Asymmetric uncertainty and strategic interaction: An info-gap analysis

Abstract

Protagonists in strategic interactions face uncertainty about their own and their adversaries’ intentions and capabilities, and about the general situation. Uncertainty is often represented probabilistically. However, probability distributions are often unknown due to complexity, limited knowledge or dynamic change. We employ info-gap theory to model and manage uncertainty non-probabilistically. We consider asymmetric uncertainty: both protagonists face significant uncertainty, but one faces much more than the other. Different degrees of uncertainty can motivate different behavior. We explore two responses to uncertainty. Robust satisficing ameliorates pernicious uncertainty to obtain essential outcomes. Opportune windfalling exploits propitious uncertainty to obtain wonderful outcomes. We explore these strategies both quantitatively and qualitatively. We prove two propositions: Protagonists facing greater uncertainty have worse robustness for achieving critical outcomes (Proposition 1) but have better opportuneness for achieving windfall outcomes (Proposition 2). We apply these ideas quantitatively to a generic example, and qualitatively to the Cuban Missile Crisis and to the Japanese attack on Pearl Harbor.

Keywords

asymmetric uncertainty strategic interaction info-gap decision theory Cuban Missile Crisis Pearl Harbor

1. Introduction

A strategic interaction between two protagonists, with conflicting interests, is an interaction that impacts the understandings of risks and opportunities, leading to the formulation, alteration or adoption of general, far-reaching plans of action. A strategic interaction occurs through the exchange of information between the parties that causes new understandings. In contrast, a tactical interaction occurs by direct contact, causing a new situation on the ground: exchange of hot words, blows or bullets.

Strategic interactions can lead to tactical interactions: understandings of risks or opportunities often lead to actions on the tactical level. For instance, Russian strategic understanding of NATO’s encroachment on Russia’s geographical domain of interests was one factor motivating Russia’s attack on Ukraine in 2022. Likewise, tactical interactions can alter strategic understandings, resulting in new plans. For instance, the bullets that killed Archduke Franz Ferdinand and his wife in 1914 contributed to understandings by world leaders that resulted in World War I. Strategic interactions occur in the realm of leaders and high-level decision-makers, while tactical interactions involve operators in the field. We focus on strategic interactions.

Uncertainty is prevalent in strategic interactions in military and defense affairs; the modeling and management of uncertainty is important for realistic modeling and simulation in these domains. Uncertainty in strategic interactions arises especially around the capabilities and intentions of the adversary. In addition, asymmetric interactions are common in military affairs, for instance, between a state and a non-state actor, or between democratic and autocratic states with greatly different degrees of informational openness. This paper demonstrates the methodology for implementing info-gap decision theory in modeling and managing asymmetric uncertainty and strategic interactions, and explains the practical implications of the analysis.

Uncertainty is usually multifaceted in strategic interactions. Understandings by one party to an interaction focus on the intentions, plans, capabilities and uncertainties of the other party, as well as on their own plans, abilities and uncertainties, and on the context in general. For instance, Atkinson et al.¹ argue that rivalries between states persist due to one side’s uncertainty—modeled probabilistically—about the resolve of their protagonist. These understandings are based on information, prior understanding and past experience, all of which can be incomplete, contradictory or simply wrong. Each party acknowledges some extent of uncertainty in its understanding of the situation. We focus on asymmetry in these recognized uncertainties: one side faces and acknowledges greater uncertainty than the other side.

Uncertainty is often represented probabilistically. However, in new and innovative situations, one may not know relevant probabilities. Knight² distinguished between “risk” for which probabilities are known, and “true uncertainty” for which probabilities are unknown. Knight stressed that true uncertainty arises in economic contexts from innovation or other dynamic changes. Knightian true uncertainty arises in strategic interactions when intentions, plans or capabilities of one protagonist are unknown or imperfectly known to the other protagonist. This can occur for several reasons: limited information, situational complexity and dynamic change.

For instance, when Japan attacked Pearl Harbor on 7 December 1941, the Americans did not know if this was intended to establish an outer border of Japan’s Greater East Asia Co-Prosperity Sphere, was intended to support the German war effort in Europe, or was intended as the beginning of a total war with the United States, or perhaps something else. Similarly, when the Arab coalition attacked Israel on 6 October 1973, the Israelis did not know if the coalition intended to reconquer limited sections of land, or intended the thorough and final destruction of the State of Israel, or maybe something else. Probability distributions are inaccessible here, and realistic likelihoods are unknowable, in part because the full space of possible events is unknown. These are examples of non-probabilistic Knightian true uncertainty.

Non-probabilistic uncertainty is important in many strategic interactions. We use info-gap decision theory to model and manage this uncertainty.

It is important to distinguish between uncertainty and ignorance. A protagonist may be ignorant of new technology developed by the adversary. The protagonist is not uncertain about this unknown technology; the protagonist simply has no knowledge of its existence. In contrast, the protagonist may be uncertain about new technologies available to the adversary, but that is different from ignorance of a specific new adversarial capability. Stated differently, awareness of the possibility of ignorance gives rise to uncertainty, but this uncertainty is not itself ignorance.

Section 2 discusses responses to asymmetric uncertainty in a wide range of disciplines, primarily (though not exclusively) in the realm of national security. Section 3 provides a first look at concepts that are developed systematically later in the paper. Section 4 develops a quantitative characterization of strategic interaction under asymmetric non-probabilistic uncertainty based on info-gap decision theory. This section provides a precise and thorough explanation of the methodology of info-gap theory and presents a quantitative example. Two propositions are proven in an online appendix (file: asym-unco25proofs.pdf). We begin with a quantitative study because it facilitates a concise graphical manifestation of the responses to asymmetric uncertainty. We then proceed to qualitative analysis. Section 5 discusses the evolution of strategic interactions, and sections 6 and 7 present examples based on the Cuban Missile Crisis and on the Japanese attack on Pearl Harbor in World War II. Section 8 presents a concluding discussion. In short, this paper demonstrates a comprehensive methodology for modeling and managing non-probabilistic uncertainty in strategic interactions between protagonists.

2. Responses to asymmetric uncertainty

Asymmetric uncertainty between protagonists in a conflict arises in many areas of national security and elsewhere. We review some of that literature. Some of the analyses we discuss are purely qualitative, while some employ quantitative treatments of uncertainty. In the latter case, uncertainty is usually modeled probabilistically.

2.1. Deterrence

Many scholars have recognized that uncertainty is an important factor in understanding deterrence between nations.

Betts³ discusses NATO strategies for deterrence of a Soviet attack on Western Europe. He explains that “asymmetries between attacker and defender in political constraints and leverage over the conditions of engagement” enhance the uncertainties facing NATO governments (p. 154). These substantive asymmetries may translate into asymmetric uncertainty between the protagonists, although Betts does not discuss asymmetric uncertainty explicitly. Betts asserts that “Defense planners cannot control the uncertainties of future scenarios” (p. 165), due in part to “the unpredictable effects of new technology” (p. 169). Betts concludes that “uncertainty is more conducive to partially nuclear than to purely conventional deterrence” by NATO (p. 179). While uncertainties are central to Betts’ analysis of deterrence, he does not discuss asymmetry in the uncertainties that may hold between NATO and Soviet planners.

Altfeld⁴ criticizes the use of uncertainty as a strategy for deterring Soviet attack on the United States. He argues that Soviet uncertainty about US retaliatory capability may deter risk-averse leaders, but not more risk-prone leaders. Altfeld writes that “uncertainty may deter but it also may not, depending on the character of the leaders making the choice” (p. 17). Altfeld does not discuss the implications of asymmetry in uncertainty between the United States and the Soviet Union. While Altfeld recognizes that probabilities may be unknown, he assumes that the protagonists must estimate these probabilities.

Nicholson⁵ discusses deterrence between two states with a multi-step rational-actor game-theoretic formulation based on the prisoners’ dilemma. He provides a probabilistic interpretation of discounting over time in international relations. He concludes that deterrence is more likely, and cooperation more prevalent, when the states can move quickly between alternative strategies and when mutual uncertainty is low. High uncertainty or a delayed transition between strategies diminishes deterrence.

Kilgour and Zagare⁶ develop a quantitative probabilistic game-theoretic model for describing bilateral deterrence between nations when each side is uncertain about the preferences of the adversary. They identify conditions on the credibility of each agent in which “deterrence can emerge in an uncertain world.”

Baker et al.⁷ argue that “uncertain sanctions [in civil and criminal law] may be preferable on efficiency grounds because they achieve more deterrence than certain sanctions of the same expected value.” (p. 445). A potential offender’s uncertainty about the severity of punishment deters criminal behavior more than certainty. While commission of crime and enforcement of the law is an interaction between two parties—offenders and the state—the different uncertainty of each party about the other is not discussed.

Similarly, DeAngelo and Charness⁸ argue, based on laboratory experiments, that “uncertainty about the enforcement regime yields a significant reduction in violations committed” (p. 73).

Chilton and Weaver⁹ discuss the use of “deliberate ambiguity” about the US response to an attack by the Soviet Union as a strategic device in the Cold War. The United States and the USSR had similar uncertainties and, more importantly, viewed those uncertainties with similar risk aversion in assessing deterrence. Symmetry in uncertainty and in the “propensity to take risks” is inherent here. In contrast, the United States and a terrorist group view uncertainties differently. Explicitly, the terrorists are more “risk-acceptant” than the United States, and the terrorists may view uncertain capabilities of the United States as an opportunity to be exploited, rather than as a source of deterrence.

Ben-Haim¹⁰ discusses a state’s decision to either initiate war or not to initiate, when facing deep uncertainty about the situation, the adversary’s capabilities, potential outcomes and more. Uncertainty is modeled with info-gap decision theory,¹¹ which is non-probabilistic, and employs the info-gap methodology of robust satisficing: maximizing robustness against uncertainty while satisfying critical requirements. The analysis is qualitative because much of the relevant understanding eludes quantitative representation. It is demonstrated that a putatively less desirable option may be preferred due to its greater robustness against uncertainty. This reflects the potential for a reversal of preference between the putatively optimal alternative and a putatively sub-optimal alternative that has greater robustness to uncertainty. The analysis supports decision-making by a state and thus does not consider asymmetric uncertainties of the state and its adversary.

Kaplow¹² discusses the impact of asymmetric uncertainty on state compliance with international security agreements such as the nuclear nonproliferation regime. States are willing to comply provided that other states comply as well. Each state knows whether it complies or not, but faces uncertainty about the compliance of other states. This asymmetric uncertainty diminishes or grows over time, impacting the willingness for compliance.

Mazarr et al.¹³ discuss strategic competition between the United States and other major powers, especially China and Russia. They write (p. 3) that “The primary risk in defense policy” arises because:

military competitions tend to shift with inflection points in how wars are fought, as well as the operational concepts and supporting technologies associated with those changes. Emerging technologies…create the potential for just such an inflection point over the next decade—and U.S. rivals are working to take advantage of the change to undermine the effectiveness of traditional U.S. military concepts of operations.

A “priority area” is to “Avoid vulnerability to novel packages of technologies and operational concepts used to generate decisive military effects” (p. 3). New and emerging technologies are uncertain, and that uncertainty is asymmetric when an adversary conceals their innovations.

2.2. War

Asymmetric uncertainty occurred between the British and the Germans in World War II. Handel¹⁴ explains that British scientific intelligence about German offensive capabilities provided the British with “early warning of the development of many new and even revolutionary German weapons, while the Germans could seldom respond in kind.” This enabled the British to take preventive or counteractive actions against German capabilities. The Germans discounted British military capability and thus made little effort to learn about that capability. As a result, there was a great disparity between what the Germans knew about the British and the actual British capability. Had the Germans attempted preventive or counteractive measures against British innovations, the Germans would have faced great uncertainty. British uncertainty about the Germans was substantially less than German uncertainty about the British.

Punla-Green et al.¹⁵ study an interdiction problem in which the evader travels from a source node to a sink node by the shortest path. The interdictor maximizes this shortest path by interdicting arcs in the network. The evader knows the nominal (estimated) values of all the relevant parameters of the system and ignores uncertainty in these values, while the interdictor recognizes the uncertainty in these estimates, creating asymmetric uncertainty between the protagonists.

Hwang and Ma¹⁶ study the detection of camouflaged objects in a military context using artificial intelligence [AI]. Camouflage is intended to enhance the uncertainty in the detection or identification of the object. Uncertainty in detection and identification arises in applying AI in the military domain due to a lack of military-specific data sets for training the AI algorithms. This uncertainty is inherently asymmetric between the camouflager and the adversary due to their different knowledge.

Kenkel and Schram¹⁷ discuss the impact of “Incomplete or asymmetric information” on the choices of protagonists in international crises. They emphasize that “states in crises possess a diverse range of coercive intermediate policy options besides war and peace” including “sanctions or tariffs,”“support to an adversary’s enemies,”“cyberattacks,”“drone strikes” or “low-level conflict” (p. 194). A state that assesses its war-making capability as exceeding that of its adversary may nonetheless choose an alternative coercive policy. We note that different levels of uncertainty may accompany a state’s assessment of its capabilities for the various options, thus influencing the state’s choice.

Fearon¹⁸ studies the question of how a state leader can make a threat of force credible, when presumably the leader would prefer to refrain from military action. The primary response is by making threats that entail costly signals. Fearon’s analysis is based on a Bayesian assessment of the decision process in which relevant probabilities and probability distributions are known for both the challenger and the defender.

Slantchev¹⁹ studies the question of how wars end, focusing on the role of protagonists’ expectations about the likelihood of victory or defeat. War is viewed as a “stochastic process of attrition modeled as “a homogeneous Markov chain” (p. 623). Considering two protagonists, the probability that “Player 1” wins is $p$ . Asymmetric information arises when one side knows the value of $p$ and the other side is uncertain about the value (p. 624).

Fey and Ramsay²⁰ explore the “mutual optimism” hypothesis that wars occur when both protagonists believe they can gain by war. Their critique of this hypothesis arises from an analysis of the uncertainty that surrounds war. Their analysis employs a probability distribution on states of the world, upon which protagonists base their decisions. They do not posit asymmetry of information, although each side may have “private information.”

2.3. Terrorism

Asymmetric uncertainty can arise when an established actor, like a nation-state, is confronted by a new actor, such as a new terrorist organization. The terrorists may be quite familiar with the state, while the state has little or no familiarity with the terrorist organization. The asymmetry of uncertainty can also arise from differences in the collection, analysis and dissemination of knowledge and understanding within a state’s large bureaucracy, as distinct from a small and centralized terrorist organization.

Various state institutions are responsible for protecting citizens and infrastructure against terrorist attacks. These institutions may face deep and highly asymmetric uncertainty about the identity of terrorist operatives as well as terrorist intentions, plans, capabilities and vulnerabilities. For instance, Jakovlev et al.²¹ discuss the deep uncertainties in detecting hidden radioactive materials in stacked containers on ships or in storage areas, the intended use of which is unknown. These uncertainties arise from diverse possible origins of the material (p. 45), from variable sensor reliability (p. 46), or from the impact of data from different sources (p. 52). As another example, Benjamin and Simon²² express uncertainty about how US military action in Iraq, and specifically strikes intended to kill Zarqawi, would impact jihadist action. Likewise, Zarqawi’s jihadist organization was uncertain about US capabilities and vulnerabilities, and also uncertain about reactions in various Iraqi populations to terrorist action against the Americans. Clarke²³ stresses that the challenge facing US action against Islamic terror “is a battle not only of bombs and bullets, but chiefly of ideas” between the West and much of Muslim society that, at least initially, did not identify with jihadist action. Likewise, English²⁴ stresses “the profound uncertainty of terrorism achieving its central goals.” Even understanding those goals is a major source of uncertainty for both sides, as Hayden²⁵ explains in describing conversations with international anti-terrorism partners in many countries. Understanding, on both sides of the terrorism divide, is impacted by cultural conceptions and biases, thus augmenting uncertainty differently on both sides.

2.4. Cyber security

Cybersecurity refers to protection against digital attacks on critical national infrastructure, corporate organizational resources and private interests such as personal security and privacy, as explained by Dunn Cavelty and Wenger.^26,27 Uncertainty abounds in the cyber domain due to rapid technological development, great diversity of potential targets and their respective vulnerabilities, and institutional dispersion of responsibility for cyber security between governments, corporations and private individuals as discussed by Bob and Evyatar.²⁸ Deterrence of potential cyber attackers is one form of security. However, a strong asymmetry of uncertainty may exist between defender and attacker. The attacker knows who is being attacked and can know existing security tools. The target, however, may know little about the attacker, their goals and capabilities.²⁹

Albanese et al.³⁰ discuss a probabilistic “moving target defense” strategy against malicious botnets. They explain that cyber attacks are typically preceded by a reconnaissance phase in which the adversary collects information about the target. The defensive strategy advocated by the authors is to continuously modify the configuration of the system and the network, thereby creating “asymmetric uncertainty, providing the defender with a tactical advantage over the attacker” (p. 92). Each side has knowledge about the other side, but each side’s uncertainty is no less important. The asymmetric uncertainty that the authors emphasize is that the defender augments the attacker’s uncertainty as a defensive mechanism. Further development of this issue appears in the work by Albanese et al.,³¹ Jajodia et al.,³² Albanese et al.³³ and Connell et al.³⁴

Ding et al.³⁵ consider the strategic interaction, in a communication channel, between an attacker and a defender using a probabilistic Markov chain approach. The attacker attempts to corrupt the communication by jamming the channel. The defender attempts to prevent this by adjusting the “transmission power” and the “real-time information,” creating “asymmetric uncertainty to mislead the attacker and thus mitigate attacks” (p. 1). The defender employs this “deception trick to manipulate the attacker’s belief” (p. 10).

Similarly, Chen et al.³⁶ study defense against attacks in cyberspace using a probabilistic Markov chain analysis. They create diverse functional equivalents of the defended target by sensing the attacker and then altering the defended target. This creates asymmetric uncertainty to favor the defender.

2.5. Economics and management

Asymmetric uncertainty arises in economic interactions. Li et al.³⁷ study the pricing strategy of a monopolist who sells substitutable products. Consumers face probabilistic uncertainties about these products, while the seller does not. The authors explore the question of whether the seller should reduce this asymmetric uncertainty—and if so, by how much and for which products—to maximize the revenue.

Maskin³⁸ studies the use of enhanced production capacity by an incumbent firm to deter the entry of competitors. He argues that uncertainty in demand and costs requires the incumbent to install greater production capacity than without uncertainty. This uncertainty thus acts to deter the incumbent from enhancing production. Enhanced production deters competitors, but uncertainty deters enhanced production by the incumbent.

Mazarr et al.¹³ discuss uncertainty in economic statistics such as gross domestic product and its rate of change when using data from an authoritarian regime such as China. “Some observers believe that official [Chinese] statistics for both are substantially overstated.” (p. 5) Here is the potential for an asymmetric uncertainty because US statistics on these issues are not subject to deliberate manipulation.

Asymmetric uncertainties can arise in the management and use of civilian infrastructure systems: roads and highways, electric power production and supply, water purification and supply, waste management, telecommunication, Internet systems, educational institutions and more. Rangrazjeddi et al.³⁹ stress that these infrastructure systems are highly interconnected in many ways: telecommunication depends on electricity supply, waste management depends on roads and highways, schools depend on water supply, and so on. Furthermore, distinct actors influence the operation of these infrastructures, including utility operators, political leaders, state and local governments, public interest groups and more. These groups have different interests and face different uncertainties of different degrees of severity. Utility operators may be uncertain about future demand for their product or future supply of essential components to their operations. Political leaders may be uncertain about political implications and public understandings of various production technologies. Public interest groups may be uncertain about infrastructure impacts on the environment, and so on. In short, asymmetric uncertainties abound.

3. Two quantitative results: A first look

Protagonists in a conflict can respond in various ways to asymmetric uncertainty in strategic interaction. In this paper, we explore two complementary aspects of the response to asymmetric uncertainty in situations where probability distributions are inaccessible. Our analysis is based on two decision functions developed in info-gap decision theory:¹¹robustness against pernicious uncertainty and opportuneness from propitious uncertainty. These functions underlie two quantitative propositions on the implications of asymmetric uncertainty for understandings (including plans and decisions) by protagonists in strategic interactions. The two propositions underlie our understanding of the strategic interaction between parties with asymmetric uncertainty and conflicting interests. The concepts are briefly and intuitively described in this section and are then formulated rigorously in the next section.

Consider an agent who faces great uncertainty about the adversary and about the situation in general, and believes that their adversary faces more moderate uncertainty. This agent may abandon the attempt to guarantee critical goals as unrealistic (Proposition 1) and yet may believe that the achievement of a wonderful windfall is possible (Proposition 2). There is no contradiction between these two seemingly conflicting approaches. These approaches arise from responding to different aspects of uncertainty. The attempt to guarantee critical goals is a response to pernicious uncertainty, while the attempt to facilitate windfall is a response to propitious uncertainty. Uncertainty is a vast and diverse realm, not a single homogeneous entity. Furthermore, this agent may anticipate that their adversary will focus on confidently achieving their critical goals (converse of Proposition 1) and would not venture into risky attempts for a wonderful windfall (converse of Proposition 2).

Similarly, consider an agent who faces moderate uncertainty about their adversary and about the situation in general, and believes that their adversary faces great uncertainty. This agent may feel confident in achieving critical goals (converse of Proposition 1) and see the aspiration for a wonderful windfall as risky (converse of Proposition 2). Furthermore, this agent may anticipate that their adversary will not focus on achieving critical goals for which they can have little confidence (Proposition 1) and could venture into attempts for windfall that their adversary would consider to be feasible (Proposition 2).

The reasoning in the last two paragraphs suggests a cognitive match between the adversaries: each side correctly anticipates the preferred strategy of their adversary. However, we will see that cognitive mismatch between adversaries occurs when one or both sides incorrectly anticipate their adversary’s strategy. We will explore the implications of both possibilities.

4. Robustness and opportuneness with asymmetric uncertainty

Uncertainty entails the potential for pernicious failure, and also the potential for propitious windfall. The response to uncertainty is far from unique, and different actors may respond differently. When two interacting protagonists confront substantially different levels of uncertainty, they may respond differently, impacting their interaction. A protagonist facing great uncertainty may recognize that adequate outcomes are not confidently attainable, but may seek to exploit the propitious uncertainty in attempting to obtain an unanticipated but wonderful outcome. In contrast, a protagonist facing more moderate uncertainty may feel that the responsible reaction is to protect against pernicious uncertainty in achieving critical goals. The info-gap robustness function assists in managing the pernicious side of uncertainty, and the info-gap opportuneness function assists in managing the propitious side. We employ these functions in analyzing the strategic interaction of protagonists with asymmetric uncertainty about each other and about the situation in general. We are considering situations where probabilities are unknown, and the protagonists face non-probabilistic Knightian true uncertainty. In this section, we formulate and discuss these functions and present two propositions and a numerical example.

4.1. Formulation of robustness and opportuneness functions

We consider the strategic interaction between two protagonists whom we refer to as Blue and Green. Their understandings of the situation are denoted generically $u_{b}$ for Blue’s understanding and $u_{g}$ for Green’s understanding. We use the term “understanding” broadly to specify assessments, plans and capabilities (their own and their understanding of the adversary’s), as well as understanding of the general context in which these other elements reside. Understanding reflects the protagonist’s experience, knowledge and insight of the situation and how to exploit it. The Blue and Green understandings may be fundamentally different. For instance, one side’s understandings may arise from widespread resentment of the other protagonist and underlie military preparations, and the other side’s understandings may arise from a widespread sense of satisfaction and underlie economic development. That is, any given Blue understanding, $u_{b}$ , may be irrelevant to Green as an understanding, and vice versa: any given $u_{g}$ may be irrelevant to Blue. Furthermore, comparison of these understandings is limited, recalling the broad meaning ascribed to the term “understanding.” For instance, one can’t say that Green’s understanding is bigger or better than Blue’s understanding if their topics, attributes or contexts are fundamentally different.

Each side can articulate its understanding, but each side also recognizes that its understanding may be wrong. That is, each side has a current best estimate of its knowledge and insight into the situation, but each side also recognizes that that knowledge may err. Blue’s best understanding is denoted ${\tilde{u}}_{b}$ , where the correct understanding of the situation may be different, denoted $u_{b}$ . Likewise, Green’s best understanding is denoted ${\tilde{u}}_{g}$ , while the correct version may differ and is denoted $u_{g}$ . Note that the subscript indicates the “owner” of the information. This convention is also used subsequently for the info-gap models and the robustness and opportuneness functions. Recall that the term “understanding” refers broadly to assessments, plans and capabilities of each side, respectively, and to the general situation in which these elements exist.

Each side has its own understanding and is uncertain about the accuracy of that understanding. The understandings are subtle and complicated, addressing a diverse array of knowledge, including the capabilities and intentions of each side. This knowledge may involve numerical assessments, functional relations and subjective assessments. Probability distributions are lacking because of the Knightian nature of the uncertainties, as discussed earlier. Non-probabilistic info-gap models of uncertainty quantify each side’s uncertainty about its understanding.¹¹ Blue is highly uncertain about its understanding, while Green faces more modest (but still significant) uncertainty about its understanding. This expresses the asymmetric uncertainty to be quantified subsequently.

We represent this uncertainty with info-gap models.¹¹ An info-gap model of uncertainty, $U (h)$ for $h \geq 0$ , is a family of sets of possible realizations of the uncertain entity. The family of sets obeys two axioms: nesting and contraction.

The nesting axiom asserts that the uncertainty sets $U (h)$ become more inclusive as $h$ increases:

h < h^{'} \to U (h) \subseteq U (h^{'})

(1)

This axiom endows $h$ with its meaning as the horizon of uncertainty.

Let $\tilde{u}$ denote the best estimate of the uncertain entity. The contraction axiom asserts that $\tilde{u}$ is the only possibility in the absence of uncertainty:

U (0) = {\tilde{u}}

(2)

That is, the uncertainty set contains only the estimated value when the horizon of uncertainty is zero. $\tilde{u}$ is called the centerpoint of the info-gap model.

Each side quantifies its uncertainty about its understanding with an info-gap model, $U_{b} (h)$ and $U_{g} (h)$ for Blue and Green, respectively.

The elements of $U_{b} (h)$ are values of $u_{b}$ , possible realizations of Blue’s understanding. The center point of $U_{b} (h)$ is ${\tilde{u}}_{b}$ , Blue’s best estimate of its understanding. The contraction axiom implies that Blue’s estimate, ${\tilde{u}}_{b}$ , is the only possibility in the absence of uncertainty. The nesting axiom implies that Blue recognizes an increasing range of possible $u_{b}$ values as the horizon of uncertainty, $h$ , increases.

Likewise, the center point of $U_{g} (h)$ is ${\tilde{u}}_{g}$ , Green’s estimated understanding. The elements of $U_{g} (h)$ are values of $u_{g}$ , possible realizations of Green’s understanding. The nesting axiom implies that Green considers an increasing range of possible $u_{g}$ values as the horizon of uncertainty, $h$ , increases.

The structures of these info-gap models may differ, one from the other, and their elements (values of $u_{b}$ and $u_{g}$ ) may be fundamentally different as explained earlier. Thus, neither info-gap model can be expressed as a subset of the other. The asymmetric uncertainty between Blue and Green will be defined precisely later in Equations (7) and (8).

The outcome of the interaction between Blue and Green bears a significant impact on both sides. Each side can assess the outcome quantitatively in terms of duration, cost in human lives or cost in financial or material resources. These assessments may be different for each side, but the subject of the assessment, for example, duration or cost, is the same. In this study, the outcome is a real scalar value whose assessment is based on models or simulations of the relevant issue. For any given realization of Blue’s understanding, $u_{b}$ , the outcome for Blue would be $f_{b} (u_{b})$ . Thus, Blue anticipates the outcome $f_{b} ({\tilde{u}}_{b})$ but this may err because Blue’s estimate, ${\tilde{u}}_{b}$ , may err. Blue prefers a small value of the outcome over a large value. For instance, Blue prefers low economic damage to its economy. Likewise, for any given realization of Green’s understanding, $u_{g}$ , the outcome for Green would be $f_{g} (u_{g})$ , and Green anticipates the outcome $f_{g} ({\tilde{u}}_{g})$ . Green prefers a large value of the outcome over a small value. For instance, Green would prefer large damage to Blue’s economy. The functions $f_{b} (\cdot)$ and $f_{g} (\cdot)$ have the same units, although their functional forms differ because $u_{b}$ and $u_{g}$ differ. Each party knows its respective outcome function. (We are considering only a single outcome, for example, cost or duration, etc., while there can be multiple outcomes of interest. The subsequent analysis can be repeated for each different type of outcome.)

Blue’s robustness to its uncertainty about its understanding is the greatest horizon of uncertainty, $h$ , up to which all realizations of Blue’s understanding result in an outcome that is no greater than the largest “critical” value, $f_{cb}$ , that is acceptable to Blue. Blue’s robustness function is defined:

{\hat{h}}_{b} (f_{cb}) = \max {h : (\max_{u_{b} \in U_{b} (h)} f_{b} (u_{b})) \leq f_{cb}}

(3)

“Reading” this equation from left to right: the robustness, ${\hat{h}}_{b} (f_{cb})$ , to uncertainty in Blue’s understanding, when Blue’s greatest acceptable outcome is $f_{cb}$ , is the maximum horizon of uncertainty, $h$ , up to which the maximum outcome $f_{b} (u_{b})$ for all possible realizations of Blue’s understandings $u_{b}$ at the horizon of uncertainty $h$ , is no greater than Blue’s critical requirement $f_{cb}$ . Equation (3) shows that Blue’s robustness function, ${\hat{h}}_{b} (f_{cb})$ , depends on Blue’s info-gap model $U_{b} (h)$ that quantifies Blue’s uncertainty about its understanding and Blue’s estimated understanding ${\tilde{u}}_{b}$ (which is the centerpoint of $U_{b} (h)$ ).

Likewise, Green’s robustness to its uncertainty is the greatest horizon of uncertainty, $h$ , up to which all realizations of Green’s understanding result in an outcome that is no less than a smallest critical value, $f_{cg}$ , that is acceptable to Green. Green’s robustness function is defined:

{\hat{h}}_{g} (f_{cg}) = \max {h : (\min_{u_{g} \in U_{g} (h)} f_{g} (u_{g})) \geq f_{cg}}

(4)

Equation (4) shows that Green’s robustness function, ${\hat{h}}_{g} (f_{cg})$ , depends on Green’s info-gap model $U_{g} (h)$ that quantifies Green’s uncertainty about its understanding and Green’s estimated understanding ${\tilde{u}}_{g}$ (which is the centerpoint of $U_{g} (h)$ ).

The inner extrema in Equations (3) and (4) express an important difference between the Blue and Green robustness functions, reflecting the conflictual nature of the interaction. Blue desires a small outcome, so its robustness evaluates bad outcomes by exploring the maximum outcome as a function of the horizon of uncertainty. Green, in contrast, desires a large outcome, so its robustness evaluates bad outcomes by exploring the minimum outcome as a function of the horizon of uncertainty.

There is a fundamental epistemic and methodological distinction between info-gap robust satisficing, Equations (3) and (4), and Wald’s⁴⁰ maximin. Wald’s approach presumes knowledge of a worst case and then chooses a decision to minimize the impact of that worst case. Info-gap robust satisficing presumes that a worst case is not known: that the horizon of uncertainty is unbounded. One then satisfices the outcome and maximizes the robustness against uncertainty. This is a fundamentally different strategy from maximin and is particularly suited to practical decision-making in many fields confronting deep uncertainty, including military affairs. The info-gap approach provides an implementable methodology for dealing with non-probabilistic Knightian uncertainty in which realistic worst cases are unknown.

Opportuneness is the complement of robustness. Opportuneness attempts to exploit favorable uncertainty, unlike robustness, which protects against pernicious uncertainty.

Blue desires a small value for the outcome function, and a very small value would be wonderful. Blue’s opportuneness from its uncertainty (about its understanding) is the lowest horizon of uncertainty, $h$ , up to which there is at least one possible realization of Blue’s understanding that results in an outcome that is no greater than a wonderfully small value, $f_{wb}$ , that is highly desirable to Blue. Blue’s opportuneness function is defined:

{\hat{β}}_{b} (f_{wb}) = \min {h : (\min_{u_{b} \in U_{b} (h)} f_{b} (u_{b})) \leq f_{wb}}

(5)

Analogously, Green desires a large value for the outcome function, and a very large value would be wonderful. Green’s opportuneness from its uncertainty is the lowest horizon of uncertainty, $h$ , at which there is at least one possible realization of Green’s understanding that results in an outcome that is no less than a wonderfully large value, $f_{wg}$ , that is highly desirable to Green. Green’s opportuneness function is defined:

{\hat{β}}_{g} (f_{wg}) = \min {h : (\max_{u_{g} \in U_{g} (h)} f_{g} (u_{g})) \geq f_{wg}}

(6)

Comparing the robustness functions in Equations (3) and (4) with the opportuneness functions in Equations (5) and (6), we see an inversion of the optimization operators: Each max in a robustness function becomes a min in the corresponding opportuneness function, and each min becomes a max. The robustness function evaluates the maximum uncertainty up to which a requirement is guaranteed; hence, the outer maxima in Equations (3) and (4). The opportuneness function evaluates the minimum uncertainty at which a wonderful outcome is possible; hence, the outer minima in Equations (5) and (6). The inner optimizations express worst cases in Equations (3) and (4) and express best cases in Equations (5) and (6).

We again note the fundamental epistemic and methodological distinction between info-gap opportune windfalling and Wald’s⁴⁰ maximin. Wald’s approach presumes knowledge of a worst case and then chooses a decision to minimize the impact of that worst case. Info-gap opportune windfalling presumes that a best case is not known: that the horizon of uncertainty is unbounded. One then windfalls the outcome and optimizes the opportuneness from uncertainty. This is a fundamentally different strategy from maximin.

4.2. Interpretation of robustness and opportuneness functions

Schematic robustness and opportuneness curves for Blue and Green are shown in Figure 1. The robustness curves (solid lines) show the two universal properties of all robustness curves: zeroing and trade-off.

Figure 1.

Schematic robustness curves (solid) and opportuneness curves (dashed) for Blue (left) and Green (right).

Zeroing of the robustness curve asserts that the robustness is zero if the critical value, $f_{cb}$ or $f_{cg}$ , is chosen as the corresponding best estimate of the outcome, $f_{b} ({\tilde{u}}_{b})$ or $f_{g} ({\tilde{u}}_{g})$ , for Blue or Green, respectively. For instance, suppose that Blue’s best understanding is ${\tilde{u}}_{b}$ . Then, Blue’s best estimate of the outcome is $f_{b} ({\tilde{u}}_{b})$ . If Blue adopts this best estimate as its critical value, $f_{cb} = f_{b} ({\tilde{u}}_{b})$ , then Blue’s resulting robustness to Blue’s uncertainty will vanish. Best estimates exploit all available knowledge (Blue’s best estimate of its understanding in the present case), some of which may err. The zeroing property asserts that there is no robustness to uncertainty of outcome predictions based on best estimates. Hence, Blue’s robustness curve reaches the horizontal axis of Figure 1 (left) precisely at Blue’s estimated outcome. Similar reasoning applies to Green, whose robustness curve in Figure 1 (right) reaches the horizontal axis precisely at Green’s estimated outcome.

Trade-off of the robustness curve asserts that the robustness improves as the required outcome becomes less demanding or less desirable: robustness trades off against performance. This is manifested in the positive slope of Blue’s robustness curve in Figure 1 (left). Blue’s robustness, ${\hat{h}}_{b} (f_{cb})$ , improves (gets larger) as Blue’s largest acceptable outcome, $f_{cb}$ , gets less demanding (larger). Likewise, the trade-off between robustness and outcome requirement is manifested in the negative slope of Green’s robustness curve in Figure 1 (right). Green’s robustness, ${\hat{h}}_{g} (f_{cg})$ , improves (gets larger) as Green’s smallest acceptable outcome, $f_{cg}$ , gets less demanding (smaller).

The dashed opportuneness curves in Figure 1 also show zeroing and trade-off properties whose meanings are complementary to robustness zeroing and trade-off.

Zeroing of the opportuneness curve asserts that no uncertainty is required to enable the predicted outcome: in the absence of uncertainty, the outcome will precisely equal the predicted value.

Trade-off of the opportuneness curve asserts that greater uncertainty is required to enable more wonderful outcomes. The negative slope of ${\hat{β}}_{b} (f_{wb})$ in Figure 1 (left) means that greater uncertainty is required to enable better (smaller) values of Blue’s windfall aspiration for a small outcome, $f_{wb}$ . The positive slope of ${\hat{β}}_{g} (f_{wg})$ in Figure 1 (right) means that greater uncertainty is required to enable better (greater) values of Green’s windfall aspiration for a large outcome, $f_{wg}$ .

4.3. Asymmetric uncertainty

Asymmetric uncertainty between protagonists—when it exists—is manifested in two ways. First, the asymmetry appears as differences between the protagonists’ info-gap models, expressing different knowledge and different levels of uncertainty about the protagonists’ understandings. In particular, asymmetric uncertainty is significant when one protagonist faces much greater uncertainty than the other. Second, the asymmetry is manifested in the different values of the outcome function when evaluating maximal and minimal responses as a function of the horizon of uncertainty. Consequently, asymmetric uncertainty results in different relations between the robustness and opportuneness functions of the protagonists, as illustrated schematically in Figure 1.

Comparing the Blue and Green robustness curves, ${\hat{h}}_{b} (f_{cb})$ and ${\hat{h}}_{g} (f_{cg})$ , in Figure 1 (left) and (right), we see that Blue has substantially lower robustness than Green. Blue is more vulnerable to its uncertainty than Green is to its uncertainty. This is an expression of the asymmetric uncertainty between Blue and Green.

In contrast, comparing the Blue and Green opportuneness curves, ${\hat{β}}_{b} (f_{wb})$ and ${\hat{β}}_{g} (f_{wg})$ , in Figure 1 (left) and (right), we see that Blue has substantially better opportuneness than Green. (Recall that a low value of the opportuneness function is desired because it implies that wonderful windfall is possible even with low uncertainty.) Propitious uncertainty is more accessible to Blue than to Green, again expressing the asymmetric uncertainty between the protagonists.

The following two propositions generalize this complementarity between robustness and opportuneness when considering asymmetric uncertainty.

We quantify the concept of asymmetric pernicious uncertainty with the inequality in Equation (7), based on the robustness functions of Equations (3) and (4) and combining the outcome functions and both info-gap models of uncertainty. Recall that Blue requires a small outcome, so large outcomes are contrary to Blue’s interest. The quantity on the left-hand side of Equation (7) is the magnitude of the maximum deviation above Blue’s anticipated outcome, at the horizon of uncertainty $h$ of Blue’s info-gap model. Green requires a large outcome, so small outcomes are contrary to Green’s interest. The quantity on the right-hand side of Equation (7) is the magnitude of the maximum deviation below Green’s anticipated outcome, at the horizon of uncertainty $h$ of Green’s info-gap model. Both sides of Equation (7) equal zero at $h = 0$ , and both sides increase as $h$ increases. The inequality expresses the asymmetric pernicious uncertainty: Blue faces a greater range of pernicious outcome-deviation than Green, at all positive horizons of uncertainty.

\max_{u_{b} \in U_{b} (h)} f_{b} (u_{b}) - {\tilde{f}}_{b} \geq {\tilde{f}}_{g} - \min_{u_{g} \in U_{g} (h)} f_{g} (u_{g}) for all h > 0

(7)

In an analogous fashion, we quantify the concept of asymmetric propitious uncertainty with the inequality in Equation (8), based on the opportuneness functions of Equations (5) and (6) and combining the outcome functions and both info-gap models of uncertainty. Recall that Blue aspires to a very small outcome, so a tiny outcome would be a wonderful windfall for Blue. The quantity on the left-hand side of Equation (8) is the magnitude of the maximum deviation below Blue’s anticipated outcome, at the horizon of uncertainty $h$ of Blue’s info-gap model. Green aspires to a very large outcome, so a huge outcome would be a wonderful windfall for Green. The quantity on the right-hand side of Equation (8) is the magnitude of the maximum deviation above Green’s anticipated outcome, at the horizon of uncertainty $h$ of Green’s info-gap model. Both sides of Equation (8) equal zero at $h = 0$ , and both sides increase as $h$ increases. The inequality expresses the asymmetric propitious uncertainty: Blue faces a greater range of propitious outcome-deviation than Green, at all positive horizons of uncertainty.

{\tilde{f}}_{b} - \min_{u_{b} \in U_{b} (h)} f_{b} (u_{b}) \geq \max_{u_{g} \in U_{g} (h)} f_{g} (u_{g}) - {\tilde{f}}_{g} for all h > 0

(8)

4.4. Analytical results

For notational convenience, define ${\tilde{f}}_{b} = f_{b} ({\tilde{u}}_{b})$ and ${\tilde{f}}_{g} = f_{g} ({\tilde{u}}_{g})$ . The two propositions presented in this section are proven in an online appendix (file: asym-unco25proofs.pdf).

Proposition 1: The first protagonist has lower robustness than the second protagonist if and only if the first protagonist faces greater uncertainty.

Given:

Info-gap models for the uncertainty of Blue and Green, $U_{b} (h)$ and $U_{g} (h)$ , respectively.

Robustness functions for Blue and Green, ${\hat{h}}_{b} (f_{cb})$ and ${\hat{h}}_{g} (f_{cg})$ , respectively, defined in Equations (3) and (4).

Then: Blue faces greater uncertainty than Green, as defined in Equation (7), if and only if Blue is less robust than Green. Specifically:

\max_{u_{b} \in U_{b} (h)} f_{b} (u_{b}) - {\tilde{f}}_{b} \geq {\tilde{f}}_{g} - \min_{u_{g} \in U_{g} (h)} f_{g} (u_{g}) for all h > 0

(9)

if and only if

{\hat{h}}_{b} ({\tilde{f}}_{b} + δ) \leq {\hat{h}}_{g} ({\tilde{f}}_{g} - δ) for all δ \geq 0

(10)

Proposition 1 can also be understood as asserting that lower uncertainty is equivalent to higher robustness.

Proposition 2: The first protagonist has lower (better) opportuneness than the second protagonist if and only if the first protagonist faces greater uncertainty.

Given:

Info-gap models for the uncertainty of Blue and Green, $U_{b} (h)$ and $U_{g} (h)$ , respectively.

Opportuneness functions for Blue and Green, ${\hat{β}}_{b} (f_{wb})$ and ${\hat{β}}_{g} (f_{wg})$ , respectively, defined in Equations (5) and (6).

Then: Blue faces greater uncertainty than Green, as defined in Equation (8), if and only if Blue is more opportune than Green. Specifically:

{\tilde{f}}_{b} - \min_{u_{b} \in U_{b} (h)} f_{b} (u_{b}) \geq \max_{u_{g} \in U_{g} (h)} f_{g} (u_{g}) - {\tilde{f}}_{g} for all h > 0

(11)

if and only if:

{\hat{β}}_{b} ({\tilde{f}}_{b} - δ) \leq {\hat{β}}_{g} ({\tilde{f}}_{g} + δ) for all δ \geq 0

(12)

Proposition 2 can also be understood as asserting that lower uncertainty is equivalent to worse opportuneness.

We note, from Propositions 1 and 2, that uncertainty has opposite impacts on robustness and opportuneness.

Propositions 1 and 2 may seem, at first sight, to be similar to Proposition 1 in Ben-Haim.⁴¹ There is, however, a significant and substantial difference. The first proposition in Ben-Haim⁴¹ considers “Two states of knowledge, or two contexts of a decision.” That paper assumes nesting of the info-gap model in these two different states, for each protagonist. That paper then asserts that each protagonist’s robustness function is ranked in those two states. In contrast, the current paper assumes ranking of the upper (or lower) range of the outcome of one protagonist on its info-gap model, against the lower (or upper) range of the outcome for the other protagonist on its info-gap model, Equations (7) and (8). These rankings of extreme values on the two different info-gap models do not imply nesting of each protagonist’s info-gap model, nor the converse. The present Propositions 1 and 2 then assert the ranking of one protagonist’s robustness or opportuneness over the robustness or opportuneness of the other protagonist. In addition, it is the robustness and opportuneness functions of the two protagonists that are ranked, rather than ranking each protagonist’s robustness in two different states. In short, the current results differ substantively from those in Ben-Haim.⁴¹

4.5. Simple quantitative example of robustness and opportuneness

We now present a simple numerical example of the robustness functions for Blue and Green, ${\hat{h}}_{b} (f_{cb})$ and ${\hat{h}}_{g} (f_{cg})$ in Equations (3) and (4), and of their opportuneness functions, ${\hat{β}}_{b} (f_{wb})$ and ${\hat{β}}_{g} (f_{wg})$ in Equations (5) and (6). The example demonstrates the info-gap methodology and illustrates practical implications of these functions.

4.5.1. Formulation of the robustness functions

Blue’s estimate of the number of missiles that Green can launch daily is ${\tilde{u}}_{b}$ , while the true but uncertain value is $u_{b}$ . For simplicity, we treat $u_{b}$ as a non-negative real variable. Blue understands that its estimate may err by as much as $w_{b}$ , or more, but has no further information, probabilistic or other, about the true value $u_{b}$ . Thus, the fractional error of Blue’s estimate is unbounded, which motivates the following non-probabilistic fractional-error info-gap model for Blue’s uncertainty about Green’s capability:

U_{b} (h) = {u_{b} : u_{b} \geq 0, | \frac{u_{b} - {\tilde{u}}_{b}}{w_{b}} | \leq h}, h \geq 0

(13)

This info-gap model is an unbounded family of nested sets of possible values of $u_{b}$ .

Blue is concerned about the extent of economic damage to Blue’s interests that can be caused by Green each day, denoted $f_{b} (u_{b})$ , which is modeled as:

f_{b} (u_{b}) = s_{b} u_{b}

(14)

where $s_{b}$ is a known positive value. A positive value for $f_{b}$ is a loss; a negative value is a gain. Blue requires that the extent of economic damage to its interests not exceed a critical value, $f_{cb}$ , so Blue’s robustness requirement is:

f_{b} (u_{b}) \leq f_{cb}

(15)

Employing Equations (13)–(15) in the definition of Blue’s robustness function in Equation (3), one readily finds the following expression for Blue’s robustness to Blue’s uncertainty about Green’s capability:

{\hat{h}}_{b} (f_{cb}) = \frac{f_{cb} - s_{b} {\tilde{u}}_{b}}{s_{b} w_{b}}

(16)

or zero if this is negative. The robustness of the estimated damage is zero, that is, ${\hat{h}}_{b} (s_{b} {\tilde{u}}_{b}) = 0$ , which is the zeroing property discussed in section 4.2. The robustness to uncertainty improves (increases) as the critical damage, $f_{cb}$ , gets worse (increases), expressing the trade-off between the performance requirement and the robustness to uncertainty. The robustness decreases as the estimated error, $w_{b}$ , gets larger.

We now define and derive Green’s robustness function. Green’s estimate of the number of missiles that Green can fire each day is ${\tilde{u}}_{g}$ , while the true but uncertain value is $u_{g}$ which is a non-negative real variable. Green understands that its estimate may err by as much as $w_{g}$ , or more, but has no further information about the true value $u_{g}$ . Thus, the fractional error of Green’s estimate is unbounded, which motivates the following non-probabilistic fractional-error info-gap model for Green’s uncertainty about Green’s firing capability:

U_{g} (h) = {u_{g} : u_{g} \geq 0, | \frac{u_{g} - {\tilde{u}}_{g}}{w_{g}} | \leq h}, h \geq 0

(17)

This info-gap model is an unbounded family of nested sets of possible values of $u_{g}$ .

Green is also interested in the extent of damage to Blue’s economic interests that can be caused by Green each day, denoted $f_{g} (u_{g})$ . Thus, $f_{b} (u_{b})$ and $f_{g} (u_{g})$ have the same units. $f_{g} (u_{g})$ is proportional to Green’s firing rate:

f_{g} (u_{g}) = s_{g} u_{g}

(18)

where $s_{g}$ is a known positive value. Green requires that the economic damage to Blue be no less than a critical value, $f_{cg}$ , so Green’s robustness requirement is as follows:

f_{g} (u_{g}) \geq f_{cg}

(19)

Employing Equations (17)–(19) in the definition of Green’s robustness function in Equation (4), one readily finds the following expression for Green’s robustness:

{\hat{h}}_{g} (f_{cg}) = \frac{s_{g} {\tilde{u}}_{g} - f_{cg}}{s_{g} w_{g}}

(20)

or zero if this is negative. The robustness of the estimated damage is zero, that is, ${\hat{h}}_{g} (s_{g} {\tilde{u}}_{g}) = 0$ , which is the zeroing property. The robustness improves (increases) as the critical damage, $f_{cg}$ , gets less ambitious (lower), expressing the trade-off between the performance requirement and the robustness to uncertainty. The robustness decreases as the estimated error, $w_{g}$ , gets larger.

4.5.2. Formulation of the opportuneness functions

Blue’s critical requirement, Equation (15), is that the extent of economic damage to Blue’s interests each day not exceed the critical value $f_{cb}$ . However, it would be a wonderful windfall for Blue if the extent of daily damage did not exceed a much smaller value, $f_{wb}$ . Thus Blue’s windfall aspiration is as follows:

f_{b} (u_{b}) \leq f_{wb}

(21)

$f_{wb}$ may even be negative, implying economic improvement. Using the info-gap model in Equation (13), the extent of damage in Equation (14) and the definition of the opportuneness function in Equation (5), one readily obtains the following expression for Blue’s opportuneness function:

{\hat{β}}_{b} (f_{wb}) = \frac{s_{b} {\tilde{u}}_{b} - f_{wb}}{s_{b} w_{b}}

(22)

or zero if this is negative. No uncertainty is required to enable the estimated damage, so ${\hat{β}}_{b} (s_{b} {\tilde{u}}_{b}) = 0$ . A more wonderful outcome for Blue, namely $f_{wb} < s_{b} {\tilde{u}}_{b}$ , requires greater uncertainty.

We now define and derive Green’s opportuneness function. Green’s critical requirement, Equation (19), is that the extent of economic damage to Blue each day be no less than the critical value $f_{cg}$ . However, it would be a wonderful windfall for Green if the extent of daily damage to Blue’s economy was no less than a much greater value, $f_{wg}$ . Thus Green’s windfall aspiration is as follows:

f_{g} (u_{g}) \geq f_{wg}

(23)

Using the info-gap model in Equation (17), the extent of damage in Equation (18) and the definition of the opportuneness function in Equation (5), one readily obtains the following expression for Green’s opportuneness function:

{\hat{β}}_{g} (f_{wg}) = \frac{f_{wg} - s_{g} {\tilde{u}}_{g}}{s_{g} w_{g}}

(24)

or zero if this is negative. No uncertainty is required to enable the estimated damage, so ${\hat{β}}_{g} (s_{g} {\tilde{u}}_{g}) = 0$ . A more wonderful outcome for Green, namely $f_{wg} > s_{g} {\tilde{u}}_{g}$ , requires greater uncertainty.

4.5.3. Numerical implementation

We now numerically evaluate and discuss the implications of the robustness and opportuneness functions in Equations (16), (20), (22) and (24). The parameters take the following values:

${\tilde{u}}_{b} = 1000$ . Blue’s estimate of the number of Green missiles launched daily.

$w_{b} = 500$ . Blue’s error estimate on ${\tilde{u}}_{b}$ .

$s_{b} = $ 1500$ . Blue’s known economic loss per missile launched.

${\tilde{u}}_{g} = 3000$ . Green’s estimate of the number of Green missiles launched daily.

$w_{g} = 300$ . Green’s error estimate on ${\tilde{u}}_{g}$ .

$s_{g} = $ 1000$ . Green’s known economic loss to Blue per missile launched daily.

Consider first the robustness curve for Blue, ${\hat{h}}_{b} (f_{cb})$ , in Figure 2. As explained earlier, the positive slope expresses a trade-off: robustness increases (which is good for Blue) as the critical economic damage to Blue, $f_{cb}$ , is raised to a larger value (which is undesirable for Blue). The zeroing property is that the robustness reaches zero when $f_{cb}$ equals the estimated value, $s_{b} {\tilde{u}}_{b} = $ 1.5 \times 10^{6}$ .

Figure 2.

Robustness and opportuneness curves for Blue (solid) and Green (dash). The horizontal axis is $f_{cb}$ and $f_{wb}$ for Blue, and $f_{cg}$ and $f_{wg}$ for Green.

Compare this with the robustness curve for Green, ${\hat{h}}_{g} (f_{cg})$ , in Figure 2. The robustness reaches zero when $f_{cg}$ equals the estimated value, $s_{g} {\tilde{u}}_{g} = $ 3.0 \times 10^{6}$ . This is twice Blue’s value, indicating that Green estimates far greater economic damage to Blue. However, the zeroing property is that the robustness of this estimate is zero, so this estimate is unreliable. Only lower damage has positive robustness for Green, as expressed by the negative slope of Green’s robustness curve.

The Blue and Green robustness curves have opposite slopes because Blue prefers low economic damage (to itself) while Green prefers greater damage (to Blue). The opposite slopes of the Blue and Green robustness curves express the conflict between these parties: Blue requires low damage, while Green requires serious damage.

The slope of a robustness curve expresses the cost of robustness: how much the outcome must be worsened to obtain a positive increment of robustness. The Blue and Green slopes are $\frac{1}{s_{b} w_{b}} \approx 1.33 \times 10^{- 6}$ and $\frac{1}{s_{g} w_{g}} \approx 3.33 \times 10^{- 6}$ , respectively. Green’s robustness curve is about 2.5 times steeper than Blue’s curve.

Now consider the opportunity curves in Figure 2 for Blue and Green, ${\hat{β}}_{b} (f_{wb})$ and ${\hat{β}}_{g} (f_{wg})$ , respectively. The opportuneness function assesses the amount of uncertainty that is needed to enable—though not guarantee—the corresponding highly desirable outcome: low loss (from Blue’s perspective) and high loss (from Green’s perspective). The positive slope of Green’s opportuneness function indicates the need for greater uncertainty to facilitate greater-than-anticipated economic loss. Similarly, though in reverse, the negative slope of Blue’s opportuneness function indicates the need for greater uncertainty to enable lower-than-anticipated economic loss.

Finally, consider the numerical values spanned by the robustness and opportuneness curves in Figure 2. Numerical robustness and opportuneness values are calibrated by referring to the info-gap models in Equations (13) and (17) for Blue and Green, respectively.

A robustness of 3 means that the corresponding Blue or Green requirement is guaranteed provided that the uncertain quantity errs no more than 3 times the corresponding error estimate, $w_{b}$ or $w_{g}$ . The information represented by the uncertainty weight is that the estimated quantity may err by as much as $w_{b}$ or $w_{g}$ , or more. We have no probabilistic information, but the uncertainty weight calibrates the error. Contextual understanding will suggest that the error equal to the estimated uncertainty weight, $w_{b}$ or $w_{g}$ , is plausible, while an error several times greater is substantial, less plausible and more likely an outlier.

For instance, Blue’s robustness curve, ${\hat{h}}_{b} (f_{cb})$ , shows that, at a robustness of 3, Blue’s economic damage reaches $$ 3.7 \times 10^{6}$ , which is more than twice its estimated damage. Damage of $$ 3.7 \times 10^{6}$ can, therefore, be viewed as fairly extreme. Analogously, Blue’s opportuneness curve shows ${\hat{β}}_{b} (f_{wb}) = 3$ at $f_{wb} = - $ 0.74 \times 10^{6}$ . A negative loss is a gain, so this gain is possible only with errors three times the estimated errors; also, an outlier is not to be counted on. In between these extremes, Blue’s opportuneness curve shows ${\hat{β}}_{b} (0) = 2$ , meaning that no loss, $f_{wb} = 0$ , is possible (though not guaranteed) only with an error equal to twice the estimated error, which is substantial; Blue plausibly faces loss in the conflict with Green.

5. Evolution of strategic interaction: Qualitative analysis

We are considering the strategic interaction between two protagonists who both face non-probabilistic uncertainty about the other protagonist and the general situation, but one protagonist’s uncertainty is significantly greater than the other’s. Proposition 1 asserts that lower uncertainty is equivalent to higher robustness. That is, the protagonist who faces lower uncertainty has greater robustness than its adversary for confidently achieving specified critical goals. Proposition 2 asserts that greater uncertainty is equivalent to better opportuneness. In other words, the protagonist who faces greater uncertainty has greater opportunity than its adversary for enabling unanticipated, wonderful windfall outcomes.

We have proven these propositions with the rigor of quantitative mathematical analysis, but also with mathematics’ lack of qualitative semantic subtlety. We now discuss two implications of these results in a more general qualitative context. We will identify two possible evolutions of the strategic interaction. As before, we refer to the protagonists as Blue and Green, where Blue faces great uncertainty, while Green faces more moderate uncertainty. We will consider two examples.

As discussed in Section 3 and subsequently, uncertainty can be viewed as either pernicious (entailing the potential for unacceptable outcomes) or propitious (entailing the potential for better-than-anticipated wonderful outcomes). One possible response to pernicious uncertainty is to robustly satisfice the achievement of an essential or critical outcome. One possible response to propitious uncertainty is to exploit an unanticipated opportunity for a wonderful windfall outcome. There are many possible strategies in addition to robust satisficing and opportune windfalling. For instance, Blue could attempt to acquire information that would reduce Blue’s uncertainty. Blue could attempt to enhance Green’s uncertainty. Green could also try to reduce its uncertainty. Green could try to reduce Blue’s uncertainty to impede Blue’s opportune windfalling.

Recall that Blue and Green have conflicting interests about the outcome of their strategic interaction. Blue faces great uncertainty, while Green faces more moderate (but not negligible) uncertainty. We focus on the following strategies for Blue and Green.

Blue:

1B. Robust satisficing for an adequate outcome.

2B. Opportune windfalling for a wonderful outcome (motivated by proposition 2).

Green:

1G. Robust satisficing for an adequate outcome (motivated by proposition 1).

2G. Opportune windfall for a wonderful outcome.

There is no immediate or obvious resolution of this inherently conflictual interaction: what is adequate or wonderful for Blue is undesirable or terrible for Green, and vice versa. We can, however, recognize two alternative states of understanding by the protagonists.

Green is aware of Blue’s great uncertainty, so Green might anticipate that Blue will attempt to exploit this uncertainty by opportune windfalling (2B). Similarly, Blue is aware of Green’s more moderate uncertainty, so Blue might anticipate that Green will attempt to manage this uncertainty by robust satisficing (1G). Thus, if Blue adopts 2B and Green adopts 1G, this could be called a cognitive match between Blue and Green.

Any of the other three possible combinations of strategies (1B&1G, 1B&2G or 2B&2G) would constitute a cognitive mismatch between Blue and Green.

We now develop a generic framework for the concepts of cognitive match and cognitive mismatch. We will demonstrate how these concepts can be operationalized in the specific examples of Sections 6 and 7.

We now consider two possible evolutions of the strategic interaction between Blue and Green.

The first possible evolution of the strategic interaction, based on a cognitive match between Blue and Green, leads to subsequent convergent strategic interaction and diminishing asymmetry of uncertainty, both resulting from diminishing uncertainty of both parties. The reasoning is as follows. Recall that the term “understanding” refers broadly to assessments, plans, capabilities and context.

1. Blue faces great uncertainty and believes that Green faces only moderate uncertainty. Moderate uncertainty implies the potential for high robustness for achieving critical goals (Proposition 1). Consequently, Blue expects that Green will formulate its strategy by robust satisficing.

2. Green faces only moderate uncertainty and believes that Blue faces great uncertainty. Great uncertainty implies the potential for facilitating better-than-anticipated outcomes (Proposition 2). Consequently, Green expects that Blue will formulate its strategy by opportune windfalling.

3. Indeed, Blue attempts opportune windfalling because Blue faces great and enticing uncertainty. Green attempts robust satisficing because Green is confident in achieving critical goals under moderate uncertainty.

4. Items 1–2 constitute asymmetric uncertainty between Blue and Green. Item 3 constitutes a cognitive match between Blue and Green.

How does this cognitively-matched strategic interaction evolve, where “strategic interaction” refers to the developing understandings of risks and opportunities leading to plans and actions?

5. As the strategic interaction evolves, each protagonist observes the other’s actions, thus reducing each protagonist’s uncertainty about the other’s current and future understandings and about the general situation.

6. As Blue’s uncertainty diminishes, Blue will be inclined to move from opportune windfalling to robust satisficing (Proposition 1).

7. As Green’s moderate uncertainty is further diminished, Green will remain with robust satisficing (Proposition 1).

8. Understandings evolve, further reducing each protagonist’s uncertainty, until a stable situation emerges. This could be called convergent strategic interaction, resulting in diminishing asymmetry of uncertainty caused by diminishing uncertainty of both parties. This convergence does not imply agreement between Blue and Green; their dispute remains until a solution is found. However, the fog around that dispute has diminished.

The second possible evolution of the strategic interaction, based on cognitive mismatch between Blue and Green, leads to subsequent divergent strategic interaction and diminishing asymmetry of uncertainty, both resulting from increasing uncertainty of both parties. The reasoning is as follows.

1. Blue faces great uncertainty and believes that Green faces only moderate uncertainty. Moderate uncertainty implies high robustness for achieving critical goals (Proposition 1). Consequently, Blue expects that Green will formulate its strategy by robust satisficing.

2. Green faces moderate uncertainty and believes that Blue faces great uncertainty. High uncertainty implies great potential for facilitating better-than-anticipated outcomes (Proposition 2). Consequently, Green expects that Blue will formulate its strategy by opportune windfalling.

3. Either Blue does not adopt the anticipated strategy of opportune windfalling, or Green does not adopt the anticipated strategy of robust satisficing, or both.

4. Items 1–2 constitute asymmetric uncertainty between Blue and Green. Item 3 constitutes a cognitive mismatch between Blue and Green.

How does this cognitively mis-matched strategic interaction evolve, where “strategic interaction” refers to the developing understandings of risks and opportunities leading to plans and actions?

5. As the strategic interaction evolves, each protagonist observes the other’s actions. Those actions do not conform to the protagonist’s anticipation, so each protagonist’s uncertainty increases.

6. The protagonists may switch strategies from time to time, thus further increasing uncertainty for both protagonists with associated decreasing asymmetry of uncertainty. This could be called divergent strategic interaction. The dispute between Blue and Green remains, and a resolution becomes more elusive as the fog around the dispute grows.

In both scenarios, the asymmetry of uncertainty decreases over time. In the first scenario, the uncertainty of each protagonist decreases over time, while in the second scenario, both protagonists’ uncertainty increases over time, in both cases resulting in reduced asymmetry of uncertainty. Either convergent or divergent strategic interaction is possible. Furthermore, neither scenario needs to proceed in its entire course; events can evolve first in one scenario and then move to the other, as we will see in the Cuban Missile Crisis.

6. The Cuban Missile Crisis

The Cuban Missile Crisis started when the United States discovered Soviet bombers and Soviet nuclear ballistic missiles in Cuba in the autumn of 1962. This Soviet action resulted from an agreement between Soviet Premier Nikita Khrushchev and Cuban Premier Fidel Castro to deter any future invasion following the US Bay of Pigs invasion in April 1961.

In discussing the Cuban Missile Crisis, Allison and Zelikow⁴² discuss three models for describing decision-making by leaders.

The “Rational Actor Model” presumes that national decisions can be understood as though made by a single entity that identifies “optimal choices in narrowly constrained, neatly defined situations. In these situations, rationality refers to an essentially Hobbesian notion of consistent, value-maximizing reckoning or adaptation within specified constraints.” (Allison and Zelikow,^{42(p. 17)}, italics in the original)

Allison and Zelikow recognize the descriptive utility of the Rational Actor Model but argue that it “must be balanced by the appreciation that (1) [governmental] monoliths are black boxes…and (2) large acts result from innumerable and often conflicting smaller actions by individuals at various levels of organizations in the service of a variety of only partially compatible conceptions of national goals, organizational goals, and political objectives.” (p. 5) Following Allison and Zelikow, the present paper argues that understanding the Cuban missile crisis requires attention to uncertainty in understandings, by which we mean uncertainty in assessments, plans, and capabilities of each protagonist, as well as uncertainty in understanding the general context in which these other elements reside.

The “Organizational Behavior Model” emphasizes the distinctive logic, capacities, culture, and procedures of the large organizations that constitute a government.” (Allison and Zelikow,^{42(p. 5)}) For instance, “the causes of inadequate camouflage [of Soviet emplacements]…[are] rooted not in incompetence or a clever policy design but instead in established routines designed for settings in which camouflage had never been required” (p. 381).

The “Governmental Politics Model” focuses on the politics of a government. According to this model, events in foreign affairs are characterized neither as unitary choice nor as organizational outputs. Rather, what happens is understood as a resultant of bargaining games among players in the national government.” (Allison and Zelikow,^{42(p. 6)}) This model reveals Khrushchev’s “appreciation of the situation to have been cloudy at best, his judgments bereft of any attribute of high-quality deliberations. Relying on haphazard and often incorrect information” (p. 382).

Allison and Zelikow⁴² also comment that policy makers “must make decisions based on partly read, partially digested, uncertain information” (p. 44) on “real-world issues where information is incomplete” (p. 45).

One can tentatively understand the evolution of the Cuban Missile Crisis between the United States and the Soviet Union in the context of the following two models of the evolution of strategic interaction. Both of these models build on the uncertainty of understandings that Allison and Zelikow emphasize. The situation is complex, dynamic, and the range of possibilities is poorly understood, so probability distributions are unavailable. The missile crisis began with substantial asymmetry of uncertainty and a distinct cognitive mismatch between the United States and the USSR. The following four items correspond to the similarly numbered items in the cognitive mismatch list in Section 5.

1. The United States faces great uncertainty about the intentions underlying the USSR’s decision to place offensive capabilities in Cuba. The United States likewise believes that the USSR faces more moderate uncertainty about how the United States will respond to what the United States perceives as a strategic offensive action.

2. The USSR faces lower uncertainty about the US understanding and recognizes that the USSR’s action would induce great uncertainty in the US leadership.

3. The United States may have initially contemplated exploiting the uncertainty about Soviet intentions with a vast and overwhelming attack that could potentially lead to unanticipated favorable outcomes, such as the replacement of Castro’s regime. However, the Soviets did not actually develop a strategically sound offensive threat; limited air power and a small number of missiles were deployed against the vast offensive and defensive capabilities of the United States. Thus, the United States did not employ an opportune windfalling strategy.

4. This constitutes the asymmetric uncertainty and the cognitive mismatch between the USSR and the United States.

From this point, the crisis proceeded with a distinct cognitive match between the United States and the USSR. The following three items correspond to the similarly numbered items in the cognitive match list in Section 5.

6. The United States recognized the strategic limitation of the Soviet action and did not observe further strengthening of the Soviet presence in Cuba. The United States became more confident that a nuclear attack was not pending: United States uncertainty diminished.

7. Likewise, Soviet uncertainty about US action diminished as the United States implemented a blockade but did not indicate clear preparations for invasion.

8. The diminishing uncertainty of both sides resulted in reduced asymmetry of uncertainty. Both sides reverted to reliably protecting critical interests. The dispute remained over the Soviet placement of offensive capability, and more generally about the status of the communist regime in Cuba. However, a stable situation evolved, agreement was reached (removal of Soviet missiles from Cuba and removal of US missiles from Turkey), and a cognitive match between the protagonists emerged.

7. Pearl Harbor: The strategic interaction

The Japanese attack on Pearl Harbor on 7 December 1941 caused damage or destruction to eight US battleships. This was a tactical operation, but it had a far wider strategic message. This can be understood by describing the attack with the first five stages of a cognitive match with asymmetric uncertainty between Japan (Blue) and the United States (Green).

The military leadership of Japan planned the attack on Pearl Harbor (and on other US and British possessions) as part of the establishment of the Greater East Asia Co-Prosperity Sphere, as discussed by Wohlstetter^{43(pp. 325, 346)}. This regional Japanese ambition was by no means secret, and Japan felt that the United States would be only moderately uncertain about how Japan would act to realize this ambition. However, Japanese leadership was very uncertain about how the United States would respond to Japanese aggression. Possibilities ranged from US efforts to contain Japanese expansion (which could even align with Japanese regional intentions) to all-out and sustained war, which would lead to Japanese defeat because of the far greater production capabilities of the United States, as explained by Wohlstetter^{43(pp. 350–351)} (Item 1 in the cognitive match scenario).

By autumn 1941, the US Navy and Army leadership were anticipating war with Japan⁴⁴ despite ongoing negotiations on the future of the Pacific Ocean region. However, there was moderate US uncertainty about when, where, and under what pretext Japan would initiate war. The United States also anticipated that Japan faced great uncertainty about how the United States (and Britain) would respond should Japan initiate war, recognizing the far greater industrial war potential of the United States, balanced against the remoteness of Hawaii from the US mainland (Item 2 in the cognitive match scenario).

Indeed, Japan pursued the surprising and risky attack on Pearl Harbor, while the United States maintained a standard defensive posture (Item 3 in the cognitive match scenario).

One sees the asymmetric uncertainty between Japan and the United States before Pearl Harbor, and the cognitive match in their understandings (Item 4 in the cognitive match scenario).

Both sides’ uncertainty is reduced immediately after the attack: the United States learns the location and nature of the multiple Japanese attacks, and Japan witnesses President Roosevelt’s request, in his address to Congress on 8 December 1941, that Congress declare war on Japan (Item 5 in the cognitive match scenario).

8. Concluding discussion

Strategic interactions between protagonists with conflicting interests may be strongly influenced by uncertainties facing each side. These uncertainties cannot be quantified with probability distributions because they result from complexity and limited knowledge about the realm of possibilities. We studied the implications of asymmetric uncertainty: one protagonist faces great uncertainty about the situation, while the other protagonist faces more moderate but still significant uncertainty.

Uncertainty can induce many diverse responses by decision-makers. We explored two alternative responses to non-probabilistic uncertainty. Robust satisficing is a methodology that responds to the pernicious potential of uncertainty and attempts to assure the achievement of a critical or essential outcome. Opportune windfalling is a methodology that responds to the propitious potential of uncertainty, and attempts to facilitate the achievement of a better-than-anticipated wonderful outcome.

We have presented two mathematical propositions: The protagonist facing greater uncertainty has worse robustness for achieving a critical outcome (Proposition 1), but has better opportuneness for achieving a windfall outcome (Proposition 2). This protagonist may lack confidence in achieving a critical outcome (if the robustness to uncertainty is low), but may recognize the potential for a windfall (if the opportuneness from uncertainty is favorable). Conversely, the protagonist facing more moderate uncertainty may be confident in achieving a critical outcome (if the robustness to uncertainty is high), but may not see great opportunity for a windfall (if the opportuneness from uncertainty is unfavorable).

The mathematical analysis of robust and opportune responses to uncertainty reveals three generic properties. First, predicted outcomes have zero robustness to uncertainty in the knowledge upon which the predictions are based. Thus, predicted outcomes are not a reliable basis for assessment. Second, the robustness improves as the required outcome becomes more modest: robustness trades off against the outcome requirement. Hence, reliable sub-optimal requirements can be identified. Third, the opportuneness improves as the windfall aspiration becomes more modest: opportuneness trades off against the outcome aspiration. Hence, plausible windfall aspirations can be identified. A simple numerical example of the quantitative robustness and opportuneness functions was presented, and the interpretation of these functions was discussed.

Mathematical analysis reveals generic structural properties but is ill-suited for the semantic subtlety required in analyzing realistic situations. In preparation for qualitative analysis of strategic interaction under asymmetric uncertainty, we identified two epistemic conditions. A cognitive match occurs when each protagonist correctly assesses its adversary’s uncertainty and its adversary’s choice between robust satisficing and opportune windfalling. A cognitive mismatch occurs otherwise. We then outlined generic stages in the evolution of strategic interaction with either a cognitive match or a cognitive mismatch between the protagonists. We demonstrated that this analysis can describe the Cuban Missile Crisis of 1962 and the Japanese attack on Pearl Harbor in World War II.

Finally, we note that different types of knowledge and understanding underlie quantitative or qualitative analysis. Practical application thus depends on first deciding which type of analysis is appropriate: quantitative or qualitative. The different results of this paper can then be appropriately applied.

Footnotes

ORCID iD

Yakov Ben-Haim

Funding

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

Declaration of conflicting interests

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

Author biography

Yakov Ben-Haim initiated and developed info-gap decision theory for modeling and managing deep uncertainty. Info-gap theory is a decision-support tool, providing a methodology for assisting in the assessment and selection of policy, strategy, action or decision in a wide range of disciplines. Info-gap theory has impacted the fundamental understanding of uncertainty in human affairs and is applied in decision-making by scholars and practitioners around the world in engineering, biological conservation, economics, project management, climate change, natural hazard response, national security, medicine and other areas (see . He has been a visiting scholar in many countries and has lectured at universities, technological and medical research institutions, public utilities and central banks. He has published more than 100 articles and six books. He is a professor emeritus of mechanical engineering at the Technion—Israel Institute of Technology and held the Yitzhak Moda’i Chair in Technology and Economics during 2001–2021.

References

Atkinson

Jackson

Williford

GW.

Rivalry, uncertainty, and militarized compellent threats. J Global Secur Stud. 2021; 6(1): 1–11. https://doi.org/10.1093/jogss/ogz079

Knight

FH.

Risk, uncertainty and profit (Hart, Schaffner and Marx. Re-issued in 1965 by Harper Torchbooks). New York, 1921, p.20.

Betts

RK.

Conventional deterrence: predictive uncertainty and policy confidence. World Politics 1985; 37(2): 153–179.

Altfeld

MF.

Uncertainty as a deterrence strategy: a critical assessment. Compar Strat 1985; 5(1): 1–26.

Nicholson

Deterrence, uncertainty and cooperative behavior. Int Interact 1990; 15(3–4): 227–242.

Kilgour

Zagare

FC.

Credibility, uncertainty, and deterrence. Am J Polit Sci 1991; 35(2): 305–334.

Baker

Harel

Kugler

The virtues of uncertainty in law: an experimental approach. Iowa Law Review 2004; 89: 443–487.

DeAngelo

Charness

Deterrence, expected cost, uncertainty and voting: experimental evidence. J Risk Uncertain 2012; 44: 73–100.

Chilton

Weaver

Waging deterrence in the twenty-first century. Strat Stud Quart 2009; 3(1): 31–42.

10.

Ben-Haim

Uncertainty and deterrence. Air Space Power J 2016; 7(3): 26–49.

11.

Ben-Haim

Info-gap decision theory: Decisions under severe uncertainty. 2nd ed. Academic Press, 2006.

12.

Kaplow

JM.

State compliance and the track record of international security institutions: evidence from the nuclear nonproliferation regime. J Global Secur Stud 2021; 7(1): 1–19. https://doi.org/10.1093/jogss/ogab027

13.

Mazarr

Frederick

Crane

YK.

Understanding a New Era of Strategic Competition. RAND Corporation, 2022.

14.

Handel

MI.

War, Strategy, and Intelligence. Routledge, 1989, p.173.

15.

Punla-Green

Mitchell

Gearhart

, et al. Shortest path network interdiction with asymmetric uncertainty. Networks 2024; 83: 605–623.

16.

Hwang

Military camouflaged object detection with deep learning using dataset development and combination. J Defense Model Simul: Appl Methodol Technol 2026; 23(1): 67–78.

17.

Kenkel

Schram

Uncertainty in crisis bargaining with multiple policy options. Am J Polit Sci 2025; 69: 194–209.

18.

Fearon

JD.

Signaling foreign policy interests: tying hands versus sinking costs. J Conflict Resolut 1997; 41(1): 68–90.

19.

Slantchev

BL.

The principle of convergence in wartime negotiations. Am Polit Sci Rev 2003; 97(4): 621–632.

20.

Fey

Ramsay

KW.

Mutual optimism and war. Am J Polit Sci 2007; 51(4): 738–754.

21.

Jakovlev

Eglynas

Jusis

, et al. Developing an IoT-enabled probabilistic model for quick identification of hidden radioactive materials in maritime port operations to strengthen global supply chain security. J Defense Model Simul: Appl Methodol Tech 2026; 23(1): 41–54.

22.

Benjamin

Simon

The next attack: the failure of the war on terror and a strategy for getting it right. Henry Holt and Company, 2006, p.180.

23.

Clarke

RA.

Against all enemies: inside America’s war on terror. Free Press, 2004, p.xix.

24.

English

RJ.

Does terrorism work? A history. Oxford University Press, 2016, p.265.

25.

Hayden

MV.

Playing to the edge: American intelligence in the age of terror. Penguin Press, 2016, p.317.

26.

Dunn Cavelty

Wenger

Introduction: Cyber security between socio-technological uncertainty and political fragmentation. In: Cavelty

Wenger

(eds.) Cyber security politics: socio-technological transformations and political fragmentation. Routledge, 2022, pp.1–13.

27.

Dunn Cavelty

Wenger

(eds) Cyber security politics: socio-technological transformations and political fragmentation. Routledge, 2022.

28.

Bob

Evyatar

Target Tehran: how Israel is using sabotage, cyberwarfare, assassination—and secret diplomacy—to stop a nuclear Iran and create a new Middle East. Simon & Schuster, 2023.

29.

Lupovici

Uncertainty and the study of cyber deterrence: the case of Israel’s limited reliance on cyber deterrence. In: Cavelty

Wenger

(eds.) Cyber security politics: socio-technological transformations and political fragmentation. Routledge, 2022, pp.128–140.

30.

Albanese

Jajodia

Venkatesan

Defending from stealthy botnets using moving target defenses. IEEE Secur Privacy 2018; 16(1): 92–97.

31.

Albanese

Connell

Venkatesan

, et al. Moving target defense quantification. In: Jajodia

Cybenko

Liu

, et al. (eds) Adversarial and uncertain reasoning for adaptive cyber defense: control- and game-theoretic approaches to cyber security. Lecture Notes in Computer Science. Vol. 11830. Springer, 2019, pp.94–111.

32.

Jajodia

Cybenko

Liu

, et al. Adversarial and uncertain reasoning for adaptive cyber defense: control- and game-theoretic approaches to cyber security. Lecture Notes in Computer Science. Vol. 11830, Springer, 2019.

33.

Albanese

Jajodia

Venkatesan

, et al. Adaptive cyber defenses for botnet detection and mitigation. In: Jajodia

Cybenko

Liu

34.

Connell

Menascé

Albanese

. Performance modeling of moving target defenses with reconfiguration limits. IEEE Trans Depend Secure Comput 2021: 205–219.

35.

Ding

Ren

Quevedo

, et al. Defensive deception against reactive jamming attacks in remote state estimation. Automatica 2020; 113: 108680.

36.

Chen

Cui

Zhang

, et al. Optimal strategy for cyberspace mimic defense based on game theory. IEEE Access 2021; 9: 68376–68386.

37.

Liang

Liu

Information provision and consumer search behavior for products with asymmetric uncertainty. J Manag Sci Eng 2023; 8: 49–82.

38.

Maskin

ES.

Uncertainty and entry deterrence?

Economic Theory 1999; 14: 429–437.

39.

Rangrazjeddi

González

Barker

. Game-theoretic algorithm for interdependent infrastructure network restoration in a decentralized environment. Risk Anal 2024; 44(7): 1630–1650.

40.

Wald

Statistical decision functions which minimize the maximum risk. Annal Math 1945; 46(2): 265–280.

41.

Ben-Haim

Conflict and collaboration between innovators and practitioners: perspectives on risk and opportuneness. Working Paper, 2025.

42.

Allison

Zelikow

Essence of decision: explaining the Cuban missile crisis. 2nd ed. Addison Wesley Longman, 1999, pp.17.

43.

Wohlstetter

Pearl harbor: warning and decision. Stanford University Press, 1962.

44.

Dahl

EJ.

Intelligence and surprise attack: failure and success from pearl harbor to 9/11 and beyond. Georgetown University Press, 2013, pp.5, 6, 17, 42, 44, 45.