Safe Opponent Exploitation in Colonel Blotto: Where the Gifts Are, and What They Are Worth

We study safe opponent exploitation in discrete Colonel Blotto games with heterogeneous battlefield values. We investigate the characterization of gifts: pure opponent allocations which are not the best response to at least one equilibrium strategy and hence can be exploited without losing the equilibrium safety guarantee. First, for an even number of equal-value battlefields, we consider the weaker divisibility condition \( n\mid 2B \), rather than \( n\mid B \), under which a paired equilibrium with uniform marginals is known to exist, and we establish the corresponding gift boundary \( \max_j\;X_j>2B/n \). We then show that the natural extension of this boundary to heterogeneous battlefields is not universally valid by giving exact counterexamples in both directions. We also provide a condition on equilibrium marginals and their coupling under which the boundary is recovered, and show on small instances that the validity of the boundary depends on the budget as well as on the battlefield values. Finally using best response oracles and constraint generation over the equilibrium polytope, we apply our equilibrium and safe exploitation framework to five public Riddler Blotto populations, quantifying the tradeoff between unconstrained exploitation and exploitation subject to the equilibrium safety guarantee.