Allocating Security Expenditures under Knightian Uncertainty: An Info-Gap Approach

ABSTRACT

We apply the information gap approach to resource allocation under Knightian (non-probabilistic) uncertainty in order to study how best to allocate public resources between competing defence measures. We demonstrate that when determining the level and composition of defence spending in an environment of extreme uncertainty vis-a-vis the likelihood of armed conflict and its outcomes, robust-satisficing-expected utility will usually be preferable to expected utility maximisation. Moreover, our analysis suggests that in environments with unreliable information about threats to national security and their consequences, a desire for robustness to model misspecification in the decision-making process will imply greater expenditure on certain types of defence measures at the expense of others. Our results also provide a positivist explanation of how governments seem to allocate security expenditures in practice.

KEYWORDS:

• Defense
• knightian uncertainty
• robustness
• info-gap

JEL CLASSIFICATION:

• D81
• F51
• F52

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Antulio J. Echevaria II (2008, 113–114).

2. David Hume, writing in 1748: ‘ What is the foundation of all conclusions from experience? … All experimental conclusions proceed upon the supposition that the future will be conformable to the past … why this experience should be extended to future times, and to other objects, which for aught we know, may be only in appearance similar; this is the main question on which I insist … If there be any suspicion that the course of nature may change, and that the past may be no rule for the future, all experience becomes useless, and can give rise to no inference or conclusion.’ (Hume (1748 2004), 19, 21, 22).

3. Intrilligator and Brito (1981) introduce probabilistic uncertainty to model the impact of nuclear proliferation. The likelihood of an individual nuclear state-initiating conflict declines as the number of nuclear states rises, creating a non-monotonic relationship between the outbreak of nuclear conflict and the number of nuclear-armed states.

4. In Bier, Oliveros, and Samuelson (2007) a defender allocates resources to defend different targets to minimise expected losses. The adversary’s preferred target is characterised by a known probability distribution.

5. First elaborated during the 1980s by Soviet military theorists, in particular, Marshal Nikolai Ogarkov, then chief of the Soviet General Staff, the RMA doctrine of war has already had a profound effect on Western military planning. The move to adapt the U.S. military to RMA type warfare was spearheaded by the Department of Defense’s Office of Net Assessment, and the man who has headed it since 1973, Andrew Marshall, and the vice chairman of the Joint Chiefs of Staff between 1994 and 1996, Admiral William Owens. See Cohen (1996).

6. Info-gap models obey two axioms: (i) Nesting asserts that the range of possible pdfs increases as $\alpha$ increases:

$\alpha <{\alpha }^{\prime }\Rightarrow \mathcal{F}\left(\alpha ,\tilde{p},{\tilde{P}}_w\right)\subseteq \mathcal{F}\left({\alpha }^{\prime },\tilde{p},{\tilde{P}}_w\right)$

(ii) Contraction asserts that when $\alpha =0,$ the best estimated models are the only possibilities:

$\mathcal{F}\left(0,\tilde{p},{\tilde{P}}_w\right)=\left\{\tilde{p},{\tilde{P}}_w\right\}$

7. ‘ As the ancient retiree from the Research Department of the British Foreign Office reputedly said, after serving from 1903–50: “Year after year the worriers and fretters would come to me with awful predictions of the outbreak of war. I denied it each time. I was only wrong twice.”’ (Hughes 1969, 244).

8. ‘ An adversary can still decide to attack even though his capabilities are relatively weak (1) if he miscalculates the strength of the intended victim (as did the Germans in their attack on the Soviet Union in 1941, or the Arab States in their underestimation of Israeli capabilities in 1967); (2) if he is more interested in applying political pressure or making political gains even at the cost of military defeat; (3) if he gambles that his surprise attack will have a force multiplier effect sufficient to compensate for his inferior capabilities.’ (Handel 1989, 241.)

9. Though the Japanese were not privy to German plans to attack the Soviet Union, during the months before Operation Barborossa was launched on 22 June 1941, the Japanese actively considered beginning to prepare for a ‘northern war’ – abrogating the Neutrality Pact they had just signed with the Soviet Union on 13 April of that year (having fought unsuccessfully a brief war with it between May and September 1939), to launch an attack on that country’s far eastern territories. Throughout that summer, civilians and military people in the Japanese government weighed the various merits of attacking the Soviet Union, attacking the Western powers to the south, or continuing to rely on diplomacy while consolidating gains in China and newly acquired French Indo-China. Only at a military conference on 3 September did the more aggressive view of the Japanese Army prevail over that of the more cautious Navy to initiate a war in the south, in response to continuing United States refusal to restore normal economic relations and not impede further Japanese ambitions in China. Three days later an imperial council approved the military’s recommendation that “ if, by the early part of October, there is still no prospect of being able to attain our demands, we shall immediately decide to open hostilities against the United States, Great Britain, and the Netherlands (see Butow [1961, 250].) The decision to begin preparations for the ‘southern war’ commenced only three months and a day prior to the attack on 7 December 1941.

10. ‘The Americans – and many Japanese – had thought that any attack on the distant American base at Pearl Harbor was impossible, given the inability of the Japanese fleet to conduct such distant operations due to refuelling challenges and the need for radio silence along the long route to Hawaii.’ (Hansen [2017, 143]).

11. This figure maps the specific functional forms and parameter values used here into the analogue of Figure 1.

12. The units that had already undergone the full transformation were either not deployed or had little influence on any part of the campaign (Adams 2006).

13. As of 2018, the US Army had 58 Brigade Combat Teams, of which 16 were Armour against 9 Strykers (South 2018).

14. In our calculations we set $\gamma =.98$, so the utility is close to logarithmic.

Additional information

Funding

This research was supported by the Economics of National Security Program, Samuel Neaman Institute, grant no. 358.