Colonel Blotto with Battlefield Games

TL;DR

This work introduces a two-level Colonel Blotto framework in which battlefields host parametrized subgames and investigate equilibrium existence and computation under discrete/continuous soldiers, sum/min aggregators, and two- or one-sided settings. It develops flow-polytope and sequence-form representations to recover convex–concave structure in several cases, enabling polynomial-time NE computation via linear programs or online learning, and leverages minimax theorems (Sion, Kneser–Fan) where applicable. The paper provides max–min strategy computations using quasiconcave objective reformulations and proves convergence guarantees for subgradient-based methods, supplemented by extensive empirical evaluations on synthetic and security-inspired scenarios. The findings demonstrate scalability and practical applicability of the proposed methods, with clear guidance on which settings yield tractable NE and which require approximate or alternative approaches.

Abstract

We study a class of two-player zero-sum Colonel Blotto games in which, after allocating soldiers across battlefields, players engage in (possibly distinct) normal-form games on each battlefield. Per-battlefield payoffs are parameterized by the soldier allocations. This generalizes the classical Blotto setting, where outcomes depend only on relative soldier allocations. We consider both discrete and continuous allocation models and examine two types of aggregate objectives: linear aggregation and worst-case battlefield value. For each setting, we analyze the existence and computability of Nash equilibrium. The general problem is not convex-concave, which limits the applicability of standard convex optimization techniques. However, we show that in several settings it is possible to reformulate the strategy space in a way where convex-concave structure is recovered. We evaluate the proposed methods on synthetic and real-world instances inspired by security applications, suggesting that our approaches scale well in practice.

Figures (5)

• Figure 1: Convergence of \ref{['alg:psa']} in the one-sided min-aggregated setting with affine (left), quadratic (middle), and security-inspired (right) battlefield utilities.
• Figure 2: Layered graph structure used to compactly describe the mixed strategy space of a single player where there are 3 battlefields and 4 soldiers (thus a height of 5). Note that this edges correspond to the $h^j$ variables and not the battlefield level strategies. Each path from left to right corresponds to a unique discrete distribution of soldiers over battlefields, i.e., a pure strategy. For instance, the path in green shows represents the pure strategy where 3 soldiers are allocated to battlefield 1, 1 to battlefield 2, and none to battlefield 3. Furthermore, we can also see that the marginal distribution of soldiers on each battlefield. For example, the probability that 2 soldiers are placed in battlefield $2$ is the sum of the edges in red. Note that for simplicity in illustration we allow soldiers to be unused (paths can end at any of the rightmost vertices, not just at the vertex corresponding to $0$ soldiers remaining). This typically does not lead to issues since in most applications, payoffs are monotonically non-decreasing in the number of soldiers. However, it is entirely possible to remove edges from the second-last to last layer to enforce this. Without the battlefield level strategies (see Figure \ref{['fig:battlefield_seq_form']}, this is identical to the approach of behnezhad2023fast.
• Figure 3: Left: Sequence form representation for battlefield 2 of Figure \ref{['fig:layered_graph']}when there are 2 soldiers placed in it. Assume there are 2 actions to be taken at battlefield 2. Red edges correspond to the cases where exactly 2 soldiers are allocated to the battlefield (these are the same red edges in Figure \ref{['fig:layered_graph']}) For each of them, their values are summed, giving the total probability that battlefield 2 has 2 soldiers (see edges in blue leading to the "+" vertex). This total probability at the "+" vertex is then split up into the two battlefield actions, given in purple. These edges correspond to the $x^j$'s; in this specific case $x^j_{2,2,\alpha_2^j}$, i.e., the marginal probability that 2 soldiers are placed in battlefield 2 and action $\alpha^j_2$ was chosen. Note that the intersection between the blue and red edges do not involve splitting the values up like flow, but rather, are duplicated at the points labelled "=". We reiterate that this is only for battlefield 2 with two soldiers,; this is repeated for each other battlefield, and for every possible number of soldiers allocated to it. Right: a similar situation but for another battlefield where there is one soldier allocated to it, and with $3$ battlefield level actions for the player.
• Figure 4: Average (red) normalized objective value over 10 instances (gray) of a one-sided continuous two-level Blotto game with min aggregator and randomly generated affine battlefield utilities.
• Figure 5: Average (red) normalized objective value over 10 instances (gray) of a one-sided continuous two-level Blotto game with min aggregator and randomly generated quadratic battlefield utilities.

Theorems & Definitions (65)

• Proposition 1
• Theorem 1: Sion's minimax theorem
• Definition 1: Concavelike, Convexlike, and Concave-convexlike
• Theorem 2: Kneser-Fan minimax theorem
• Theorem 3: Kuhn's theorem under two-sided sum aggregator
• Corollary 1
• Proposition 2
• Remark 1
• Theorem 4
• Theorem 5: Kuhn's theorem under one-sided min aggregator