The Colonel Blotto game (Q2505532)

The author considers the following generalization of the classical \textit{Colonel Blotto game}: Two players \(1\) and \(2\), simultaneously allocate their forces of amount \(X_1\) and \(X_2\), respectively, across the finite number \(n\geq 3\) of homogeneous battlefields (\(X_1\) and \(X_2\) are positive real numbers). It is assumed that player \(i\) will capture a battlefield \(j\), \(1 \leq j\leq n\), when he allocates there more forces than player \(B\). Otherwise, player \(B\) will capture that battlefield. The payoff of a player amounts \(\frac{s}{n}\), where \(s\) is the number of captured battlefiels by him. The model described above is equivalent to the two-person constant-sum game with normal form \(\langle {\mathcal{B}}_1,{\mathcal{B}}_2, K_1, K_2\rangle\), where for \(i=1,2\), \({\mathcal{B}}_i = \{(x_1,\dots, x_n)\in \mathbb{R}^n_+: \sum_{j=1}^n x_j\leq X_i\}\) is the space of pure strategies of player \(i\), and \(K_i\) is his payoff function on \({\mathcal{B}}_1 \times {\mathcal{B}}\) defined above. For such a game, it is shown that there exists a Nash equilibrium \((\mu_1,\mu_2)\) in mixed strategies. In particular, the distribution functions \(\mu_i\) (defined over \({\mathcal{B}}_i\)) are explicitly constructed with the help of the so-called \(n\)-copulas (some special fuctions from \([0,1]^n\) to \([0,1]\)). The paper extends the literature on the Colonel Blotto game in several important ways.