Inefficient Alliance Formation in Coalitional Blotto Games

TL;DR

This study investigates costly alliance formation in the framework of coalitional Blotto games, in which two players compete separately against a common adversary, and are able to collude by exchanging resources with one another.

Abstract

When multiple agents are engaged in a network of conflict, some can advance their competitive positions by forming alliances with each other. However, the costs associated with establishing an alliance may outweigh the potential benefits. This study investigates costly alliance formation in the framework of coalitional Blotto games, in which two players compete separately against a common adversary, and are able to collude by exchanging resources with one another. Previous work has shown that both players in the alliance can mutually benefit if one player unilaterally donates, or transfers, a portion of their budget to the other. In this letter, we consider a variation where the transfer of resources is inherently inefficient, meaning that the recipient of the transfer only receives a fraction of the donation. Our findings reveal that even in the presence of inefficiencies, mutually beneficial transfers are still possible. More formally, our main result provides necessary and sufficient conditions for the existence of such transfers, offering insights into the robustness of alliance formation in competitive environments with resource constraints.

Figures (5)

• Figure 1: A cartoon depiction of a coalitional Colonel Blotto game between Players 1 and 2 and a common adversary. Player 1, Player 2, and the adversary are equipped with budgets $0.5$, $1.5$, and $1$, respectively. Player 1 and the adversary compete on the left set of contests with cumulative value $1$, and Player 2 and the adversary compete on the right set of contests with cumulative value $1.2$.
• Figure 2: The stages of the coalitional Blotto game. In Stage 0, the game is initialized. In Stage 1, the two players perform mutually beneficial transfers. In Stage 2, the adversary decides how to divide their budget. In Stage 3, the two disjoint Blotto games are played.
• Figure 3: The change in payoff $\Delta U_i^{\rm NE}(\tau; G) \triangleq U_i^{\rm NE}(\tau; G) - U_i^{\rm NE}(\tau; 0)$ of Player $i \in \{1, 2 \}$ as a function of the budget transfer amount $\tau$ for the game $G^1 \triangleq (1, 1.2, 0.5, 1.5)$ shown in Figure \ref{['fig:coalitional_basic']}. When $\beta = 1$ (top), both players' payoffs increase for a range of budget transfers (grey shaded region). When $\beta = 0.5$ (bottom), both players' payoffs never increase simultaneously.
• Figure 4: When $\phi_1 = 1.2$ and $\phi_2 = 1$, plots of the subsets of the parameter space in which mutually beneficial budget transfers exist (top) and alliance optimal transfers are nonzero (bottom) for various values of $\beta$ (the colors and their corresponding values of $\beta$ are the same in each plot). The horizontal and vertical axes represent $X_1$ and $X_2$, respectively, and the black dot represents $G^1$. The shading indicates layers of regions of existence (e.g., when $\beta = 1$, transfers exist across all of the shaded regions). Only games where $\frac{\phi_2}{\phi_1} \leq \frac{X_2}{X_1}$ are depicted to avoid redundancy.
• Figure 5: The maximum values of $U_1^{\rm NE}$ (blue), $U_2^{\rm NE}$ (green), and $U_{1, 2}^{\rm NE}$ (orange) for the game $G^1$ (Figure \ref{['fig:coalitional_basic']}) for various values of $\beta$. The colored horizontal dashed lines depict the corresponding nominal value when no transfer occurs. The black vertical dashed line on the top left indicates the threshold for the existence of nonzero coalition optimal transfers; the one in the center indicates the threshold for the existence of mutually benefical transfers. Note that the maximizing transfer is chosen separately for each curve, subject to its respective constraints.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof