A Dynamic Markov Reformulation of the Colonel Blotto Game Under Terminal Payoffs

The Colonel Blotto game is a classical model of competitive resource allocation, but equilibrium computation becomes difficult in heterogeneous and asymmetric instances. In this research, we study a finite-horizon dynamic reformulation in which players allocate resources sequentially over publicly observed stages, while the payoff depends only on terminal cumulative allocations. The purpose of the reformulation is not to change the primitive objective, but to represent the same terminal-payoff problem as a zero-sum Markov game. We first show that the dynamic formulation admits a pathwise payoff-equivalent Markov representation through telescoping rewards. Under a known finite horizon, costless carryover, and terminal-only payoff evaluation, the dynamic game and the corresponding static Blotto game have the same minimax value at every reachable continuation state. This is a value-equivalence result; it does not imply a one-to-one correspondence between static and dynamic equilibrium strategy sets. The proof is based on terminal-deferral upper and lower bounds for the two players. We also study action-independent geometric termination, for which the discounted telescoping return coincides exactly with the expected stopped terminal payoff, and we provide a probability-controlled mismatch bound for truncated stopping rules. Numerical finite-grid experiments illustrate the value identity and report residual diagnostics. The results clarify when sequential Markov representations preserve the original Blotto objective and when additional primitives, such as carryover depreciation or primitive flow payoffs, require separate analysis.