325) → Unstable

(x* = −5.5815, y* = −5.8352) → Saddle point

(x* = 0.0993, y* = 0.7052) → Saddle point (called E₁)

(x* = 0.4980, y* = 0.5917) → Stable (node) (called E₂)

(x* = 0.6201, y* = 0.0991) → Saddle point (called E₃)

(x* = 36.3195, y* = −7.2711) → Unstable

(x* = 0, y* = 0)

Like in previous scenarios, the accepted fixed points are E₁: (x* = 0.0993, y* = 0.7052), E₂: (x* = 0.4980, y* = 0.5917), and E₃: (x* = 0.6201, y* = 0.0991), because they belong to the interval [0, 1]. The fixed point E₂ describes (more realistically than E₁ and E₃) the strategic behavior of the two opponents. If we would interpret the fixed point E₂, we could say that the Japanese gain a significant advantage as the value of the parameter $T I_{x}$ reduces. So, it is visible how important it is to decode the opponent’s messages and be prepared on time for the impending battle. Finally, if the United States did not gather the information, their defeat would be inevitable.

Figure 4 shows the strategic behavior of Americans (blue dots) and Japanese (red dots) as the parameter $T I_{x}$ varies. On one hand, concerning the blue dots, an upward trend exists, and at the point $T I_{x} \approx 0.73$ , a double-period bifurcation appears. On the other hand, regarding the red dots, we observe two fixed points, which explain the binary behavioral tactics (aggressive or defensive) that IJN could adopt as the parameter $T I_{x}$ increases. Particularly, when the parameter value is $T I_{x} = 0.5$ (initial value), there is no advantage of decrypting hostile messages; the blue dots are below the red dots (i.e., $x \approx 0.5, y \approx 0.67, and y \approx 0.53$ ). It indicates a decisive factor to USN and the corresponding results. More defensive behavior of USN would lead to an assured defeat, and Yamamoto’s strategic plan would have been accomplished.

Case 2: an increase of Japanese Intelligence

In this scenario, we have set the parameter $T I_{y} = 0.7$ , which means that the Japanese would have a considerable ability to decode messages compared to the United States, keeping all the other initial values the same as before. We have repeated the analysis process by solving Equation (1). The real equilibrium points are as follows:

(x* = −7.4135, y* = 36.0161) → Unstable

(x* = −6.7294, y* = −7.2940) → Saddle point

(x* = 0.0621, y* = 0.7584) → Saddle point (called E₁)

(x* = 0.5939, y* = 0.6448) → Stable (node) (called E₂)

(x* = 0.6958, y* = 0.0703) → Saddle point (called E₃)

(x* = 51.1242, y* = −8.8067) → Unstable

(x* = 0, y* = 0)

Like in previous scenarios, the accepted fixed points are E₁: (x* = 0.0621, y* = 0.7584), E₂: (x* = 0.5939, y* = 0.6448), and E₃: (x* = 0.6958, y* = 0.0703), because they belong to the interval [0, 1]. Once again, the fixed point E₂ describes (more realistic than E₁ and E₃) the strategic behavior of the two opponents. If we could interpret the fixed point E₂, we observe that the Japanese have begun to gain an advantage over the United States. Therefore, we can assume that the USN would no longer have the information advantage in this hypothetical scenario. If this could have happened, the Japanese would either send false messages to US espionage or the United States would not have any clue of the surprise attack at Midway islands.

In Figure 5, we can see that when $T I_{y}$ = 0.7 (initial value), the situation is completely chaotic. It means that we cannot predict the strategic behavior of both opponents. Still, we believe that in this case, due to a remarkable ability to collect information, Yamamoto’s plan would have been a complete success.

At this point, we should mention that the results from the analysis of Cases 1 and 2 show how important role had played the Intelligence in this battle.

3.2.2.2. What if … The IJN brings massive damages to the USN

In this scenario, we have changed the parameter $D_{y}$ , which means that the Japanese cause more damage to the USN than the initial approach (section 3.2.1). We have set $D_{y} = 0.8$ in Equation (1), maintaining all the other initial values. We have repeated the analysis process by solving Equation (1). The real equilibrium points are as follows:

(x* = −6.257, y* = 21.511) → Unstable

(x* = −5.7029, y* = −6.2447) → Saddle point

(x* = 0.1073, y* = 0.7018) → Saddle point (called E₁)

(x* = 0.5491, y* = 0.5878) → Stable (node) (called E₂)

(x* = 0.6774, y* = 0.0907) → Saddle point (called E₃)

(x* = 49.9598, y* = −8.7737) → Unstable

(x* = 0, y* = 0)

Like in previous scenarios, the accepted fixed points are E₁: (x* = 0.1073, y* = 0.7018), E₂: (x* = 0.5491, y* = 0.5878), and E₃: (x* = 0.6774, y* = 0.0907), because they belong to the interval [0, 1]. Once again, the fixed point E₂ describes (more realistic than E₁ and E₃) the strategic behavior of the two opponents. If we could interpret the fixed point E₂, we can observe that the values of x* and y* are very close, and the value of y* is slightly higher than x*. It means that, on one hand, the value 0.8 (of the parameter $D_{y}$ ) is a point where the battle’s outcome is in favor of the Japanese; they would have caused at least twice as much damage as they had already done to the United States to achieve a decisive victory.

In Figure 6, we depict the strategic behavior of Americans (blue line) and Japanese (red line) as the parameter $D_{y}$ is varied. When $D_{y}$ = 0.8 (initial value of the model) means that the Japanese cause more damage to the United States (than USN brings to IJN); the blue lines are between the two red lines. In particular, the values of each are as follows: $x \approx 0.55, x \approx 0.58, y \approx 0.48, y \approx 0.67$ . In addition, the initial value $(D_{y} = 0.8)$ is close to 1, which means that IJN should cause catastrophic damages to USN to achieve the decisive victory.

4. Conclusion

The Battle of Midway was a significant turning point in the history of naval warfare in the Pacific Ocean since the victory of the United States marked an end to Japanese expansionist policy. Indeed, it is widely considered one of the defining battles in history.^21,23,27 In this paper, we attempted to model and analyze their strategic behavior during the battle. Through a discrete dynamical system, its critical aspects were highlighted and the factors that determined the US victory, irrespective of its significant losses at Pearl Harbor (a few months earlier). Concurrently, some alternative hypothetical scenarios have been studied that modeled and highlighted the strategic mistakes of the two warring parties.

The presented discrete dynamical system aims not only to model a battle but also to simulate the strategic behavior of the opponents. The various solutions of the dynamical system express the possible strategic behaviors for both sides. At the same time, the parameters are the most crucial factors in analyzing the battle. Moreover, it turns out that a slight change in a parameter’s value can bring significant changes and chaos that can potentially reverse the outcome of the battle.

Initially, we tried to approach the facts of the battle through specific initial values. The extracted results prove that the battle’s outcome was shaped and based on the crucial decisions of the two admirals of IJN and USN. In particular, the existence of dilemmas and alternative strategies, which the opponents had during the battle, has been depicted through bifurcation diagrams. At the end of the conflict, some “key situations” determined the outcome of the combat. The key situations describe the critical conditions such as unforeseen situations or the factor of “luck.”

Furthermore, we studied the weather conditions (parameter G) during the battle. In particular, we observed that the weather was used as an advantage or disadvantage for each participant, i.e., depending on how each side can use (or manage) the weather for its favor. As we explained in section 3.2.1, the Japanese airwave bombed the Midway’s airbase when the weather was clear and sunny, which helped them accomplish their mission. Hence, the United States made reconnaissance flights to locate the Japanese carriers when the weather was cloudy to be covered up within the clouds, and thus unnoticed.

Regarding the hypothetical scenarios, the most significant conclusions are as follows:

Studying the change of Intelligence on both sides (Cases 1 and 2), on one hand, if IJN had better ability to collect the enemy’s information, they would conquer the airbase of Midway and they would take control of the Pacific Ocean. On the other hand, if both would have similar skills to decode messages, we cannot predict what could happen during the battle, so it would be upon them to make perceptive “key—moves” and choices.

Another way to overturn the outcome of the battle was presented. As observed, if IJN had inflicted at least twice as many losses as it had already caused, then it would be able to achieve a decisive victory.

As a final remark, let us mention that the validity of this modeling procedure has already been tested in another famous and crucial battle in the Greek–Persian war, the battle of Salamis.¹² The proposed model can be further applied to other contemporary battles and could be a useful decision-making tool for practitioners and strategic analysts. In addition, we could adopt the conceptual frameworks of the North Atlantic Treaty Organization (NATO)’s Political, Military, Economic, Social, Information, Infrastructure, Physical Environment, and Time (PMESII-PT)²⁸ and US’s Diplomatic, Information, Military, Economic, Financial, Intelligence, and Law Enforcement (DIMEFIL)²⁹ in our mathematical model. These frameworks provide a more holistic and systemic view that include political, military, economic, diplomatic, and financial aspects and are adjusted to the state-of-the-art battlefield environment. Furthermore, we could increase the nonlinearity of the system by introducing new parameters (i.e., new geostrategic factors such as interstate relations, diplomacy or institutional framework) in the model and/or changing its structure to capture the complexity of a real modern combat. An incorporation of a third actor (alliance, such as an NATO’s member) can be considered as a decisive factor that may overturn the outcome of the battle.

Finally, the model could be a useful additional tool for practitioners and strategic analysts, and it has the potential to be used in various scientific fields, including business, microeconomics, or international trade.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Konstantina Founta

Author biographies

Konstantina Founta is PhD Candidate in Department of Economics, University of Thessaly, Greece.

Loukas Zachilas is an Associate Professor on Econophysics at the University of Thessaly, Department of Economics, Volos, Greece and currently is the Chairman of this Department.

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