Alliance Mechanisms in General Lotto Games

TL;DR

This paper compares three alliance mechanisms—budget transfers ($\tau$), contest transfers ($\nu$), and joint transfers ($(\tau,\nu)$)—within the Coalitional General Lotto framework. It shows that mutual improvement opportunities differ across mechanisms: budget and contest transfers yield mutual gains only in limited regions, while joint transfers accomplish mutual gains in almost all game instances; by contrast, collectively beneficial alliances yield the same maximum total payoff regardless of mechanism. Across all mechanisms, the maximum collective payoff is identical, indicating equivalence for collective improvement. The findings imply that the choice of alliance mechanism should be guided by the alliance objective: joint transfers dominate for mutual improvement, whereas any mechanism suffices for maximizing total payoff.

Abstract

How do different alliance mechanisms compare? In this work, we analyze various methods of forming an alliance in the Coalitional General Lotto game, a simple model of competitive resource allocation. In the game, Players 1 and 2 independently compete against a common Adversary by allocating their limited resource budgets towards separate sets of contests; an agent wins a contest by allocating more resources towards it than their opponent. In this setting, we study three alliance mechanisms: budget transfers (resource donation), contest transfers (contest redistribution), and joint transfers (both simultaneously). For all three mechanisms, we study when they present opportunities for collective improvement (the sum of the Players' payoffs increases) or mutual improvement (both Players' individual payoffs increase). In our first result, we show that all three are fundamentally different with regards to mutual improvement; in particular, mutually beneficial budget and contest transfers exist in distinct, limited subsets of games, whereas mutually beneficial joint transfers exist in almost all games. However, in our second result, we demonstrate that all three mechanisms are equivalent when it comes to collective improvement; that is, collectively beneficial budget, contest, and joint transfers exist in almost all game instances, and all three mechanisms achieve the same maximum collective payoff. Together, these results demonstrate that differences between mechanisms depend fundamentally on the objective of the alliance.

Figures (6)

• Figure 1: An example of a Coalitional General Lotto Game. Players 1 and 2 are equipped with budgets $X_1 = 0.4$ and $X_2 = 1.6$, respectively, and compete against the Adversary across sets of contests with cumulative valuation $4 + 4 + 4 = 12$ and $4 + 3 + 3 = 10$; we summarize these game parameters with the tuple $(12, 10, 0.4, 1.6)$. Throughout the text, we refer to this example using the symbol $\mathbin{\color{myorange}\blacklozenge}$.
• Figure 2: The stages of the Coalitional General Lotto game. In Stage 0, the game is initialized. In Stage 1, the two Players may form an alliance through a budget transfer, contest transfer, or both. In Stage 2, the Adversary determines how to optimally split their budget between the two standard General Lotto games. In Stage 3, the agents allocate their budgets and receive their payoffs.
• Figure 3: The regions in which mutually beneficial budget (left), contest (center), and joint transfers (right) exist, shown as grey shaded regions, plotted in the $X_1$-$X_2$ space for fixed $\phi_1 = 12$, $\phi_2 = 10$. Only games where $X_1 / \phi_1 \leq X_2 / \phi_2$ are plotted to avoid redundancy. The lines in the rightmost plot depict the measure-zero subset $\mathbf{G} \setminus \mathbf{G}^{\tau, \nu}$, and the game depicted in Figure \ref{['fig:simple_example']} is indicated using $\mathbin{\color{myorange}\blacklozenge}$. Observe that there are some games (e.g., $\mathbin{\color{myorange}\blacklozenge}$) that belong to both $\mathbf{G}^\tau$ and $\mathbf{G}^\nu$, but there are also games for which there exist mutually beneficial contest transfers, but not budget transfers (e.g., any point in $\mathbf{G}^\nu$ with $X_1 > 1$ in the center plot), and vice versa.
• Figure 4: Left: an example of a Coalitional General Lotto game ($\mathbin{\color{myorange}\blacklozenge}$); Player $1$ and $2$ have budgets of $X_1 = 0.4$ and $X_2 = 1.6$ and compete for contests with valuations $12$ and $10$, respectively. The Players either perform a budget transfer $\tau = -0.69$ (mustard) or a contest transfer $\nu = 7.6$ (magenta). Right: the Players' collective payoff as a function of the transfer amount $\nu$, $\tau$. The symbol $\mathbin{\color{myorange}\blacklozenge}$ shows the collective payoff for the nominal setting with no transfers, i.e., $(\tau, \nu) = (0, 0)$. Observe that whether the Players perform a budget transfer $\tau = -0.69$ (mustard) or a contest transfer $\nu = 7.6$ (magenta), they achieve the same maximum collective payoff (dashed black line).
• Figure 5: The Cases that delineate the form of the Adversary's best response, plotted in the $X_1$-$X_2$ space for fixed $\phi_1 = 12$, $\phi_2 = 10$. The game $G^1 = (12, 10, 0.4, 1.6)$ (Figure \ref{['fig:simple_example']}) is indicated by $\mathbin{\color{myorange}\blacklozenge}$.

Theorems & Definitions (8)

• Corollary 1: Kovenock and Roberson, 2021
• Definition 1: Types of transfers
• Definition 2: Mutually beneficial transfers
• Theorem 1: Properties of $\mathbf{G}^\tau$, $\mathbf{G}^\nu$, and $\mathbf{G}^{\tau, \nu}$
• proof
• Definition 3: Collectively beneficial transfers
• Theorem 2: Equivalence of transfer types
• proof