Shield versus sword resource distribution in K-round duels

The paper considers optimal resource distribution between offense and defense in a duel. In each round of the duel two actors exchange attacks distributing the offense resources equally across K rounds. The offense resources are expendable (e.g. missiles), whereas the defense resources are not expendable (e.g. bunkers). The outcomes of each round are determined by a contest success functions which depend on the offensive and defensive resources. The game ends when at least one target is destroyed or after K rounds. We show that when each actor maximizes its own survivability, then both actors allocate all their resources defensively. Conversely, when each actor minimizes the survivability of the other actor, then both actors allocate all their resources offensively. We then consider two cases of battle for a single target in which one of the actors minimizes the survivability of its counterpart whereas the counterpart maximizes its own survivability. It is shown that in these two cases the minmax survivabilities of the two actors are the same, and the sum of their resource fractions allocated to offense is equal to 1. However, their resource distributions are different. In the symmetric situation when the actors are equally resourceful and the two contest intensities are equal, then the actor that fights for the destruction of its counterpart allocates more resources to offense. We demonstrate a methodology of game analysis by illustrating how the resources, contest intensities and number of rounds in the duels impact the survivabilities and resource distributions.

r, R:

Actors’ resources

ρ :

Ratio r/R between actor 1’s and actor 2’s resources

x, X:

Offense-defense resource distribution parameters

K :

Number of consecutive attacks

s, S:

Target survivability (probability of survival in all K attacks)

v, V:

Success probability of each attack

p _i , P _i :

Probability that the target is destroyed in the i-th attack

μ, m:

Contest intensities in attacks against the actors

θ(q|K):

Conditional survivability of target 2 in q rounds of a K-round duel

Θ(q|K):

Conditional survivability of target 1 in q rounds of a K-round duel

Authors and Affiliations

1. Faculty of Social Sciences, University of Stavanger, Stavanger, Norway

   Kjell Hausken

2. Collaborative Autonomic Computing Laboratory, School of Computer Science, University of Electronic Science and Technology of China, Chengdu, People’s Republic of China

   Gregory Levitin

3. The Israel Electric Corporation Ltd., Haifa, Israel

   Gregory Levitin

Corresponding author

Correspondence to Kjell Hausken.

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