Battlefield Transfers in Coalitional Blotto Games

TL;DR

A new method of alliance formation referred to as a joint transfer, whereby players publicly transfer battlefields and budgets between one another before they engage in their separate competitions against the adversary is studied.

Abstract

In competitive resource allocation environments, agents often choose to form alliances; however, for some agents, doing so may not always be beneficial. Is there a method of forming alliances that always reward each of their members? We study this question using the framework of the coalitional Blotto game, in which two players compete against a common adversary by allocating their budgeted resources across disjoint sets of valued battlefields. On any given battlefield, the agent that allocates a greater amount of resources wins the corresponding battlefield value. Existing work has shown the surprising result that in certain game instances, if one player donates a portion of their budget to the other player, then both players win larger amounts in their separate competitions against the adversary. However, this transfer-based method of alliance formation is not always mutually beneficial, which motivates the search for alternate strategies. In this vein, we study a new method of alliance formation referred to as a joint transfer, whereby players publicly transfer battlefields and budgets between one another before they engage in their separate competitions against the adversary. We show that in almost all game instances, there exists a mutually beneficial joint transfer that strictly increases the payoff of each player.

Figures (5)

• Figure 1: Coalitional Colonel Blotto game depiction. The items enclosed by the dashed box depict a standard Colonel Blotto game between Player 1 and the adversary.
• Figure 2: The stages of the coalitional Blotto game. In Stage 0, the game is initialized. In Stage 1, the two players publicly perform mutually beneficial transfers. In Stage 2, the adversary determines how to optimally allocate their budget to each competition. In Stage 3, the two disjoint Blotto games are played.
• Figure 3: Cartoon depictions of games $G^1$ and $G^2$, along with the change in payoff of each player as a function of the budgetary transfer for each game.
• Figure 4: For a game with initial battlefield valuations $\phi_1 = 1.2$ and $\phi_2 = 1$ (depicted on the left), plots of the subsets of the parameter space in which mutually beneficial joint transfers (middle) and budgetary transfers (right) exist. The budgets of Players 1 and 2 represent the horizontal and vertical axes, respectively, of each of the plots. The shaded regions indicate where there exists a mutually beneficial transfer. Note that the subsets are depicted only for parameters satisfying $\frac{\phi_1}{\phi_2} \geq \frac{X_1}{X_2}$ to avoid redundancy.
• Figure 5: For game $G^2$, the change in payoff of each player as a function of the battlefield transfer (left) and budgetary transfer when $\tau^v = 0.1$ (right). The change in payoff in each plot is shown with respect to the payoff in the original game $G^2$.

Theorems & Definitions (1)

• Theorem