Dynamical Systems Theory compared to Game Theory: The case of Salamis's battle

In this paper, we present an innovative non-linear discrete system trying to model the historic battle of Salamis between Greeks and Persians. September 2020 marks the anniversary of the 2500 years that have passed since this famous naval battle, which took place in late September 480 B.C. The suggested model describes very well the most effective strategic behavior between two participants during a battle (or a war). Moreover, we compare the results of the Dynamical Systems analysis to Game Theory, considering this conflict as a"war game".


Introduction
In recent years, many researchers have studied the players' behavior either through Game theory or through Dynamical Systems. Some of the notable works are Archan and Sagar [2] who present a possible evolutionary game-theoretic interpretation of nonconvergent outcomes. They highlight that the evolutionary game dynamics is not about optimizing (mathematically) the fitness of phenotypes, but it is the heterogeneity weighted fitness that must be considered. They mention that heterogeneity can be a measure of diversity in the population. In our research, this is described by the asymmetry in the conflict. In addition, Toupo, Strogatz, Cohen and Rand [3] present how important the role of the environment of the game is for the decision-makers. They suggest simulations of agents who make decisions using either automatic or controlled cognitive processing and who not only compete, as well as affect the environment of the game. Moreover, they propose a framework that could be applied in several domains beyond intertemporal choices, such as risky choice or cooperation in social dilemmas.
In other words, it's not all about the tactical behavior (aggressively or defensively), but also the impact of the location that the battle is taking place. Furthermore, the well-known evolutionary game "Hawk -Dove" has been used in several scientific fields to describe the effects of different behavioral changes in populations. Some interesting applications are presented. Altman and Sagar [4] apply this game in a flock of birds modeling their behavior. In addition, Souza de Cursi [5] examines the applicability of Uncertainty Quantification (UQ) in this game under the uncertainty of both the reward and the cost of an injury to determine the mean evolution of the system. Lastly, Benndorf, Martínez-Martínez and Normann [6] investigate the equilibrium selection and they predict a dynamical bifurcation from symmetric mixed Nash equilibrium to asymmetric pure equilibria in the hawkdove game, which depends on the frequency of interactions of the population.
Regarding the previous studies, it has been observed that there are no comparison results between dynamical analysis and game theory. The motivation of the present research refers to identify not only the connection of terms of these two scientific fields but also to apply this attempt in a battle.
The created model approaches short-term battles between two participants (players), where one is weaker than the other opponent. Also, the parameters (that we use in Eq. 1, see below) are the most crucial factors to highlight the optimal way to achieve a decisive victory. Below, the game hawkdove and its results are presented.
One of the most representative games of Evolutionary Game Theory is the so-called game "Hawk -Dove", which was originally developed by Smith and Price [7] to describe animal conflicts and is quite similar to our attempt. There are two animals (or two players) fighting for the same resource. Each of them can behave either as a hawk (i.e., fight for the resource) or as a dove (i.e., abandon the resource before the conflict escalates into a fight). Individuals have a benefit B if they win and a cost C if lose.
If a Hawk meets a Hawk, they will fight and one of them will win the resource; the average payoff is (B-C)/2. If a Hawk meets a Dove, the Dove immediately withdraws, so the payoff of the Dove is zero, while the payoff of the Hawk is B. If a Dove meets a Dove, the one who first gets hold of the resource keeps it, while the other does not fight for it; average payoff B/2. The strategic form of the game is given by the payoff matrix (1):

Results of the game "Hawk -Dove"
We set random values in the benefit B = 2 if a player wins, and in the cost C = 1 if a player loses in the payoff matrix (1). Using the Gambit a software (16.0.1), we find Nash equilibriums and the dominant strategy.

The Dynamical Model
It is widely acknowledged that the military strategy is the combination of «ends, ways and means» [8]. In our attempt to study the strategic behavior of two warring parties, we developed an innovative non-linear discrete system of two equations based on the above phrase. The main objective of the model is to simulate the way by which the two opponents behave strategically, where the one is weaker than the other.
Simultaneously, in Game Theory, the war is considered as a dynamic game where the strategies of the players are studied by calculating their optimal strategy (Nash equilibrium). In the present paper, we compare the results of the Game Theory to those from the analysis of the discrete dynamical system. At the end of the analysis, the optimum and effective strategy for both participants (players) will be suggested.
The model, which is applied in short-term conflicts and describes the strategic behavior of each participant, is given by Eq. 2: where: : The strategic behavior of the participant (player) x at the time t.
: The strategic behavior of the participant (player) y at the time t. +1 : The optimal strategic behavior of the participant (player) x at the (next moment of) time t + 1. +1 : The optimal strategic behavior of the participant (player) y at the (next moment of) time t + 1.
We consider , , +1 , +1 ∈ [0,1], because the logistic equation is defined in [0,1], which is derived from the study of biological populations reproduced in discrete time [9]. It's the evolution of the population model of Malthus [10] and shows that the exponential growth cannot tend to infinity, but there is a critical point, i.e., a saturation.
In other words, it is not possible for someone to win and the other to continuously lose.
Also, each optimal strategic behavior, at the time t, affects the next movestrategic behavior, at the time t + 1, of the opponent.
In addition, we can interpret the values of variables (and parameters, as shown below) as percentages or probabilities, which help us to explain the results; these are also explained through the Game Theory.
Moreover, if the value of +1 (or +1 , respectively) equals to 0, it indicates the fully defensive strategic behavior of participant x (or y respectively), while if it equals to 1, then it indicates the fully aggressive behavior of participant x (or y respectively).
The parameters of Eq. 1 are the main and most important factors that could affect the strategic behavior of x (or y, respectively). In particular: The parameter represents the strength (economic, military, population, territorial) of x and is the strength of y, respectively. These two parameters indicate the substance of each form of social organization compared to the other. and represents the Technological Naval capability and evolution of x and y respectively. These two parameters are also defined in comparison with the technological capability and evolution of the other participant and describe the means mentioned by Lykke [8].
The parameter G represents the geographical location (geophysical terrain) of the area where the battle or the war is taking place. We believe that this is another part of the military strategy, namely the ways [8]. Trying to emphasize the importance of this parameter and how it can be an advantage or disadvantage for each participant, we set in the first equation as G and in the second equation as 1 -G. The closer to the 1 the value of the parameter, the easier the geophysical terrain of the area is.
The parameter represents the damages caused by x to y and respectively, represents the damages that y brings to x. The damages which we refer to may be economic, territorial, military, etc. or even deception and damaging of the psychological part of the opponent. Moreover, these two parameters complete the last part of the military strategy, namely the ends [8].
The parameter represents the expenses of participant x and the expenses of participant y, respectively. In other words, these denote the preparation costs of each participant for a battle (or war), compared to each other.
In the next section, we present the dynamic analysis and the results from the application of Eq. 1 in naval battle of Salamis.

The case of (naval) Battle of Salamis
The naval battle of Salamis was an important battle of the second Persian invasion in Greece and has been estimated to being held on September 28 th , 480 BC in the Salamis straits (in the Saronic Gulf near Athens). The two warring parties were the Greeks (Hellenic alliance) and the Persian Empire [11].
After the fall of Thermopylae, the Persians went ahead to Athens. The Greeks had been advised by the Oracle of Delphi, that only the "wooden walls" would save them, and they considered that this referred to a fight in the sea [12].
A few days before the battle, the war council of the and periplous, (flanking or enveloping move, which generally gave an extra benefit against superior numbers in open water). The purpose of both was to ram the enemy in the side. In this way, they achieved serious damages or even the complete destruction of the Persians ships. On the contrary, the Persian tactic was "ramming and boarding" [14].
b Aeschylus, writing decades earlier, also gives 1,207 triremes, but Herodotus writes, shortly before battle took place, that the Persian fleet wasn't much bigger than Greek. Because of a weather phenomenon (storms) 600 ships sank (400 at the coast of Magnesia, north of Artemisium and 200 in Euboea). Herodotus reports that "the Greeks fought with discipline and held their formation, but the Persians did not seem to be following any plan, so things were bound to turn out for them as they did". Also, Aeschylus mentions that Themistocles must be given the credit for their battle and the winning tactics. The turning point of the battle came as the Persians "suffered their greatest losses when the ships in their front line were put to fight and those following, pressing forward to impress the King (i.e., Xerxes) with their deeds, became entangled with them as they tried to escape", as Herodotus comments [13].
The naval battle evolved rapidly and by the noon it was visible that the Greeks would win. The Persian fleet had crushed, while the Greek fleet continued to haunt them, killing the helpless, non-swimming soldiers. This brought the battle to an end, leaving the Greek force in full control of the straits [14].
When the battle was over, a Roman source mentions that Greeks lost more than 40 triremes and Persians more than 200 ones [14]. The victory of the Greek force was of major importance, since they managed to cause the collapse of the Persian morale, which is evidenced by the abandonment of the battle. In addition, the right decision of Themistocles for the geographic location of the naval battle was one of the most intelligent movements to bring the Greek victory. the Persian ships were similar in shape, so we assume that the cost of each ship was similar. Thus, it is obvious that the Persians spent more money to support their expedition to the Greek territories than the Greeks.
With these initial conditions, we solve the system (Eq.1), by using the mathematical software Maxima c (5.39.0), calculating the equilibrium points. Then, we study more extensively the behavior of the model, and we present bifurcation diagrams and timeseries diagrams using the software E&F Chaos d .
The Jacobian matrix is: The trace of * is trace(J * ) = 0.
Therefore, the equilibrium point E1 is a stablecenter.
Therefore, the equilibrium point E2 is a saddle point.
Thus, we continue the analysis for the fixed point E1. Interpreting this equilibrium point, we confirm the aggressive (strategic) behavior of Greeks; since the value of x * is close to 1 and the mild (strategic) behavior of Persians; since they thought it would be an "easy win".
Indeed (historically), the courage of the Greeks, their technological naval skills, and the advantageous geographical location contributed to this aggressive behavior. As far as the Persians are concerned, their mild (strategic) behavior is due to the fact that they underestimated their enemy, since they regarded that the Greeks are an easy target, and they would achieve a decisive victory.
Connecting the game "Hawk -Dove" to the naval battle of Salamis, player 1 (red) is "Persians" and player 2 (blue) is "Greeks" (Fig. 4)       In this section, we will study two alternative scenarios, that were indeed discussed in Greek Generals' meeting. In the first scenario we changed the place of the naval battle, which now is supposed to take place in a more open sea, (i.e., openly the Saronic Gulf), while in the second scenario we changed the place to a mixed battle (part of the battle takes place in Saronic straits, and part at the mainland of Isthmus of Corinth).

(a) Openly the Saronic Gulf
In Eq. We solve Eq. 1, and we get two equilibrium points: E1 (x* = 0.667, y* = 0.67) and E2 (x** = 0.904, y** = 0.963). According to Game Theory, these fixed points are Nash equilibriums. The stability of these points is studied, again, by the Jacobian matrix: We calculate the Jacobian matrix at the fixed point E1: The determinant of * is det ( * ) =  0.314 < 0.
Therefore, the equilibrium point E1 is a saddle point.
Studying the second fixed point E2, the Jacobian matrix at the equilibrium point is: The determinant of * * is det ( * * ) =  2.068 < 0.
The trace of * * is trace ( * * ) = 0. and y ** areagainvery close, while very close to 1. This means that both opponents have a very aggressive behavior (we may assume that it would be a conflict between two Hawks, according to game "Hawk -Dove"). This is a non-effective scenario (strategy) for Greeks, because the Greek's benefit of such a conflict would be lower than the scenario: "Greeks / Hawk" versus "Persians / Dove". Nevertheless, it can be classified as an unstable situation for the same reasons to the first fixed point.  We solve Eq. 1, and we get (again) two equilibrium points: E1 (x * = 0.7, y * = 0.77) and E2 (x ** = 0.85, y ** = 0.92). According to Game Theory, these fixed points are Nash equilibriums. We have studied (again) the stability of these points by calculating the Jacobian matrix: The Jacobian matrix at the fixed point E1: * = ( 0 0.778 0.708 0 ) (10) The determinant of * is det ( * ) =  0.583 < 0.
Therefore, the equilibrium point E1 is a saddle point.
Therefore, the equilibrium point E2 is also a saddle point.
We can observe for the first fixed point E1 that the values of x * and y * are very close, while, for the second fixed point E2 the values of x ** and y ** have a slight deviation. In both fixed points, the value of y is greater than x, which means that in both cases the Persian fleet could win this conflict anyway. Therefore, since in both cases the fixed points are saddle ones, the situation is unstable and Persian dominance in this mixed battle is indisputable.

Studying bilateral damages of two opponents
In this section, we present the results of three hypothetical scenarios concerning the damages caused by Greeks to Persians and vice versa. We will keep all the rest parameters at their initial values, representing that the battle took place in Salamis's straits.
In the first scenario, we assume that = 0.5 and = 0.5, i.e., the damage that Greeks caused to Persians is 50% (of their total armament) and vice versa. In the second scenario, we set = 0.3 and = 0.7, i.e., Greeks cause 30% damage to Persians and Persians cause 70% damage to Greeks, respectively. In the last scenario, we assume that = 0.8 and = 0.2, i.e., Greeks cause 80% damage, while the Persians cause 20% damage to Greeks. The time series diagrams for each scenario are presented in   and Persians (red), respectively. The supremacy of the Greeks is evident from the beginning, since, on the one hand, they had a better technological ability and an advantageous geographical location and, on the other hand, the fact that 50% of the damage to the opponent would be capable of bringing the Greeks a decisive victory.    at the beginning, in the Salamis straits is owing to naval tactic "diekplous" and the advantageous geographical location. In this way, Greeks achieved the decisive victory against to Persian empire.

Conclusions
In this paper, an innovative nonlinear discrete model has been presented, which simulates the optimum strategic behavior of two warring parties for short-term battles.
In addition, we try to compare this model with the classical Game Theory, applying this attempt in the naval battle of Salamis. Based on the results we have extracted, we (mathematically) proved the historical events of this conflict, and we concurrently studied some alternative hypothetical scenarios. Specifically, by changing the geographical location of the conflict, we prove that the optimum location for Greeks was the Salamis straits and, on the contrary, the worst location for achieving the decisive victory would be Isthmus of Corinth. Moreover, we study various scenarios of damages that could be caused by Persians to Greeks and vice versa. The third scenario (80% damage by Greeks to Persians and 20% damage by Persians to Greeks) was the most realistic version, confirmed by historical texts.