Putting all eggs in one basket: some insights from a correlation inequality

We give examples of situations – stochastic production, military tactics, corporate merger – where it is beneficial to concentrate risk rather than to diversify it, i.e., to put all eggs in one basket. The examples admit a dual interpretation: as optimal strategies of a single player (the “principal†) or, alternatively, as dominant strategies in a non-cooperative game with multiple players (the “agents†). The key mathematical result can be formulated in terms of a convolution structure on the set of increasing functions on a Boolean lattice (the lattice of subsets of a finite set). This result generalizes the well-known Harris inequality from statistical physics and discrete mathematics; we give a simple self-contained proof of this result, and then prove a further generalization based on game-theoretic ideas.