Denial and punishment in war

Abstract

Formal models of war termination have been developed along two major approaches: in one, war is interpreted as a series of battles, where nations exchange denials that aim to destroy enemy forces; in the other, war is illustrated as a process of bargaining with mutual punishments that inflict costs on opponents. By integrating these two approaches, I build a dynamic model of war, where two nations choose between military force and civilian value as the targets of their attacks in every battle. The war proceeds along one of the following three paths in equilibrium. First, either nation immediately gives in to the other’s threat of punishment such as nuclear strikes. Second, both the nations continue to conduct counterforce denial campaigns to weaken each other until either side is fully disarmed (i.e. all-out war). Third, after significant military imbalance is generated by the exchange of denials in an early battle, the winner of the battle switches to a countervalue punishment campaign, to which the loser responds by capitulation (i.e. limited war). My equilibrium analyses suggest that while denials largely determine a war’s outcome, punishments can influence its duration. Unlike existing models, mine illuminates the two-way causal relationship, where past battle outcomes can influence the choice of military strategy, whereas military strategies also shape how the war will further evolve.

Keywords

aerial bombardment choice of target denial vs. punishment military strategy

The Fighters are our salvation but the Bombers alone provide the means of victory.

– Winston Churchill

Introduction

Theorists in International Relations have modeled the process of war along two major approaches. Models in one approach depict war as a political affair, where two nations strive to compel each other’s surrender, or concession at least, by exerting the power to hurt (Schelling, 1966). By nature, this approach consorts with bargaining models of war (Fearon, 2007; Leventoğlu & Slantchev, 2007; Powell, 2004, 2012; Slantchev, 2003a; Wagner, 2000). In many of these ‘punishment’ models, wartime costs – in addition to the revealed information as to the opponent’s resolve and/or strength – constitute a major motivation for nations to terminate war; in the meanwhile, the relative strength between nations – namely, the per-period probability that either side defeats its opponent – remains fixed (but often uncertain) throughout war. As a logical consequence, either nation collapses probabilistically but suddenly unless a peaceful agreement is reached.

Models in the other approach portray war as a military affair, where two nations struggle to overpower each other by wielding the ability to diminish the enemy’s military strength, or what might be called the power to weaken (St Petersburg Declaration of 1868).¹ The latter approach presumes a war to be a series of battles, where the relative strength can change through fighting, as found in random-walk models (Slantchev, 2003b; Smith, 1998; Smith & Stam, 2003, 2004) and also in combat models by Lanchester, ([1916] 2009) and his followers (Bellany, 1999; Langlois & Langlois, 2009, 2012). According to these ‘denial’ models, a downward shift of military balance – as informed through battlefields and diplomatic tables – and associated decline in the prospect of war can lead to a concession and restoration of peace. While the punishment models have been inspired by von Clausewitz’s ([1832] 1976: 87) renowned definition of war as ‘merely the continuation of policy by other means’ (Reiter, 2003; Valentino, Huth & Croco, 2006; Wagner, 2000), the denial models may better reflect von Clausewitz’s alternative definition of war as ‘nothing but a duel in a larger scale’ ([1832] 1976:75).²

This dichotomy in models of war may mirror the long-lasting debate over the use and effectiveness of airpower (Overy, 1992). Early proponents of military aviation, on one side of the debate, were affirmative of punishment campaigns to destroy civilian targets such as economic infrastructures, commercial districts, residential areas, and in particular ‘the will of the people’ (Douhet, [1921] 1983).³ On the other side of the debate, a majority of contemporary scholars contend that the political objectives of war can be better pursued by denial campaigns – consisting of air support and interdiction – that are projected onto the enemy’s military capacity (e.g. fielded forces, ordnance, arsenals, and weapon plants).⁴

Despite the fact that the problem of the choice between denial and punishment has been long acknowledged and disputed since the rise of airpower (Douhet, [1921] 1983), we still lack a proper theoretical instrument with which to address the problem. I thus develop a formal model in a synthetic approach that incorporates both denial and punishment as alternatives.⁵ Major questions are the following. How does (and should) a nation choose between military force and civilian value as the target of attacks during war?⁶ How does (and should) the targeted nation react against these attacks? How do their choices influence the outcome and duration of war? To the best of my knowledge, my research is the first attempt to study the determinants and effects of target choice with an analytical, game-theoretic model.

This model is built upon the standard war-of-attrition model (Maynard Smith, 1974) but critically differs from it in that: (i) each of two nations possesses some military units at a war’s onset; (ii) they fight a series of battles, where a loss results in the reduction of one military unit; (iii) they choose one out of ‘denial’, ‘punishment’, and ‘surrender’ in every battle; (iv) while ‘denial’ can weaken the enemy by destroying its military units, ‘punishment’ hurts it more heavily by inflicting larger costs.⁷ The war ends when one nation exhausts all its military units or when it surrenders. While my model abstracts away the bargaining aspect of war, it does allow strategic termination of war in light of developments on the battlefield.

I found that three patterns emerge as equilibrium. If punishment is as serious as nuclear strikes, war immediately ends, as either side gives in to the other side’s punishment. Otherwise, war evolves in one of the two ways: the nations continue to exchange denials until either side is made incapable of fighting (i.e. all-out war); or after mutual denials generate military imbalance in an early battle, the winner of the battle finishes the war with punishment (i.e. limited war).

The innovation of my work is threefold: (a) unlike existing formal models of war, among which the majority have focused on the political process of bargaining and paid little attention to military strategies (for exceptions, Baliga & Sjöström, 2008; Lindsey, 2015; Meirowitz & Sartori, 2008; Slantchev, 2010; Tarar, 2016), my model explores the determinants and effects of military strategies;⁸ (b) unlike existing doctrines of airpower, which have commonly treated denial and punishment as mere alternatives (for an exception, Allen & Martinez Machain, 2019), my theory suggests that denial and punishment function as strategic complements and their dynamic combination can best bring about a victory, as observed in the Pacific War;⁹ (c) unlike existing quantitative empirical studies of military strategy, most of which have assumed military strategies to be fixed throughout war (for an exception, Bennett & Stam, 1998), this study allows the strategies to be changed during war. This flexibility enables us to illuminate the two-way causal relationship between military strategies and war – military strategies influence war, and vice versa.

The rest of the article proceeds as follows. First, the literature on military strategy is reviewed. Second, the model of war is presented, solved, and interpreted to depict the dynamic of nations’ target choices. Third, theoretical findings are summarized. Fourth, implications for empirical studies are discussed. Fifth, agendas for future research are proposed. Technical materials are concisely shown in the Appendix at the end of the article, while mathematical proofs and discussions on air strategies during the Pacific War are in the Online appendix.

The literature on military strategy

Empirical studies have explored the determinants and effects of military strategies. According to them, military strategies are influenced by a variety of both international and domestic factors. The international factors include institutionalized alliances, international laws, military strength, high stakes, war duration, and battle-deaths (Allen & Martinez Machain, 2018; Downes, 2008: Ch. 2; Valentino, Huth & Croco, 2006; Wallace, 2008). The domestic factors are political regime, military culture, advanced industry, and war experience (Downes, 2007, 2008: Ch. 2; Goemans, 2000; Legro, 1995; Reiter & Meek, 1999).

The empirical literature has also investigated the effects of military strategies on the outbreak of war (Epstein, 1987; Mearsheimer, 1983; Reiter, 1999), its duration (Bennett & Stam, 1996; Martinez Machain, 2015; Reiter & Stam, 2002; Stam, 1996), and the outcome (Allen & Martinez Machain, 2019; Arreguin-Toft, 2001; Reiter & Stam, 1998, 2002; Stam, 1996).

While the empirical literature has grown and matured as shown above, the theoretical studies on military strategy have remained sparse. Among the extant theoretical studies, those closest to mine are Snyder (1961) and Intriligator & Brito (1984), both of whom explored the target choice between force and value.¹⁰ However, in contrast to the focus here on the process and termination of war, their primary concerns are deterrence and outbreak of war. Into bargaining models with incomplete information, some forms of military strategy have been incorporated such as concealment of strength (Baliga & Sjöström, 2008; Meirowitz & Sartori, 2008; Slantchev, 2010), indirect strategy (Lindsey, 2015), and fait accompli (Tarar, 2016). Unlike these bargaining models, my model illuminates the shift of military balance by incorporating the state vector of strength. In the context of nuclear deterrence, some strategies have been formally studied, such as the risk strategy (Powell, 1987, 1988), limited retaliation (Powell, 1989), and counterforce first strike (Wagner, 1991). Also in the context of counter-terrorism, the choice between pre-emption and deterrence has been investigated (Nakao, 2019; Sandler & Siqueira, 2006).

In contrast to some qualitative studies on wartime changes of military strategies (Downes, 2008; Gartner, 1997; Goemans, 2000), most of the existing quantitative studies have commonly presumed military strategies to be chosen at war’s onset and fixed throughout (for an exception, Bennett & Stam, 1998). Although the presumption of fixed military strategy sounds reasonable for certain (especially land and sea) warfare, where an operation is difficult to modify once invoked, it has limited applicability to other (aerial) warfare, where nations flexibly adjust their means and plans to evolving strategic circumstances (Byman & Waxman, 2002: 88–99). For this reason, I develop a dynamic model that allows the choice of targets (‘campaign’ in my words) to be changed during war. The setting of flexible military strategy enables us to capture the two-way causal relationship between military strategies and war, which in turn suggests that the existing large-N empirical studies may suffer from selection effects (Valentino, Huth & Croco, 2006: 376), as elaborated toward the end of the article.

The model of war as exchange of denials and punishments

My model aims to analyze how nations choose and change the targets of their attacks during war and also how they react to these attacks. In the model, two nations attack each other on either force or value over time. While an attack on force can reduce the opponent’s military capabilities, an attack on value inflicts heavier costs on the opponent.¹¹ Recall that this model is built upon the model of war of attrition (Maynard Smith, 1974).

Denial vs. punishment

The model depicts a war fought between two nations (A and B) across infinite time periods $t = 1, 2, 3, \dots$ . Each $i \in \{A, B\}$ possesses two military units at the war’s onset, or $m_{i} = 2$ , where $m_{i} \in \{0, 1, 2\}$ denotes i’s strength.¹² The pair of the nations’ strengths $M \equiv (m_{A}, m_{B})$ functions as a state vector. Because $m_{i} = 0$ means i’s complete disarmament, there are four states of war: $(2, 2)$ , $(2, 1)$ , $(1, 2)$ , and $(1, 1)$ .

In each state of war $M \in \{1, 2\} \times \{1, 2\}$ , they simultaneously choose one of the three options: to attack the opponent’s force (‘denial’), to attack its value (‘punishment’), or to ‘surrender’ ( $σ_{i} \in \{D, P, S\}$ ).¹³ While a denial campaign weakens the opponent, a punishment campaign hurts it.

If i adopts denial, it can destroy one of the opponent’s military units with a (later-defined) probability that depends on i’s strength m_i relative to the opponent’s m_j . A denial by i also inflicts a per-period cost $c^{D} \geq 0$ on the opponent.

On the other hand, if i adopts punishment, he can hardly destroy the opponent’s military unit but can impose a heavier cost $c_{m_{i}}^{P}$ on the opponent. The size of $c_{m_{i}}^{P}$ increases with i’s absolute strength m_i and is surely larger than c^D (i.e. $c^{D} < c_{1}^{P} < c_{2}^{P}$ ).¹⁴ In addition, punishment is assumed to be sufficiently costly on the opponent, or $c_{2}^{P} > 2 c^{D}$ , which substantially simplifies the analysis of equilibrium.

All-out war and limited war

The war (and also the game) end with i’s victory when its opponent j is fully disarmed ( $m_{j} = 0$ ) or when j surrenders ( $σ_{j} = S$ ). The former form of war is referred to as all-out war, while the latter form is limited war. If i wins the war, i seizes a lump-sum benefit $W > 0$ , while j gains nothing. By surrendering, j can avoid any additional costs of fighting (i.e. costs c^D if $σ_{i} = D$ and $c_{m_{i}}^{P}$ if $σ_{i} = P$ ). If both the nations surrender simultaneously, they both gain nothing.¹⁵

Figure 1.

Dynamics of war

Battles

Depending on the nations’ military capabilities $(m_{A}, m_{B})$ , I define battles of the war:

Definition 1: A state is called battle $(m_{A}, m_{B})$ if A has m_A military unit(s), while B has m_B unit(s).¹⁶

A battle results in one of i’s ‘winning’, ‘losing’, and ‘indecisive’ outcomes in a period. If i ‘wins’ a battle, it succeeds in reducing its opponent force by one unit. In any battle, a nation’s ‘win’ is identical to the opponent’s ‘loss’. If a battle remains ‘indecisive’ in a period, the same battle will be fought again in the next period, that is, a battle can last (or be thought of as being repeated) over multiple periods. Then, the war proceeds in at most three battles – from $(2, 2)$ through $(2, 1)$ or $(1, 2)$ to $(1, 1)$ (Figure 1).

If nation i adopts denial in period t, it can ‘win’ with per-period probability $\frac{(1 - δ) {(m_{i})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$ , where $δ \in (0, 1)$ denotes a baseline per-period probability of the ‘indecisive’ outcome, and $L \geq 0$ determines the relative advantage to the stronger side in a battle. If both the sides adopt denial, the per-period probability of the ‘indecisive’ outcome is $δ$ .

If i adopts punishment, it can still ‘win’ but only with much less per-period probability $\frac{ε (1 - δ) {(m_{i})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$ , where $ε$ is positive but near zero – a counter-civilian strike can hardly destroy the opponent’s force. Instead, a battle is more likely to remain ‘indecisive’, that is, the complementary probability $\frac{(1 - ε) (1 - δ) {(m_{i})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$ is added to the baseline probability $δ$ of the ‘indecisive’ outcome, as i adopts punishment. Per-period probabilities of outcomes in battles are summarized in Table I.

Nation i’s continuation payoff from $(σ_{A}, σ_{B})$ in battle M – denoted as $U_{i | M}^{σ_{A} σ_{B}}$ – can be derived recursively. For instance, A’s continuation payoff from $(D, D)$ in battle $(1, 1)$ is:

U_{A | (1, 1)}^{D D} = \frac{(1 - δ) {(1)}^{L}}{{(1)}^{L} + {(1)}^{L}} W + δ U_{A | (1, 1)}^{D D} - c^{D}

= \frac{W}{2} - \frac{c^{D}}{1 - δ} .

Equilibria in battles

Given state-contingent actions $σ_{i} \in \{D, P, S\}$ , each battle M (with ‘surrender’ as an outside option) can be represented in normal form (Table II for battle $(1, 1)$ ).¹⁷ With the Markov Perfect Equilibrium as the solution of the game, equilibria for any battle M (as a subgame) can be derived from the comparison of payoffs in normal form.¹⁸ Because equilibria depend on the size of W, I introduce thresholds of W below which a nation surrenders in battle M:

Definition 2: In battle M, $W_{M}^{D}$ ( $W_{M}^{P}$ ) is the threshold of W below which a nation surrenders to its opponent’s denial (punishment).¹⁹

Table I.

Per-period probabilities in battle M

Action profile	$P r (A ’ s) (‘win’)$	$P r (‘indecisive’)$	$P r (B ’ s ‘win)$ ’)
$(D, D)$	$\frac{(1 - δ) {(m_{A})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$	$δ$	$\frac{(1 - δ) {(m_{B})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$
$(D, P)$	$\frac{(1 - δ) {(m_{A})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$	$δ + \frac{(1 - ε) (1 - δ) {(m_{B})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$	$\frac{ε (1 - δ) {(m_{B})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$
$(P, D)$	$\frac{ε (1 - δ) {(m_{A})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$	$δ + \frac{(1 - ε) (1 - δ) {(m_{A})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$	$\frac{(1 - δ) {(m_{B})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$
$(P, P)$	$\frac{ε (1 - δ) {(m_{A})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$	$δ + (1 - ε) (1 - δ)$	$\frac{ε (1 - δ) {(m_{B})}^{L}}{{(m_{A})}^{L} + {(m_{B})}^{L}}$

Table II.

Normal form for battle $(1, 1)$

		Nation B
		D	P	S
Nation A	D	$\frac{W}{2} - \frac{c^{D}}{1 - δ}$ , $\frac{W}{2} - \frac{c^{D}}{1 - δ}$	$\frac{W}{1 + ε} - \frac{2}{1 + ε} \frac{c_{1}^{P}}{1 - δ}$ , $\frac{ε W}{1 + ε} - \frac{2}{1 + ε} \frac{c^{D}}{1 - δ}$	$W, 0$
	P	$\frac{ε W}{1 + ε} - \frac{2}{1 + ε} \frac{c^{D}}{1 - δ}$ , $\frac{W}{1 + ε} - \frac{2}{1 + ε} \frac{c_{1}^{P}}{1 - δ}$	$\frac{W}{2} - \frac{c_{1}^{P}}{ε (1 - δ)}$ , $\frac{W}{2} - \frac{c_{1}^{P}}{ε (1 - δ)}$	$W, 0$
	S	$0, W$	$0, W$	$0, 0$

Figure 2.

Equilibria in battle $(1, 1)$

The sizes of $W_{M}^{D}$ and $W_{M}^{P}$ indicate how coercively denial and punishment can alter an opponent’s behavior. For instance, a larger $W_{(1, 1)}^{P}$ implies that punishment can coerce even a more resolved opponent (possessing a larger W) to surrender in battle $(1, 1)$ .²⁰ $W_{M}^{D}$ and $W_{M}^{P}$ for each M are derived in the Appendix.

Conditions for equilibria in general battle M

By utilizing $W_{M}^{D}$ and $W_{M}^{P}$ , I outline the conditions for equilibria in M. Equilibria can be categorized into three kinds: (a) the two nations durably fight; (b) either nation immediately surrenders; (c) both the nations probabilistically surrender. For illustration, $W_{M}^{D}$ , $W_{M}^{P}$ , and equilibria (a, b, c) of battle (1,1) appear in Figure 2.²¹

For (a), the only possible candidate for equilibrium is $(D, D)$ , as none of $(D, P)$ , $(P, D)$ , and $(P, P)$ can be an equilibrium. This is because D weakly dominates P in any M.²² $(D, D)$ forms an equilibrium if $W > W_{M}^{D}$ .

For (b), the only reasonable candidates are $(P, S)$ and $(S, P)$ in the sense that $(P, S)$ and/or $(S, P)$ can be equilibria whenever $(D, S)$ and/or $(S, D)$ are.²³ This is because punishment has a greater coercive effect than denial within each battle:

Lemma 1: Within any battle M, punishment is more coercive than denial, i.e.

W_{M}^{D} < W_{M}^{P} for any M .

Lemma 1 holds, because in addition to inflicting larger costs, punishment tends to make battles less decisive, longer, and thus costlier. $(P, S)$ and/or $(S, P)$ form equilibria if $W < W_{M}^{P}$ .

For (c), as in the war-of-attrition model (Maynard Smith, 1974), the nations randomize their actions D and S such that they are indifferent between these actions. This mixed-strategy equilibrium will be excluded from my analysis, because it can be an equilibrium only when W is even smaller than in (b), or when $W < W_{M}^{D}$ .

Among $(D, D)$ , $(P, S)$ , and $(S, P)$ in (a, b), which ones emerge as equilibria depends on the size of W relative to $W_{M}^{D}$ and $W_{M}^{P}$ . To summarize the equilibrium conditions, (a) $(D, D)$ is an equilibrium in M if $W > W_{M}^{D}$ ; (b) $(P, S)$ and/or $(S, P)$ are equilibria in M if $W < W_{M}^{P}$ .

Equilibria in specific battles

Next, I deliver equilibria and their conditions in specific battles by backward induction – from battle $(1, 1)$ through $(2, 1)$ and $(1, 2)$ to $(2, 2)$ . For battle $(1, 1)$ , I rule out uninteresting ‘no-fight’ equilibria $(P, S)$ and $(S, P)$ by imposing the following restriction:

U_{A | (1, 1)}^{D P} = U_{B | (1, 1)}^{P D} = \frac{W}{1 + ε} - \frac{2}{1 + ε} \frac{c_{1}^{P}}{1 - δ} > 0,

which is equivalent to:

Assumption 1: $W > W_{(1, 1)}^{P}$ .²⁴

This assumption holds for all subsequent lemmas and propositions.

Proposition 1: In battle $(1, 1)$ , the nations exchange denials.

By making $(D, D)$ the unique equilibrium in battle $(1, 1)$ , Assumption 1 greatly simplifies the analysis of battles $(2, 1)$ and $(1, 2)$ .

Proposition 2: In battle $(2, 1)$ , A never surrenders.

By Proposition 2, equilibria in $(2, 1)$ are either $(D, D)$ , $(P, S)$ , or both, depending on the size of W. Proposition 2 holds, because A has more military units and thus is more advantageous than B. Trivially, equilibria in battle $(1, 2)$ are identical to those in $(2, 1)$ except that the players swap sides (i.e. equilibria in $(1, 2)$ are only $(D, D)$ , $(S, P)$ , or both), because the game is symmetric with respect to them.

Multiple equilibria in battles $(2, 1)$ and $(1, 2)$ suggest that the war may proceed along one of two major paths. In one, where the booty is sufficiently large ( $W > W_{(2, 1)}^{D}$ ), the nations exchange denials in battles $(2, 1)$ and $(1, 2)$ – the weaker side is willing to fight even a disadvantageous battle. In the other path (with $W < W_{(2, 1)}^{P}$ ), the weaker side surrenders in battles $(2, 1)$ and $(1, 2)$ . The latter path is plausible if the stronger side is tactically so advantageous (with a large L) and/or if the damage by punishment is so heavy (with a large $c_{2}^{P}$ ). In the latter path, punishment functions as a ‘knockout punch’ in the sense that the weaker nation immediately gives in once it is threatened with punishment.

The equilibria in battle $(2, 2)$ of the latter path are identical to those in $(1, 1)$ . The next proposition focuses on the former path:

Proposition 3: In battle $(2, 2)$ followed by $(D, D)$ in battles $(2, 1)$ and $(1, 2)$ , whether the nations fight or not depends on the size of W relative to costs c^D and $c_{2}^{P}$ ; i.e. equilibria in $(2, 2)$ are: $(D, D)$ if $W > W_{(2, 2)}^{P}$ ; $(D, D)$ , $(P, S)$ , and $(S, P)$ if $W_{(2, 2)}^{D} < W < W_{(2, 2)}^{P}; (P, S)$ and $(S, P)$ if $W < W_{(2, 2)}^{D}$ .

Among the equilibria in Proposition 3, $(D, D)$ may depict a war with conventional weapons – conventional wars start with exchange of denials, because a punishment has limited effectiveness (with $c_{2}^{P}$ so small that $W > W_{(2, 2)}^{P}$ ). In contrast, $(P, S)$ and $(S, P)$ may represent the threat of nuclear strikes, which can compel the

Figure 3.

Thresholds that demarcate equilibria

targeted nation to immediately surrender (with

c_{2}^{P}

so large that

W < W_{(2, 2)}^{P}

).²⁵

Dynamic of target choice during war

The dynamic of target choice depends on how coercively denial and punishment can alter the opponent’s behavior in each of the three battles. Namely, it is determined by the transition of the thresholds that demarcate equilibria ( $W_{(1, 1)}^{σ}, W_{(2, 1)}^{σ}$ , and $W_{(2, 2)}^{σ}$ for $σ \in \{D, P\}$ ). In addition to the relative effectiveness between denial and punishment (Lemma 1), these thresholds have the following relationships:

Lemma 2: Each of denial and punishment is most coercive in the second battle; i.e. regardless of L,

W_{(1, 1)}^{D} < W_{(2, 2)}^{D} < W_{(2, 1)}^{D}

W_{(1, 1)}^{P} < W_{(2, 2)}^{P} < W_{(2, 1)}^{P} .

Lemma 2 compares the conditions on W with which A can coerce B into surrender by each of denial and punishment across the three battles – the condition is mildest in the second battle, where B is disadvantaged. That means, a nation who would refuse to surrender in the first and third battles can surrender in the second.

Table III.

Three equilibrium paths

	1st battle $(2, 2)$	2nd battles $(2, 1)$ & $(1, 2)$	3rd battle $(1, 1)$
Nuclear threat	$(P, S)$ or $(S, P)$	–	–
All-out war	$(D, D)$	$(D, D)$	$(D, D)$
Limited war	$(D, D)$	$(P, S)$ in $(2, 1)$ $(S, P)$ in $(1, 2)$	–

For punishment, the relationships among

W_{(1, 1)}^{P}

W_{(2, 2)}^{P}

, and

W_{(2, 1)}^{P}

can be seen in Figure 3.²⁶ For denial,

W_{(2, 2)}^{D}

and

W_{(1, 1)}^{D}

do not appear in the figure, as

W_{(2, 2)}^{D}

is below

W_{(2, 2)}^{P}

, and

W_{(1, 1)}^{D}

below

W_{(1, 1)}^{P}

(Lemma 1).

The relationships among the thresholds in Lemma 2 determine how the war evolves:

Proposition 4: (i) If punishment is so destructive, or if $c_{2}^{P}$ is so large that $W \leq W_{(2, 2)}^{P}$ , the war can end immediately as either side yields to the other’s punishment. (ii) Otherwise, for $W \in (W_{(2, 1)}^{D}, W_{(2, 1)}^{P})$ , the war evolves in one of the two paths: (ii-a) denials are exchanged in all three battles; (ii-b) denials are exchanged in battle $(2, 2)$ , whose winner initiates punishment in battle $(2, 1)$ or $(1, 2)$ , and the loser gives in.

Proposition 4(i) may illustrate a nuclear threat to which either side immediately gives in – because nuclear weapons can destroy enemy homeland without defeating enemy forces, a nuclear punishment could end the war before military advantage is established (Schelling, 1966: 22–23).²⁷ Proposition 4(ii), in contrast, better portrays a conventional war that takes the path of either: all-out war, where denials are exchanged throughout; or limited war, which ends with the weaker side’s surrender to the other’s punishment (Table III).²⁸ Among the three paths in Table III, which path is likely to emerge depends on the size of W – loosely speaking, a war tends to end sooner if W is smaller, or equivalently if the cost of being punished $c_{2}^{P}$ is larger (Figure 3).

The two conventional-war equilibria match the two forms of war defined by von Clausewitz: if the aim of warfare is to ‘render the enemy powerless’ ([1832] 1976:75), armed forces and other war potentials should be targeted through denial, leading to all-out war; in contrast, if the military objective is to ‘increase the enemy’s suffering’ (93), non-military assets can also be targeted through punishment, ending up with limited war. Echoing von Clausewitz, Wagner (2007: 133–137) also suggests that a contest in forcible disarmament (by denial) tends to shape war in the all-out manner, while killing and destruction (by punishment) can make war limited.²⁹

In limited war, a nation that refused to surrender at the war’s onset surrenders if it loses the first battle – punishment will have more coercive power as the targeted nation becomes disadvantaged. This equilibrium offers a novel reason for why denials often precede punishments in conventional wars (Arreguín-Toft, 2001: 108–109; Downes, 2008: 115–155; Legro, 1995; Schelling, 1966: 12–18).³⁰ As far as the nations are evenly matched, punishment cannot coerce the targeted nation into surrender. Without the possibility of surrender, punishment merely wastes the opportunity of denial to weaken the opponent. Thus, exertion of punishment in the early stage of war disadvantages the attacker itself in future battles. Only after the attacker overpowers the opponent, can punishment be effective, implying that timing matters for the success in coercive punishment. The coercive effect of punishment depends on the relative military balance, but because this balance changes over time, a nation should redirect its attacks from denial to punishment as it begins to prevail in war. This result may explain why nations rarely resort to punishments at the onset of (conventional) war. The futility of early punishment can be explained by the lack of the military advantage in M, or the ‘air superiority’ (Pape, 1996: 58), without which the punisher must undergo unbearable damages by denial counterattacks.³¹

The time lag between denial and punishment can also be explained by the payoff analysis in a reduced form as shown below. The targeted nation is forced to surrender if its expected payoff from fighting falls below zero, or if

[\begin{matrix} Benefit \\ from winning \end{matrix}] \times [\begin{matrix} Probability \\ of winning \end{matrix}] - [\begin{matrix} Cost of \\ fighting \end{matrix}] < 0

for which denial has a future impact on the probability of winning the war, while punishment has an immediate impact on the cost of fighting. Because denial generates tardier effects than punishment, denial must be introduced earlier for successful coercion.

Moreover, my model could deliver implications toward a war’s outcome and duration. It suggests that the war’s outcome (i.e. which side wins) is largely determined by the exchange of denials in battle $(2, 2)$ . Once the first battle ends, its loser has little chance to make a recovery and win the war. In limited war, the outcome is determined solely by mutual denials in $(2, 2)$ . Even in all-out war, such a recovery is difficult.³² It also suggests that the war’s duration (i.e. the timing of surrender) hinges on punishment – the punisher can shorten the war by threatening the targeted nation if punishment is severe enough.

Theoretical findings

This article addresses an agenda which has been overlooked by the formal-theory literature – the dilemma between force and value as targets of attacks in war. Under what kind of circumstances, which campaign better serves the purposes of war between denial and punishment? How do these choices shape war? One of my model’s novelties is that it allows nations to flexibly change their targets during war. This flexibility in targeting enables the model to depict the two-way causal relationship between military strategies and war that existing studies have ruled out – in one way, the nations’ choices of targets determine how the war evolves; in the other, past battles can influence their choices in upcoming battles.

Through equilibrium analyses, I have produced the following theoretical findings. If punishments are as destructive as nuclear bombings, a war ends immediately, as either side threatens with punishment, to which the other side gives in (Proposition 4(i)). Otherwise, nations initiate war with an exchange of denial campaigns, which consequently brings about military imbalance in favor of the winner of the early battle. The war then proceeds along one of the two paths (Proposition 4(ii)): one is of all-out war, where the nations continue to exchange denial campaigns until either side is fully disarmed; the other is limited war, where the winner of the early battle switches to a punishment campaign (‘knockout punch’), to which the loser responds by capitulation.³³ The adoption of strategies in these ways match Wagner’s (2007: 133–137) interpretation of the two forms of war defined by von Clausewitz ([1832] 1976).³⁴

Whether a conventional war is all-out or limited depends on the severity of punishment, but along either path, the nations refrain from punishment in the early stage of war, because punishment has no or little coercive effect on an opponent with equal or similar strength. In other words, the coercive effect of punishment depends on the relative military balance, but this balance is determined by the exchange of denials. Only after the opponent is sufficiently weakened through denial, can punishment have a coercive effect (Schelling, 1966: 12–18). In military terms, one must secure the ‘air superiority’ through denial before introducing punishment – an attempt of punitive bombing without the superiority would inevitably fail, because it could be hindered by deadly denial interception. My theory thus holds that denials precede punishments in limited war (Arreguín-Toft, 2001: 108–109; Downes, 2008: 115–155; Legro, 1995).³⁵ This pattern resembles MacNamara’s ‘no city’ doctrine of 1962 that refrains from attacks on cities, but the reason I offered above is different from his doctrine’s that aims to keep civilians as ‘hostages’.³⁶ My theory also indicates that denial and punishment function as strategic complements (Allen & Martinez Machain, 2019) – they generate the most coercive power when combined dynamically.

The Pacific War was waged in ways my theory predicts: (i) until the US military victory became evident, denials had been exchanged in successive battles including critical air-sea battles of Midway, Philippine Sea, and Layte Gulf; (ii) toward the end, punishments were adopted by the winning United States, such as carpet bombings on cities, starvation blockades on ports, and nuclear attacks on Hiroshima and Nagasaki; (iii) few punishments were undertaken by losing Japan.³⁷

Moreover, to draw implications for air strategies, one of my major findings comports with Pape’s (1996) assertion that denial constitutes a primary determinant of war’s outcome, while punishment has little impact on it. When the prevailing side initiates punishment, the tide of war has already been determined – once the first battle ends in my model, its loser has little chance to make a recovery and win the war. Another implication drawn from this model is that punishment can shorten a war by charging the targeted nation with the extra burden of continuing the war. In sum, denials largely determine war’s outcome (i.e. which side wins), while punishments can influence its duration (i.e. when to surrender).

Implications toward empirical studies

My findings above raise some concerns toward empirical procedures in existing quantitative studies of military strategy. Assuming that each nation chooses a single military strategy at a war’s onset and adheres to it throughout, most of the existing large-N studies have abstracted away the possibility that nations’ choices of strategies are influenced by the ongoing war’s battles.³⁸ Put differently, when the existing studies analyzed the effects of military strategies on war, they treated military strategies as explanatory variables and regarded war’s outbreak, duration and outcome as dependent variables. However, if the effectiveness of a certain military strategy changes during a war, it is quite plausible that the nations condition their choices of strategies on how the war was waged. That is, past battle outcomes of the ongoing war can influence the choice of military strategy, while military strategies shape how the war will further evolve. If battle outcomes do have such causal effects on military strategies, a certain strategy might be adopted only in particular circumstances, implying the selection effect (Valentino, Huth & Croco, 2006: 376).

For instance, if punishment is undertaken only by the side which has fought advantageously and is almost winning a war, it cannot be concluded that punishment is an effective strategy to bring about victory (Reiter & Stam, 1998, 2002; Stam, 1996: 138). Conversely, it could be rather the advantageous military imbalance that prompted nations to adopt punishment. Other studies determined punishment to be ineffective (Horowitz & Reiter, 2001; Pape, 1996), but it seems so merely because these studies included punishments used in wrong timing and situation. As my theory holds, punishment is not particularly coercive on its own, but it can be made coercive by preceding successful denial – punishment can be effective but only conditionally. Analogously, the seemingly causal finding that punishment tends to make a war longer (Bennett & Stam, 1996) contradicts my prediction that punishment can shorten a war if the punisher has overpowered the opponent. As a way to reconcile this contradiction, it could be inferred that punishment is more likely to be employed in prolonged wars (Downes, 2006, 2008: 243).

Given the mutual influence between military strategy and war, one cannot properly investigate a causality in either direction without paying attention to the causality in reverse. This simultaneity problem arises even in the very simple setting of my model, where the choice of strategy depends only on the vector of strength. In more complex scenarios, where the choice of strategy is affected by a variety of factors (Allen & Martinez Machain, 2018; Downes, 2007, 2008; Goemans, 2000; Legro, 1995; Reiter & Meek, 1999; Valentino, Huth & Croco, 2006; Wallace, 2008), it is quite plausible that strategies flexibly change during a war, and the simultaneity problem would be more significant.

Agendas for further theoretical research

Finally, I close my discussion in suggesting agendas for further theoretical research, which can be pursued in one of the following two directions. One is to delve deeper into the model’s focus on target choice – in such ways as to increase targets as in Blotto games, further to diversify the targets in value and vulnerability (Pape, 1996: 56, 200), and even to allow continuity in choice and time (Intriligator & Brito, 1984). Another possible extension would be to incorporate Schelling’s (1966: 105–109) risk strategy, or punishment with random severity, which may enable the portrayal of prolonged punishments (e.g. the US strategic bombings during the Pacific War) and the transition from punishment to denial (e.g. the US Operations from Rolling Thunder to Linebacker I during the Vietnam War).

The other direction is to broaden the model’s scope by integrating military strategy into the extant theoretical frameworks such as bargaining, deterrence, and domestic politics. Unlike models of fixed military strategy (Lindsey, 2015; Slantchev, 2010; Tarar, 2016), my model allows flexibility in choosing a strategy in war and instead abstracts away the bargaining process. Thus, a possible extension would be to develop a model in which nations choose military strategies in the battlefields, while placing offers on the diplomatic table. In such a model, while military strategies influence the process of bargaining, bargaining interactions opposingly alter the choices of strategies. This mutual interplay between military and diplomacy might ultimately decide how a war evolves and ends.

Furthermore, the inclusion of military strategy into bargaining models could contribute to the causes of war on the ground that military strategy matters for conventional (Mearsheimer, 1983; Snyder, 1961) and nuclear deterrence (Intriligator & Brito, 1984; Powell, 1987, 1988, 1989; Wagner, 1991). Then the bargaining process before and during war would be more complicated than previously considered.

My model also abstracts away the roles of domestic politics to frame military strategies (Allen, 2007; Downes, 2007, 2008; Goemans, 2000; Reiter & Meek, 1999; Stam, 1996). The debate over strategic bombing stemmed from the disagreement about the reaction of targeted citizens. Early proponents of airpower (e.g. Douhet, [1921] 1983) maintained that strategic bombing can provoke a rebellion among citizens against their own government, while recent critiques insist an opposing and galvanizing effect – bombings on cities can strengthen the civilian morale to prosecute war (Carr, 2003: 190–191, 248–251; Kocher, Pepinsky & Kalyvas, 2011; Pape, 1996: 24–25). Formal modeling could serve the debate by offering some clues to the problem.

Footnotes

Online appendix

The Online appendix, can be found at .

Acknowledgments

I thank Erik Gartzke, Yasutomo Murasawa, Hisashi Sawaki, Alastair Smith, Masatoshi Tsumagari, and three anonymous reviewers for valuable comments. Earlier versions of this article were presented at Konan University, Osaka University of Economics, the 2017 Annual Meeting of the Midwest Political Science Association, and the 2018 Pacific International Politics Conference. All errors are my own.

ORCID iD

Keisuke Nakao

Notes

Appendix

Derivation of $W_{M}^{D}$ and $W_{M}^{P}$ : The thresholds $W_{M}^{D}$ and $W_{M}^{P}$ in battle M can be derived from the conditions $U_{B | M}^{D D} = 0$ and $U_{B | M}^{P D} = 0$ , respectively. For $(1, 1)$ , by $U_{B | (1, 1)}^{D D} = \frac{W}{2} - \frac{c^{D}}{1 - δ} = 0$ and $U_{B | (1, 1)}^{P D} = \frac{W}{1 + ε} - \frac{2}{1 + ε} \frac{c_{1}^{P}}{1 - δ} = 0$ , it is immediate that

For $(2, 1)$ , by plugging $U_{B | (1, 1)}^{D D} = \frac{W}{2} - \frac{c^{D}}{1 - δ}$ into $U_{B | (2, 1)}^{D D} = \frac{1}{2^{L} + 1} U_{B | (1, 1)}^{D D} - \frac{c^{D}}{1 - δ} = 0$ and $U_{B | (2, 1)}^{P D} = \frac{1}{2^{L} ε + 1} U_{B | (1, 1)}^{D D} - \frac{2^{L} + 1}{2^{L} ε + 1} \frac{c_{2}^{P}}{1 - δ} = 0$ ,

Similarly to those in $(2, 1)$ , $W_{(2, 2)}^{D}$ and $W_{(2, 2)}^{P}$ can be derived by plugging $U_{B | (1, 2)}^{D D} = \frac{2^{L + 1} + 1}{2^{L + 1} + 2} W - \frac{2^{L} + 2}{2^{L} + 1} \frac{c^{D}}{1 - δ}$ and $U_{B | (2, 1)}^{D D} = \frac{1}{2^{L} + 1} \frac{W}{2} - \frac{2^{L} + 2}{2^{L} + 1} \frac{c^{D}}{1 - δ}$ into $U_{B | (2, 2)}^{D D} = \frac{1}{2} U_{B | (1, 2)}^{D D} + \frac{1}{2} U_{B | (2, 1)}^{D D} - \frac{c^{D}}{1 - δ}$ and $U_{B | (2, 2)}^{P D} = \frac{2^{L}}{2^{L} + 2^{L} ε} U_{B | (1, 2)}^{D D} + \frac{2^{L} ε}{2^{L} + 2^{L} ε} U_{B | (2, 1)}^{D D} - \frac{2^{L} + 2^{L}}{2^{L} + 2^{L} ε} \frac{c_{2}^{P}}{1 - δ} = 0$ :

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