1 Introduction

The Peloponnesian War, 431–404 (all dates BCE), between democratic Athens and oligarchic Sparta was a long, destructive, “global” war extending over Greece, Sicily, and Asia Minor. It ended with a crushing defeat of Athens, the loss of her empire and termination of her Golden Age. The conflict was chronicled and analyzed by Thucydides (c.460–c.400), the ancient historian who also participated in the War as a military commander. Scholars in politics and international relations who followed Thucydides’ analysis have advanced two different theories of the causes of the conflict: Sparta’s fear of rising Athenian power, and unwise leadership. The present study re-examines the causes of the War in the light of the economics of conflict. This approach points to other, sometimes compatible, causes of war that, upon inspection, match well with the historical evidence.

The fear of Athens hypothesis arises from the statement of Thucydides,Footnote 1 that “the real reason, true but unacknowledged, which forced the war was the growth of Athenian power and Spartan fear of it”, 1.23.6, (Thucydides 2009, translated by  Hammond, 2009:13) reiterated in 1.88 and 1.118; see Rhodes (1987), Dillon and Garland (2013), Crawley (2012), and Ober (2015). Theses scholars construe Thucydides as reasoning that when a ruling power is threatened by a rising power, she will inevitably launch a preventive war and fight until it vanquishes the opponent. The deeper cause of the preventive war is traced to the lack of credible commitment to peace. This inevitability of war has been termed “Thucydides’ Trap.” Modern scholars, commentators and politicians alike see the Trap as applying to several later conflicts including the post-WWII US-USSR relations. Most recently the Trap has come to mean that conflict between China and the United States is inevitable; see Levy (1987), Gilpin (1988), Levy (2008), Copeland (2000), Allisson (2017), Weisiger (2013), and Blattman (2022).

The unwise leadership hypothesis rebuffs the inevitability explanation for ignoring human agency and for equating the war to destiny; see Kagan (1969, 1987), Lendon (2010), Samons (2016), and Platias and Trigkas (2021).Footnote 2 These authors reject that the War was an unavoidable consequence of a structural power shift. On the contrary, they emphasize that Thucydides blamed unwise leadership for the conflict, and focused on how leaders applied the power of the state to diffuse a crisis or precipitate war.Footnote 3

A closer look at how the War unfolded reveals problems with both theories. Critically for the inevitability hypothesis, in a peace treaty of 445 ending a previous confrontation during 460–445, Sparta had accepted Athens as a great power (Thucydides 1.80, 1.89–92). This raises the question of what had changed in 431 for Sparta to attack Athens. The hypothesis is further contradicted by the fact that during the War, Sparta, the attacker seeking to eliminate Athens, made several peace offers all rejected by Athens, the defender. On the other hand, the unwise leadership hypothesis requires an explanation of the channels through which leadership affects the decision to go to war. In this connection, Thucydides directs attention to a host of factors. These included the effect of the internal politics of the Greek city-states on decisions for war and peace, the expectations of rival powers to win, and the pursuit of honour from victory. Thucydides wrote that in 432 (just before the outbreak of the Great War) the Athenians declared that “we were forced to advance our dominion … as chiefly for fear, next for honour, and lastly for profit” (1.75.3, Hobbes’ translation). The statement demonstrates that not just Sparta, but Athens too chose war for a multiplicity of reasons, including her security, the prestige conferred to a position of power, and economic gain. More generally, such considerations point to the usefulness of applying the economics of conflict to study the causes of the War, its timing, failures of peace attempts, its length and ending.

The premise of the economics of conflict is that despite the costs and uncertain outcome of war, rational actors will fight to acquire an asset when the expected net gain from victory exceeds the gain from sharing the asset in peace (fully articulated by Thucydides 2,500 years ago in the opening quotation). The literature has shown that in addition to the acquisitive motive, war breaks out because of a host of causes. They include miscalculations of the probability of winning the war which originate from incomplete information about military strengths and weaknesses; principal-agent problems where unaccountable political leaders benefit more from war rather than peace; non-material objectives or ideological values; and non-credible commitments to future peace as posited by Thucydides’ Trap. This paper explores a formal model of a leader’s choice for war which incorporates the above factors and applies its predictions to the long conflict between Sparta and Athens. It is shown that, different combinations of the above causal factors drove the conflict at different phases, and that, to some extent, a “double Trap” was operating, where not only Sparta feared Athens, but Athens too was afraid of Sparta and chose war.

The study contributes to two strands of research, the political economy of Ancient Greece, and conflict economics. The former has generated significant insights into the emergence of democracy in some Greek city-states against oligarchy in others, mechanisms of democratic governance, public finances, legal arrangements, slavery systems, and economic development, (Fleck & Hanssen, 2018 offer a survey). However, it has left unexplored the wars between the Greek city-states. This study adds a rational choice model of the epoch-defining fight of Sparta versus Athens. Regarding the economics of conflict, it is noted that, surprisingly, neither ideological incentives nor Thucydides’ Trap have been formally analysed. The model explored here addresses both these shortcomings. First, it formalizes ideological objectives by adding the utility gained from satisfying the ideological objective upon winning the war. Second, since the Trap relates to a future uncertain development, it models a future war as probable rather than certain event. This implies that depending on their discount rates different leaders make different decisions about war and peace, which also shows that war is not inevitable.

The paper is structured as follows. The next Section offers an account of the conflict between Sparta and Athens during the Fifth century. The two powers fought four different wars: the First Peloponnesian War, 460–445; the Archidamian War, 431–421; the Sicilian Expedition, 415–413; and the Ionian War, 413–404. After summarizing theories of the causes of war, Sect. 3 presents a formal game-theoretic model of the decision of leaders to fight. It features the size of the disputed asset net of destruction costs, the fighting effectiveness of the belligerents, the possibility that they hold inconsistent expectations about defeating their enemy, the prospect that leaders derive higher personal gains from victory at war than peace, the non-material objective of honour from victory, and the factor by which the future is discounted (which relates to the fear of future developments). Section 4 shows that with different intensity the above causal factors explain how the conflict unfolded during its separate phases. Section 5 concludes.

2 Synopsis of the Peloponnesian war

This section draws on Kagan (1969, 1974, 1981 and 1987), Andrewes (1992a, b), Lewis (1992), and Hornblower (2011).

Militaristic Sparta was governed by two kings reigning simultaneously and assuming supreme military command, an aristocratic council of elders, and a board of five annually selected ephors with wide executive and judicial powers including responsibility for foreign policy. Her economy was agricultural, and all work was carried out by the helots, a type of public serfs. Sparta led the Peloponnesian alliance and was the traditional political and military powerhouse of Greece. Athens with an economy dependent on international trade was a democracy where the assembly of the demos and the popular courts were the sovereign bodies. An annually elected board of ten generals served as military commanders and led policy debates at the assembly. After final victory against the Persian invasion of 480–479, Athens established her empire over the islands and coastal cities of the Aegean Sea challenging Sparta’s position as the topmost power in Greece.

2.1 The first Peloponnesian war

During 460–445, the two powers fought the so-called First Peloponnesian War; see the Appendix for details. Athens enjoyed early gains by defeating Corinth, a naval power and Sparta’s most powerful ally. Sparta defeated Athens in the battle of Tanagra (458/457), but after winning against the Boeotians (Sparta’s allies) at Oenophyta, Athens established a “land empire”. She also took the island of Aegina, a longstanding naval rival. However, Athenian dominance of Boeotia ended in 446 after losing the battle of Coronea. Then Euboea revolted against Athens, but the revolt was suppressed. Megara also revolted and Sparta, allied to Megara, invaded Attica but her army drew back. In 445 Athens and Sparta agreed a Thirty Years’ Peace. Athens accepted Spartan hegemony over continental Greece in exchange for Sparta’s recognition of the Athenian naval empire. Members of each block were not permitted to switch sides, neutrals could join either side, and disputes were to be resolved by arbitration (although how such arbitration would work was not specified).The terms reflected the strengths of the two sides, but their mutual suspicion remained (Kagan, 1969).

figure a

Source: https://upload.wikimedia.org/wikipedia/commons/8/8e/Pelop_war_en.png

2.2 The great Peloponnesian war

Peace ended early because of an “inadvertent escalation” whose significance for the Great Peloponnesian War has been debated at length by scholars. In 435 the island of Corcyra entered a bloody dispute against Corinth, her ex-colonial metropolis. Athens agreed a defensive alliance with Corcyra which in 433 prevented Corinth from defeating Corcyra. In 432, Athens passed a Decree imposing economic sanctions on Megara (allied to Corinth); Megara appealed to Sparta. Although for Thucydides the Decree was not important, it was common belief in Athens that the refusal of Pericles, “the leading Athenian at the time and a man of the greatest ability both with words and actions” (Thucydides 1.139), to rescind it brought the War. Most modern opinion agrees that it was a significant driver of the war. In 433/2 Athens issued an ultimatum to Potidaea, tribute-paying ally of Athens but a colony of Corinth, to demolish her walls and expel Corinthian magistrates. Potidaea rebelled but was defeated in 430 after a costly siege. Aegina also complained that Athens had deprived her of autonomy. Thucydides considered these events as the immediate cause of the Great Peloponnesian War. In 432, amidst intense Corinthian pressure, Sparta declared war against Athens on the pretext of the liberation of Greece from Athenian dominance. The ensuing War is divided into three periods according to the theatre of operations (see the accompanying map and the Appendix).

2.2.1 The Archidamian war, 431–421

The Spartans invaded Attica in 431. The Athenians withdrew behind their Long Walls and sent their fleet to raid Laconia. In 430 a deadly plague struck Athens, whose worst was over by 427 after it had killed one third of the Athenian population including Pericles. In 425 an Athenian force took the island of Sphacteria at the west of the Peloponnese capturing a score of Spartans. Sparta sued for peace, but Athens refused. Fighting moved to Northern Greece where the pro-war leaders of Athens and Sparta, Cleon and Brasidas, were both killed. Then, a Fifty-Year Peace, negotiated by the Athenian Nicias, was agreed in 421 broadly based on the pre-war status quo. Most historians opine that Athens emerged victorious because Sparta failed to destroy the Athenian empire. A fragile peace followed.

2.2.2 The Sicilian expedition 415–413

In 415 the Athenians swayed by Alcibiades, a flamboyant politician, launched a grand armada against Sicily, coveted for her abundant grain and timber. Syracuse, a Corinthian colony, was the strongest Greek city in Sicily and since the 460s a democracy. Alcibiades and Nicias led the expedition, but after the fleet had sailed, Alcibiades was recalled facing charges for profanities. Worried that he would be condemned, he fled to Sparta. Sparta sent to Sicily a token force commanded by Gylippus. The campaign turned to be a disaster for Athens as her grand armada was annihilated at the harbour of Syracuse.

2.2.3 The Decelean or Ionian war, 413–404

In 413, on Alcibiades’ advice, the Spartan king Agis, fortified Decelea a village north of Athens, striking a severe blow to the Athenian economy (Thucydides 7.27). In 411 a coup overthrew the Athenian democracy and the new oligarchic government sued for peace, but Sparta refused demanding Athens to surrender her naval empire. The Athenian navy rejected the oligarchy and after defeating the Spartan fleet in a series of engagements culminating at Cyzicus, it restored the democracy in 410. Sparta offered peace, but Athens refused. Sparta then secured Persian funding for her fleet in exchange for Persian sovereignty of the Greek cities of Asia Minor, and in 407 Lysander commanding the Spartan fleet defeated the Athenians at Notium. A year later at the sea battle of Arginousae the Athenians trounced the Spartans. Once more, Sparta offered peace, but Athens rejected. In 405, in a final battle at Aegospotami of the European shore of the Hellespont Lysander’s fleet destroyed the Athenian forces. Without a fleet to put at sea, besieged, cut-off from her grain supplies, and with large numbers of citizens dying of starvation Athens surrendered in 404. Harsh terms were imposed including the demolition of the Long Walls, dissolution of her alliances, limiting her navy to twelve ships only, following wherever Sparta led, and replacement of democracy by a cruel oligarchy dubbed “The Thirty Tyrants.”

All in all, Athens suffered the loss of hundreds of ships, massive property damage, and the demolition of reputedly impregnable city walls. The human toll from war, disease and famine was enormous; the number of male citizens fell from approximately 60,000 in the 430 s to 30,000 at the beginning of the fourth century (Hansen 1999). Large numbers of Spartan citizens had also been killed, so that fourth century Sparta suffered from population shortage (oliganthropia, Aristotle, Politics, 1270a 29–32), although war losses were not the only factor. A year later and after much bloodshed, the Athenians restored their democracy. Athens gradually recovered to become an important political player again, but never reached her previous height. Sparta’s supremacy ended in 371 after defeat by Thebes, a previous ally.

3 An economic model of conflict

3.1 Overview of theories of causes of war

In the economics of conflict, war is a choice to fight over a resource, but it is a painful, costly, and uncertain mode of acquisition. To avoid the losses from violence, a bargaining equilibrium may be identified where the rivals are better off by sharing the disputed asset; see among others, Hirshleifer (1989), Fearon (1995), Powell (2006), Anderton and Carter (2009), Weisiger (2013), and Blattman (2022). Hence, war breaks out when the expected benefit from victory exceeds the benefit from settlement. The causes of the breakdown of peace include information problems, lack of accountability of political leaders, non-material objectives, and lack of commitment to the peace settlement.

In the information mechanism, war erupts because the two sides misperceive strengths and weaknesses and overestimate the probability of victory. Leaders who hold inconsistent expectations about victory start wars thought as winnable. As the course of fighting reveals information, the rivals may update their expectations. Fighting then serves as a costly but credible signal of strength. Other things being equal, with the two sides better informed, an acceptable peace agreement arises shortening the duration and destructiveness of war (Weisiger, 2013). Nevertheless, the learning process may be slow prolonging the war. It may be even longer when new leaders take over (since they may also overestimate the probability of military success).

Wars redistribute wealth not only across but also within states because the costs and gains from victory may differ across separate domestic groups. Politicians backed by groups expecting to benefit may then precipitate war. The cause of war here is traced to a political principal-agency problem, where weak checks and balances reduce leaders’ accountability to the public. With weak accountability, leaders may pursue wars for profit and for diverting attention from domestic problems.Footnote 5 However, public pressure against this type of war will mount when the leaders’ private motives and adverse developments in the battlefield become known. If political change is possible, leaders will be forced to settle or will be replaced by others amenable to negotiations. Hence, wars motivated by the private interests of leaders will continue only for as long as principal-agent problems persist.

Wars are also fueled by the pursuit of non-material objectives, like political ideology, religious beliefs, and quest for glory and status, what Blattman (2022) calls “intangible incentives.”Footnote 6 Ideology is often seen as the cause of indivisibility, for example, the exclusive right to an asset or an absolute religious doctrine, in which case no compromise may be possible. Surprisingly, no formal models of ideological wars are found in the economics literature. Contrary to wars fought only for acquisition of resources, wars motivated by highly valued non-material objectives will tend to last long and probably bring widespread destruction.

Wars may also break out because existing peace arrangements lack credibility as argued by the Thucydides’ Trap hypothesis. If the currently dominant power cannot trust that a rising adversary will keep the peace in the future, she worries that unacceptable concession will be demanded. She may then start a preventive war and keep fighting until it vanquishes the opponent (or, on the contrary, she is conquered by the opponent). Note that lack of credibility may originate from high valuation of the future and/or reflect imperfect information about future developments. Such wars tend to be long and result in cumulative losses.Footnote 7

It bears noting that although political leadership is featured explicitly only in the context of political accountability (in a bad light for that matter), it is a critical determinant of other factors too. A leader’ s judgments, rhetoric, and actions may be instrumental for articulating objectives, promoting a particular ideology, affecting morale, preparing the country for war, and signalling intentions to the enemy. Therefore, leadership is a factor encompassing the probability of victory, intensity of ideological benefits, valuation of the future, and confidence (or lack thereof) in the preservation of peace.

3.2 Model setup

Failure of leadership can be studied using either a static or dynamic model, but since the Trap relates to a fear of future events, its formal examination requires a two-stage game of conflict. At the first stage, two rival states, Sparta and Athens denoted by \(1\) and \(2\) respectively, may fight over an asset of size \(Y\), or settle peacefully by sharing it. War destroys a fraction \(\phi\) of the asset. The victor gets the entire net asset and the defeated gets nothing. Each state spends a sum \({X}_{i}, i=\text{1,2}\) to fight the war. Decisions about war are taken by the leaders of the states. To capture the principal-agent problem, we assume that a victorious leader receives a share \(a\) of the net asset but contributes a proportion \(b<a\) to the war cost.Footnote 8

In addition to the material spoils, the two sides may be motivated by non-material objectives and fight over values and ideas. The Greek city-states drawn by the quest for glory often fought seeking victories that in effect denoted symbolic dominance (Lendon, 2010). According to van Wees (2007: 287–288) “… for the Greeks repeated acts of disrespect—especially widely witnessed snubs and challenges at international festivals or large gatherings of allies—were serious and valid reasons to go to war … The competition for honour and profit was widely recognised as the greatest cause of war, with honour and profit carrying equal weights.” Only the war victor satisfies the non-material objective which generates utility \(G\) and has a relative marginal utility of \(k\); when the sides settle, their utility from the non-material objective is zero.

States \(1\) and 2 win the first stage war with probabilities \({P}_{1}\) and \({P}_{2}\) respectively. With Sparta the strongest Greek power at the beginning of the Fifth century, the probability she wins the first stage war is \({P}_{1}>{P}_{2}\). Following standard treatment, \({P}_{1}\) increases with military expenditure \({X}_{1}\) and the relative effectiveness of using it denoted by \(\gamma\) with \(0<\gamma \le 1\); it decreases with the military expenditure of its rival \({X}_{2}\), and the latter’s military effectiveness \(1-\gamma\). With state \(1\) stronger than \(2\), we have \(\gamma >\frac{1}{2}>1-\gamma\). To account for the effect of erroneous expectations from incomplete information we follow Anderton and Carter (2009) and assume that the states overestimate the probabilities of victory by a quantity \({\varepsilon }_{i}>0\), \(i=\text{1,2}\). We may then write the conflict success functions as

$$P_{1} = \frac{{\gamma X_{1} }}{{\gamma X_{1} + \left( {1 - \gamma } \right)X_{2} }} + \varepsilon_{1} ;\;\,P_{2} = \frac{{\left( {1 - \gamma } \right)X_{2} }}{{\gamma X_{1} + \left( {1 - \gamma } \right)X_{2} }} + \varepsilon_{2}$$
(1)

With inconsistent expectations leading to overestimation of success \({P}_{1}+{P}_{2}>1\). In the absence of information failures, \({\varepsilon }_{1}={\varepsilon }_{2}=0\) and \({{P}_{1}=1-P}_{2}>{P}_{2}\). Assuming risk neutral leaders as per standard practice their first stage or static payoffs from war are written as

$$U_{1}^{W} = P_{1} \left( {aY\left( {1 - \phi } \right) + kG} \right) - bX_{1}$$
(2.1)
$$U_{2}^{W} = P_{2} \left( {aY\left( {1 - \phi } \right) + kG} \right) - bX_{2}$$
(2.2)

We posit that the sides can raise the utility-maximising size of military expenditures, so that the probability of victory is determined endogenously. The Nash equilibrium of the war game is obtained when \(i=\text{1,2}\) maximizes (2.i) subject to (1) treating \({X}_{j}, j\ne i\) as given. Solving, and denoting the prize of victory by \(Z\equiv \left(aY\left(1-\phi \right)+kG\right)>0\), the equilibrium values of the probabilities of victory and expected payoffs are:

$$P_{1} = \gamma + \varepsilon_{1} ;\;P_{2} = 1 - \gamma + \varepsilon_{2}$$
(3)
$$U_{1}^{W} = \left( {\gamma^{2} + \varepsilon_{1} } \right) Z$$
(3.1)
$$U_{2}^{W} = \left( {\left( {1 - \gamma } \right)^{2} + \varepsilon_{2} } \right)Z$$
(3.2)

If instead of fighting the sides settle peacefully, they divide the asset taking respectively shares \(s\) and \(1-s\). As the leaders cannot profit from war, their shares are \(bs\) and \(b\left(1-s\right)\), that is, equal to their war cost shares. Ideological benefits, destruction costs and military expenditure are zero and the corresponding payoffs are

$$U_{1}^{S} = bsY$$
(4.1)
$$U_{2}^{S} = b\left( {1 - s} \right)Y$$
(4.2)

We assume that when the rival powers settle, they share the asset according to the “split-the-surplus” formula, where each side obtains the same gain by avoiding war, that is, \({U}_{1}^{S}-{U}_{1}^{W}={U}_{2}^{S}-{U}_{2}^{W}\) (see Garfinkel & Skaperdas, 2007). Substituting from Eqs. (3.i) and (4.i) into the latter, the peace equilibrium shares, and payoffs are respectively

$$s^{*} = \frac{1}{2}\left[ {1 + \left( {2\gamma - 1 + \varepsilon_{1} - \varepsilon_{2} } \right)\frac{Z}{bY}} \right];1 - s^{*} = \frac{1}{2}\left[ {1 + \left( {1 - 2\gamma - \varepsilon_{1} + \varepsilon_{2} } \right)\frac{Z}{bY}} \right]$$
(5)
$$U_{1}^{S*} = \frac{1}{2}\left[ {bY + \left( {2\gamma - 1 + \varepsilon_{1} - \varepsilon_{2} } \right)Z} \right]$$
(5.1)
$$U_{2}^{S*} = \frac{1}{2}\left[ {bY + \left( {1 - 2\gamma - \varepsilon_{1} + \varepsilon_{2} } \right)Z} \right]$$
(5.2)

Comparing (5.i) and (3.i), the first stage net benefit from settlement is

$$D^{*} \equiv U_{1}^{S*} - U_{1}^{W} = U_{2}^{S*} - U_{2}^{W} = \frac{1}{2}\left[ {bY - \left( {\gamma^{2} + \left( {1 - \gamma } \right)^{2} + \varepsilon_{1} + \varepsilon_{2} } \right)Z} \right]$$
(6)

According to (6) war breaks out (settlement prevails) when \({D}^{*}<\left(>\right) 0\). War is more likely the lower the cost of war \(\phi\); the higher the fighting effectiveness of the stronger state \(\gamma\); the greater the misperceptions about victory \({\varepsilon }_{1}\) and \({\varepsilon }_{2}\); the higher the leader’s share of war spoils \(a\) and the lower his share of cost \(b\); and the higher the utility conferred by satisfying the non-material objective \(k\) and \(G\). On the contrary an increase in the size of the asset \(Y\) has an ambiguous effect because a large \(Y\) implies large shares under settlement but also large payoffs from victory. These results formalize the hypotheses that wars are caused by material and non-material incentives, misperceptions of probabilities of victory, and principal-agent problems.

3.3 Second stage outcomes

The first stage choices constrain second stage decisions. Garfinkel and Skaperdas (2007) demonstrated that when a side values the future highly, its payoff may be higher by fighting a first-stage war. The reason is that the first-stage winner has exclusive use of the entire net asset over both stages. To capture the logic of preventive war motivated by fear of the future, the present analysis builds on their baseline model by adding two new points. First, the first-stage peace does not necessarily collapse in the future (as also noted by Allisson, 2017). Second, at the second stage of the game, the first-stage weaker power is stronger than its rival.

Specifically, to approximate Sparta’s fear from a rising Athens, we assume that \({D}^{*}>0\), so that there is peace at the first stage. At the second stage there is a \(Q\) probability that the first-stage settlement remains, and a \(1-Q\) probability that to avoid war a new peace settlement will be agreed with disadvantageous terms for state 1 (Sparta). Let the second stage fighting effectiveness of state \(1\) be \(\theta <\frac{1}{2}<\gamma\), and thus \(\theta <1-\theta\) (smaller than the fighting effectiveness of \(2\)). Thus, state \(1\) wins a second-stage war with a probability \({F}_{1}\) smaller than the first stage \({P}_{1}\) and smaller than the probability of second-stage defeat; that is \({F}_{1}<{P}_{1}\) and \({F}_{1}<1-{F}_{1}\). We then have (where without loss of generality we abstract from further misperceptions),

$$F_{1} = \frac{{\theta X_{1} }}{{\theta X_{1} + \left( {1 - \theta } \right)X_{2} }} < P_{1}$$
(7)

Working as before, the payoffs from second-stage war are

$$U_{1}^{WW} = \theta^{2} Z$$
(7.1)
$$U_{2}^{WW} = \left( {1 - \theta } \right)^{2} Z$$
(7.2)

The corresponding payoffs from split-the-surplus settlement are

$$U_{1}^{S**} = \frac{1}{2}\left[ {bY + \left( {2\theta - 1} \right)Z} \right]$$
(8.1)
$$U_{2}^{S**} = \frac{1}{2}\left[ {bY + \left( {1 - 2\theta } \right)Z} \right]$$
(8.2)

Assuming that the payoff from second-stage peace exceeds that from war, \({U}_{1}^{S**}-{U}_{1}^{WW}={U}_{S}^{S**}-{U}_{2}^{WW}>0\), the expected second-stage payoffs are

$$\tilde{U}_{i}^{S} = QU_{i}^{S*} + \left( {1 - Q} \right)U_{i}^{S**} ,\;i = 1, 2$$
(9.i)

3.4 1st stage choices

First-stage promises not to fight are not credible because there is no third power to enforce the settlement. The first-stage choices depend on the expected intertemporal payoffs of war and peace over the two stages. Let \(\delta\) denote the discount factor. If at the first stage the sides fight, the victor eliminates her rival and takes the net asset at both stages. The respective two-period payoffs from first-stage war are

$$V_{i}^{W} = U_{i}^{W} + P_{i} \delta Z;$$
(10.i)

The two-period payoffs from a first-stage peace are

$$V_{i}^{S} = U_{i}^{S*} + \delta \tilde{U}_{i}^{S} ,\;i = 1, 2$$
(11.i)

Figure 1 illustrates the extensive form of the game for state 1. The game is solved by backwards induction.

Fig. 1
figure 1

A two-stage war game

3.5 Equilibrium choices

State \(i=1, 2\) chooses preventive war (peace) at the first stage when \({V}_{i}^{W}- {V}_{i}^{S}>\left(<\right) 0\). Substituting from (11.i), (10.i), (8.i), (5.i), and recalling (6), we obtain

$${\text{For}}\;{\text{ state}}\;{ 1}\;:V_{1}^{W} - V_{1}^{S} = \frac{1}{2}\delta \left( {\left( {2\left( {1 - Q} \right)\left( {\gamma - \theta } \right) + 1 + \varepsilon_{1} + \varepsilon_{2} } \right)Z - bY} \right) - D^{*}$$
(12.1)
$${\text{If}}\;\left( {2\left( {1 - Q} \right)\left( {\gamma - \theta } \right) + 1 + \varepsilon_{1} + \varepsilon_{2} } \right)Z - bY < 0 \Rightarrow V_{1}^{W} < V_{1}^{S}$$
(12.1A)

The leader of state 1 will not start a war at the first stage.

$${\text{If}}\;\left( {2\left( {1 - Q} \right)\left( {\gamma - \theta } \right) + 1 + \varepsilon_{1} + \varepsilon_{2} } \right)Z - bY > 0 \Rightarrow V_{1}^{W} > ( < ) V_{1}^{S} \;{\text{for}}\;\delta > \left( < \right)\frac{{2D^{*} }}{{\left( {2\left( {1 - Q} \right)\left( {\gamma - \theta } \right) + 1 + \varepsilon_{1} + \varepsilon_{2} } \right)Z - bY}} \equiv \delta_{1}$$
(12.1B)

The leader of state 1 will start a preventive if he discounts the future by a factor higher than the threshold \({\delta }_{1}\).

$${\text{For}}\;{\text{ state}}\;{ 2}:\;V_{2}^{W} - V_{2}^{S} = \frac{1}{2}\delta \left( {\left( {2\left( {1 - Q} \right)\left( {\theta - \gamma } \right) + 1 + \varepsilon_{1} + \varepsilon_{2} } \right)Z - bY} \right) - D^{*}$$
(12.2)
$${\text{If}}\;\left( {2\left( {1 - Q} \right)\left( {\theta - \gamma } \right) + 1 + \varepsilon_{1} + \varepsilon_{2} } \right)Z - bY < 0 \Rightarrow V_{2}^{W} < V_{2}^{S}$$
(12.2A)

The leader of state 2 will not start a war at the first stage.

$${\text{If}}\;\left( {2\left( {1 - Q} \right)\left( {\theta - \gamma } \right) + 1 + \varepsilon_{1} + \varepsilon_{2} } \right)Z - bY > 0 \Rightarrow V_{2}^{W} > ( < ) V_{2}^{S} \;{\text{for}}\;\delta > \left( < \right)\frac{{2D^{*} }}{{\left( {2\left( {1 - Q} \right)\left( {\theta - \gamma } \right) + 1 + \varepsilon_{1} + \varepsilon_{2} } \right)Z - bY}} \equiv \delta_{2}$$
(12.2B)

The leader of state 2 will start a preventive war if he discounts the future by a factor higher than the threshold \({\delta }_{2}\), where \({\delta }_{1}<{\delta }_{2}.\)

Figure 2 illustrates the relationship between \({\delta }_{i}\) and \({V}_{i}^{W}- {V}_{i}^{S}\), and the threshold values for choosing war. When fearing future developments, the leader of state 1 is more likely to choose war (see Eq. 12.1) the lower the probability that the first stage settlement survives in the future \(Q\); the lower the second stage fighting efficiency \(\theta\); and the higher the discount factor \(\delta\). We then obtain:

  1. (a)

    For values \(\delta <\) \({\delta }_{1}<{\delta }_{2}\) both sides abide by the first stage settlement, a circumstance that for avoiding war may be labelled “wise leadership.”

  2. (b)

    For \({\delta }_{1}<\delta <{\delta }_{2}\), the leader of state \(1\) launches a war but the leader of state \(2\) still prefers peace. This can be thought as the essence of the original Thucydides’ Trap: Sparta’s fear of future decline made her declare war, even though Athens preferred peace. Thucydides’ Trap then emerges as a special case of the general model.

  3. (c)

    For \({\delta }_{1}<{\delta }_{2}<\delta\), both leaders choose war at the first stage, a case of “unwise leadership” afflicting both sides. Formally, the values of \(\gamma\), \({\varepsilon }_{1}\), \({\varepsilon }_{2}\), \(a\), \(\theta\), and \(\delta\) are high, and of \(b\), and \(Q\) are low.

Fig. 2
figure 2

Discount factor and the choice for war

Equations (6) and (12.i) imply that if large enough, each one of the causal factors is sufficient to cause war. On the other hand, their sizes may be such that the settlement payoff is higher, and peace prevails. As none of them is predetermined and all are affected by human agency, it follows that (a) war is not inevitable, (b) different factors may drive war at different circumstances, but also (c) peace is fragile.

4 Different causes explain the different phases of the Peloponnesian wars

The historical narrative showed that the long conflict between Sparta and Athens comprised separate phases of distinct circumstances under different protagonists. The contention here is that different causes were the principal drivers at the different parts of the conflict. Table 1 sets the theoretical causes examined above against the four different wars between Sparta and Athens, the First Peloponnesian War, 460–445; the Archidamian War, 431–421; the Sicilian Expedition, 415–413; and the Ionian War, 413–404.

Table 1 The causes of the Peloponnesian war(s)

4.1 Net material gains

For the Athenians the material gains included security of the sea lanes for their exports and unimpeded imports of grain and timber for ship construction, enrichment from war booty, setting up cleruchies (estates for the benefit of poorer Athenians in dependent territories), and tribute paid by the subjects of the empire. These acquisitive motives feature in the first three wars. In the last phase Athens fought for her survival as an independent polity rather than imperial possessions. Note also that the wars implied asymmetrical benefits and costs (a factor that fed into domestic politics and principal-agent problems). Athens’ land aristocrats and ordinary farmers lost their crops and had their estates looted by the invading Spartans. The wealthiest among them also paid the taxes to finance the war. It is indicative that the oligarchs who in 411 overthrew the democracy and approached Sparta for peace were members of the land aristocracy. On the other hand, poorer rowers, toolmakers, tradesmen, and those with urban properties benefited from the increase in demand for their services.

Sparta with her hinterland naturally protected by a range of mountains, remained unwalled and was safe from invasion (until the ascend of Thebes in the 360s). Her material motive for war was continued control of the subjugated helots and dominance of the Peloponnesian alliance. Sparta had no direct economic interest during the Sicilian campaign, but in the last phase of the War she was motivated by conquest and tribute.

4.2 Fighting effectiveness

The First Peloponnesian War demonstrated that Sparta, a militarized society, had the strongest infantry but could not match the Athenian navy. This justified Pericles’ strategy of withdrawing behind the Long Walls, refusing a pitched battle against the Spartans, and naval raids against the Spartan coastline. The Sicilian campaign also relied on Athenian naval power. Remarkably, after the 413 Sicilian catastrophe Athens was again able to dispatch a new fleet. It was only after receiving Persian money that Sparta operated a fleet able to challenge the Athenian. Without Persia bankrolling Lysander, it is doubtful that the war would have lasted that long and ended with Athens surrendering.

4.3 Misperceived probabilities of victory

Both sides miscalculated grossly in choosing war. In 433 Corinth attacked Corcyra dismissing the possibility that Athens would intervene and escalate the conflict (Kagan, 1969). In the Archidamian War, Sparta was wrong to expect a quick victory by invading Athens. Her expedition force could campaign only for 40–45 days during the dry months for lack of supplies and the overwhelming need to guard the helots at home. In addition, Athens’ refusal to give a pitched battle neutralised Sparta’s infantry advantage. But Pericles was also wrong to expect a short, defensive war, and believe that the Athenian treasury was large enough to see the war out. Throughout the conflict Athens had to increase tax levies. More spectacularly, the Athenian expedition to Sicily was a case of an “irrational gamble” (Ober, 2015), a colossal overestimation of the probability of victory.

Further, the probability of an existing peace surviving in the future (\(Q\) in the model) depended on the signals provided by the leaders of the sides about their intentions. Athens with her powerful navy, economic strength, authoritarian hold of the empire, and expansion to Western Greece and Sicily (areas in the Corinthian sphere of interest), magnified Sparta’s anxiety about her own security and position in the “hierarchy of prestige” of Greek city-states. Such signals increased the perceivable threat leading to war.

4.4 Principal-agent

Throughout his narrative Thucydides demonstrates that domestic politics mattered greatly for foreign policy. Political principal-agent relationships were present and significant in both Sparta and Athens. A warrior elite imbued in martial ideals governed Sparta. War offered ambitious men, like Brasidas, Gylippus and Lysander, the opportunity to advance their careers. In a militarized society, the military is more sensitive to security threats and henceforth more willing to fight (Levy, 1987). Even without Corinthian inciting, there was a substantial group in Sparta that could not tolerate falling behind Athens (Kagan, 1969). In 432 the pro-war faction of ephors prevailed and initiated the Archidamian War. In the later years of the Ionian War, king Agis and Lysander represented the pro-war faction and sought personal glorification (Kagan, 1987).

The quintessential direct democracy of Athens was also fraught with principal-agent problems, as it needed to delegate to various magistrates, most importantly the ten elected generals. The latter were political entrepreneurs competing relentlessly for influence and recognition; increasingly, they took higher risks. Thucydides (2.65), an admirer of Pericles, considered his successors, like Cleon and Alcibiades, inferior statesmen. He regarded them as unprincipled demagogues manipulating the demos to pursue insatiable personal ambitions. Thucydides was especially critical of Alcibiades attributing to him the eventual defeat of Athens. As already noted, in a democracy citizens punish recalcitrant leaders and wars not in the public interest will end quickly. We know that in the adversarial Athenian politics generals were punished for their decisions and conduct.Footnote 9 With such controls available, one asks why the principal-agent problem persisted throughout the wars. The answer is that the Fifth century Athenian democracy lacked checks and balances. The political leaders debated policy at the meetings of the assembly of the demos, whose composition could change from meeting to meeting, and motions passed with only a simple majority. Hence, policy decisions that could have been swayed by emotion rather than reason were not sufficiently scrutinized resulting in the pursuit of “unproductive” wars (Ferejohnh & McCall Rosenbluth, 2008).Footnote 10

4.5 Non-material objectives

All four wars feature the motivation for the honour of supremacy over the Greek city-states. Historians attest that for the ancient Greeks protecting one’s honour (timē) was always a cause to fight. Timē related to the degree of admiration for a warrior at the individual level, and the recognition of the status of a city-state in the ranking of the Greek city-states. In Greek thinking, death was followed by a bleak, dreary, afterlife of ghostly existence in the underworld to which nobody aspired. Thence, being honoured in life and remembered after death for one’s achievements was of the highest order (Sears, 2019:19–20). For individuals honour was won in the battlefield by showing andreia (courage, or manliness). It was shameful for a Spartan to flee from the battlefield, and a Spartan found glory in death for his polis (Sears ibid., Wheeler & Strauss, 2007). Athenians could be “accused of cowardice not only if they retreated from a battle but also if they failed to endorse a war that appeared to be necessary” (Pritchard, 2015:149).Footnote 11 Such values favoured war instead of peace.

The honour incentive is highly visible in the Peloponnesian War. As mentioned in the Introduction, Thucydides 1.75.3 explicitly refers to it as a motive for the Athenians to preserve their empire. For Kagan (1969:353) “the deep and emotional hatred between Corinth and Corcyra” was a crucial factor behind the crisis that started the Peloponnesian War. More emphatically, “Pericles inspired the Athenians to make war on the Spartans and numerous other Greek states … he held up the acquisition of future glory for the state as the citizenry’s highest possible goal” (Samons, 2016:5). Fighting for honour tended to prolong and escalate the conflict. This fed a spiral of mutual hatred providing in turn an additional explanation for the long duration of the Peloponnesian War.

It bears noting that political ideology did not motivate the Peloponnesian War.Footnote 12 Neither Sparta nor Athens desired to spread their political ideologies, and both made alliances for pragmatic rather than ideological reasons. For Thucydides Greek city-states were divided not by political ideology but by a host of traditional factors. The latter included ethnicity, specifically Dorian-Spartan against Ionian-Athenian, colonial relationships, that is, colony with (or against) the metropolis (as with Corinth and Corcyra), and relationships of friendship or enmity for past injuries. Pericles, the greatest democratic Athenian statesman no less, did not advocate that Athens should promote democracy outside Athens. On the contrary, he urged his fellow citizens to pursue war and deprive others of the freedom that was so precious to the Athenians (Samons, 2016). Further, the Athenian attack against Syracuse denies the applicability of the “democratic peace”, the proposition that democracies do not fight each other, to Ancient Greece.Footnote 13 But as later conflicts demonstrate Athens was not an outlier of the democratic peace. “For better or worse, democratic foreign policy is driven by public desires rather than fundamental pacifism” (Reiter & Stam, 2002: 145).

4.6 The discount factor and non-credible commitment to peace

Both Sparta and Athens were proud of their long histories. This made them players with high discount factors, which as the model showed increased the propensity to war. Sparta contemplating a future of inferiority to Athens fought a preventive war to force the surrender of Athens, as per the Trap hypothesis. This may be a valid explanation of the First Peloponnesian War where Sparta fought to stop the Athenian overland imperial expansion. In the period of Athenian economic growth and imperial expansion, 479–460, a stagnating Sparta must have been even more bellicose than otherwise. Fear of Athens as a cause of war was also behind Sparta’s launch of the Archidamian War in 431, and again in her refusal of peace in 411 after the oligarchs overthrew the Athenian democracy. In the latter occasion Sparta did not believe that a peace offer was credible and demanded that Athens would give up her empire. But the Athenian oligarchs would not accept that (or could not, for fear of internal reaction).

However, there are several problems with fear as an explanation for the entire conflict. Though fear of Athens was behind the First Peloponnesian War, Sparta accepted Athenian hegemony in the Aegean with the Thirty Year’s Peace. Second, in the Archidamian phase, rather than avoiding the clash, Pericles of Athens was a keen advocate of war against Sparta. He did not repeal the Megarian Decree and he had Potidaea besieged. That is, not only Sparta but Athens too chose war. Third, the fear explanation does not apply to the post 413 war (after the Sicilian disaster) since Athens was no longer able to threaten Sparta militarily. Finally, the hypothesis that Sparta launched a preventive attack ill-sits with her repeated peace offers. If fear of decline necessitates war to eliminate the rival, the attacker will not seek peace. But Sparta sued for peace after her heavy defeats in Sphacteria (425), Cyzicus (410) and Arginousae (406). An explanation is that the defeats forced Sparta to update expectations and concluded that the wars were not winnable. But why did Athens, the defender, refuse? Two reasons spring to mind. First, different Athenian leaders at the separate phases thought erroneously that they were at the brink of victory, a view consistent with the misperceptions explanation of the conflict. Second, Athens did not trust the sincerity of Spartan peace offers and calculated that Sparta was bidding for time to regroup and attack later. If so, Thucydides’ Trap applied to Athens too; that is, fearing that Sparta was going to get stronger, Athens dismissed peace proposals and continued fighting.

The latter implies that mutual fear and distrust drove the wars, so that a “double” Trap applied. Neither Athens nor Sparta trusted each other to preserve the peace. They were locked into a security dilemma, where actions taken for the security of one undermined the security of the other. Sparta feared Athens’ rise, and Athens saw Sparta as fixated at destroying her naval power vital for her security. Athenian engagement at Corcyra was also driven by Athenian fear; had Corinth been allowed a free hand against neutral Corcyra would have annulled the Thirty Years’ Peace. After the Sicilian catastrophe Athens was no longer able to dominate Greece. Yet, Sparta insisted on the total surrender of Athens and the installation of the oligarchic regime because she could not trust that a democratic Athens would not become a threat again. Distrust meant that peace treaties “failed because their purpose was hegemony through peace rather than peace per se … Competitiveness and a fierce spirit of independence on the one side, imperial ambitions to be realised by war on the other, made every agreement and treaty temporary. Peace was observed until one power believed it could gain more by going to war. War between communities, the Greeks believed, was an unalterable condition of human society” (Raaflaub, 2009:240).

Summarizing, it is easy to see that material and non-material objectives and fighting strengths drove all wars. Leadership, in the sense of perceptions of the probability to win and advantage of politicians against the public (political agency problems), is a prominent cause in the Archidamian war and the Sicilian expedition. Sparta’s fear of Athenian superiority caused the First Peloponnesian, the Archidamian and the Ionian wars; nevertheless, it was not only Sparta that did not trust Athens; Athens too considered Sparta’s offers of peace as disingenuous. Although lack of commitment to peace characterizes three out of the four wars, it was only the Ionian war that was consistent with the view that commitment wars are fought until one of the sides capitulates.

If a preventive war is not inevitable one may also wrestle with the question of whether the Great Peloponnesian War would have been avoided or stopped before its bitter end for Athens. In 432 and 431 events conspired so that both Sparta and Athens chose war over peace. Might cooler heads have avoided the conflagration that followed? Admirers of Pericles might think that had the wise leader lived longer, he would have negotiated peace with Sparta in 425. Indeed, a persistent accusation against Pericles’ successors is that they ignored his policy advice that Athens should cease further imperial expansion. Again, had the Athenians not suffered the Sicilian disaster, either by not starting the campaign, or more importantly, defeating Syracuse, the Greek world in the Fourth century could have looked significantly different. Athens would have retained her empire and preeminence, most probably Macedon would have remained weak and not conquered the Persian empire. A different culture would then have emerged in the post-Fourth century Hellenistic world (although we are in no position to evaluate the benefits from such developments).

5 Conclusions

Scholars have interpreted Thucydides’ History of the Peloponnesian War differently and offered two conflicting explanations of its causes. Thucydides’ Trap argues that the War was inevitable because Sparta feared that the rising power of Athens would displace her as the foremost Greek city-state. On the other hand, the unwise leadership view sees the main protagonists as lacking foresight and making bad judgments. The present study investigated the causes of the conflict using the formal perspective of the economics of conflict.

The model of eclectic decision-makers rejects the inevitability of Thucydides’ Trap, if inevitability means that only structural shifts of power matter regardless of the actions of human players. It also improves nebulous notions of unwise leadership by offering a systematic account of the decision of political leaders to go to war. The causes of the conflict were sought in the acquisitive incentive of possessing imperial possessions, the honour accorded from the highest ranking in the power relations of the ancient Greek city-states, perceived fighting strengths and erroneous expectations about probabilities of victory, insufficient constraints on the opportunities of political leaders to pursue personal interests and take risks, and the valuation of the future that fed mutual distrusts between the belligerents.

Hence, instead of subscribing to single cause explanations of the conflict between Sparta and Athens, the present work argues that a multiplicity of causes better captures the complex drivers of the wars. But rather than applying uniformly, the importance and intensity of each causal factor differed significantly over the different phases of the long conflict. As a result, the Peloponnesian Wars morphed from a war of Sparta preventing Athens to dominate Greece into a war of deep mutual distrust where each one of the two powers believed that her rival was predisposed to aggression. None of the causal factors was predetermined. On the contrary, they were shaped by specific circumstances at different times. Leaders were instrumental in rousing fears and tensions instead of building confidence, and miscalculating chances of success. We may then claim that the preventive war of the original Thucydides Trap was preventable rather than inevitable.

Perhaps there is a temptation to apply the wisdom earned from studying an old sage to what may seem like similar circumstances of later events. But comparisons between the ancient and the modern worlds are hazardous. This study suggests that the intuitions gained from Thucydides’ monumental work cannot be reduced to a single proposition; nor can they be extrapolated uncritically to the conflicts of different eras.More discreetly, it presents a systematic framework for analysing the original Thucydides’ Trap versus unwise leadership explanation of the Peloponnesian wars and vouches for the fruitfulness of applying economic reasoning to the ancient world.