War games: the life and military ranks of Z.S. Blotto

Peter Baudains

Outline

•  Motivation •  Colonel Blotto •  General Blotto •  Field Marshall Blotto

Problems of strategic, competitive and constrained allocation of resources

Applications to military, terrorism, political campaigns, research and development races, etc.

Colonel Blotto (Borel, 1921)

Two players, red and blue, to specify numbers:

where some options for might be the n-dimensional non-negative real numbers or non-negative integers, subject to:

With payoffs to each player given by:

Borel, E. (1921), “La theorie du jeu les equations integrales a noyau symetrique”, Comptes Rendus de l’Academie 173; English translation by Savage, L (1953), “The theory of play and integral equations with skew symmetric kernels”, Econometrica, 21.

i.e. 1 point for winning a zone, -1 point for losing a zone and 0 for a draw.

Example

Blue wins

with one to spare.

What’s the best strategy?

But there do exist mixed strategy solutions.

General Blotto: generalising the payoff function

“A test of elementary conditions given here exposes as incorrect various solutions to Blotto games which appear in the classified literature”

Blackett, D. (1958). Pure strategy solutions of Blotto games, Naval Res. Logist. Quart. 5:107-110.

General Blotto (Golman and Page, 2009)

For certain values of p and under certain conditions, pure strategy optimal solutions can be found. For example, when p=1 (in which case, the payoff function is C2-differentiable) if R=B and if pairs of zones are as valuable as zones themselves, an optimal solution is to assign resources evenly.

Golman, R. and Page, S.E. (2009). General Blotto: games of allocative strategic mismatch, Public Choice 138:279-299.

Golman and Page propose a (basic) payoff function given by:

A non-zero sum Blotto game

Field Marshall Blotto – A further extension

with Hannah Fry, Toby Davies and Alan Wilson

is the ‘threat’ that is imposed on team R through the presence and location of team B across all zones. So it is not just the individual battlefields that matter but also the strategic physical distribution of each team.

Example

Example

Blue wins Colonel Blotto, BUT their team has been isolated.

Red wins Field Marshall Blotto (for p=0 and β=0.01) due to their large presence in a highly connected central zone.

•  What effect do the parameters p and β have on the result? •  How can we calibrate these parameters for real-world situations? •  Given p and β, what is a ‘good’ strategy?

Simple example: Two zone case

Blue team wins zone 1 when:

Blue team wins zone 2 when:

Zone 1 Zone 2

Blue wins Draw Draw

Dynamic games – steps towards models of war?