Double Oracle Algorithm for Game-Theoretic Robot Allocation on Graphs

TL;DR

This work models two players allocating multi-type robots on a directed graph to contest multiple sites, extending the Colonel Blotto framework to capture graph constraints and cyclic dominance among robot types. It develops a two-pronged approach: (i) a DOA-based method to compute Nash equilibria by iteratively solving a sequence of subgames with best-response MILPs, and (ii) a novel CDH-specific construction consisting of an elimination-based transformation, an outcome interface $\\pi_{oi}$, and a piecewise-linear utility $u_{CDH}$ that preserves a meaningful win/lose/draw demarcation. The authors prove the correctness of the CDH utility construction, linearize it for MILP-based best responses, and demonstrate convergence to equilibrium across homogeneous, linear heterogeneous, and CDH robot allocations on graphs. The findings enable tractable equilibrium computation in realistic, constrained, on-graph Blotto settings and offer insights into how graph structure and robot-type interactions influence optimal strategies. This has practical implications for coordinating competitive multi-robot deployments in environments with connectivity and heterogeneity constraints.

Abstract

We study the problem of game-theoretic robot allocation where two players strategically allocate robots to compete for multiple sites of interest. Robots possess offensive or defensive capabilities to interfere and weaken their opponents to take over a competing site. This problem belongs to the conventional Colonel Blotto Game. Considering the robots' heterogeneous capabilities and environmental factors, we generalize the conventional Blotto game by incorporating heterogeneous robot types and graph constraints that capture the robot transitions between sites. Then we employ the Double Oracle Algorithm (DOA) to solve for the Nash equilibrium of the generalized Blotto game. Particularly, for cyclic-dominance-heterogeneous (CDH) robots that inhibit each other, we define a new transformation rule between any two robot types. Building on the transformation, we design a novel utility function to measure the game's outcome quantitatively. Moreover, we rigorously prove the correctness of the designed utility function. Finally, we conduct extensive simulations to demonstrate the effectiveness of DOA on computing Nash equilibrium for homogeneous, linear heterogeneous, and CDH robot allocation on graphs.

Figures (12)

• Figure 1: Game-theoretic robot allocation on a graph. Two players ("red" and "blue") allocate three types of robots to compete for five sites in an area. The area is abstracted into a graph with five sites as the nodes and their connections as edges.
• Figure 2: Examples of allocation games with (a) homogeneous and (b) heterogeneous robots. The number of robots allocated on each node is demonstrated as the length of the color bar, with red representing Player 1, and blue representing Player 2. In (a), at time $t$ (left), Player 1 wins node 3 and Player 2 wins node 5. At time $t+1$ (right). Player 1 moves one robot from node 4 into node 1, and Player 2 moves one robot from node 3 to node 2. Thus Player 1 wins nodes 1 and 3 while Player 2 wins the remaining nodes. In (b), Player 1 adopts mixed strategy $\mathbf{\Delta}_X \sim \text{Binomial}(0.4, 0.6)$ and Player 2 adopts pure strategy $\mathbf{\Delta}_Y=\mathbf{S}_y^1$. The win or loss cannot be claimed solely based on the number of robots because of the CDH robots.
• Figure 3: The top panel (a) illustrates the composition of the function's definition space $\Gamma$, which is partitioned into three subspaces $Q_1$, $Q_2$, $Q_3$. In each subspace, the function $\pi_{\texttt{oi}}(\mathbf{x})$ is linear, that is, $\pi_{\texttt{oi}}(\mathbf{x}) = g_i(\mathbf{x})$ for $\mathbf{x} \in Q_i$ ($Q_i$ extending indefinitely outward into space). The plot on the right of (a) shows the positions where $g_i(\mathbf{x}) = 0$. (b) displays the relative positioning of the surface $\pi_{\texttt{oi}}(\mathbf{x}) = 0$ in the Cartesian coordinate system. The red half-plane is $g_1$ in \ref{['g_1']}, the green half-plane is $g_2$ in \ref{['g_2']}, and the yellow half-plane is $g_3$ in \ref{['g_3']}. These three half-planes together form the outcome interface, i.e., $\pi_{\texttt{oi}}(\mathbf{x}) = 0$.
• Figure 4: Illustration of utility function $u_{\texttt{CDH}}$. The surface $\pi_{\texttt{oi}}(\mathbf{S}_x-\mathbf{S}_y)$ in the middle is the outcome interface in Fig. \ref{['fig: PI']}.
• Figure 5: Three different graphs: $G_1$ (left), $G_2$ (middle), $G_3$ (right).

Theorems & Definitions (8)

• Lemma 1
• Lemma 2
• Definition 1: ovchinnikov2000max
• Theorem 1: ovchinnikov2000max
• Theorem 2
• Theorem 3
• Lemma 3: adam2021double
• Lemma 4