Allocation of Heterogeneous Resources in General Lotto Games

TL;DR

This work extends the General Lotto game to multiple heterogeneous resource types and derives full equilibrium characterizations under two primary winning rules: the weakest-link/best-shot pair and a weighted-contribution variant. It shows that, under strongest forms of competition, equilibrium payoffs depend on budget ratios across types via a function $L(\alpha)$ with $\alpha = \sum_t Y_t/X_t$ or on a weighted budget ratio $\beta$, and it provides explicit mixed strategies for both players. The paper also analyzes a two-stage, costed setting ML-C, deriving exact equilibrium investments across resource types and revealing how total expenditure remains equal despite differing type composition; a sunk-cost variant ML-M is discussed. Finally, a rigorous proof framework introduces tractable strategy classes and tight bounds that establish the equilibrium in the multi-resource WL setting and extend to WC with analogous structure, offering actionable insights for the allocation of heterogeneous assets in adversarial environments.

Abstract

The allocation of resources plays an important role in the completion of system objectives and tasks, especially in the presence of strategic adversaries. Optimal allocation strategies are becoming increasingly more complex, given that multiple heterogeneous types of resources are at a system planner's disposal. In this paper, we focus on deriving optimal strategies for the allocation of heterogeneous resources in a well-known competitive resource allocation model known as the General Lotto game. In standard formulations, outcomes are determined solely by the players' allocation strategies of a common, single type of resource across multiple contests. In particular, a player wins a contest if it sends more resources than the opponent. Here, we propose a multi-resource extension where the winner of a contest is now determined not only by the amount of resources allocated, but also by the composition of resource types that are allocated. We completely characterize the equilibrium payoffs and strategies for two distinct formulations. The first consists of a weakest-link/best-shot winning rule, and the second considers a winning rule based on a weighted linear combination of the allocated resources. We then consider a scenario where the resource types are costly to purchase, and derive the players' equilibrium investments in each of the resource types.

Figures (3)

• Figure 1: (Left) The classic General Lotto game, wherein a single resource type (e.g. money) is allocated to multiple simultaneous contests. Player $\mathcal{X}$ has a budget of $X \geq 0$ resources, and player $\mathcal{Y}$ has a budget of $Y\geq 0$ resources. Success on a contest here simply depends on sending more resources than the opponent. (Right) The multi-resource General Lotto game. There are multiple resource types available to allocate (e.g. money, advertising, and human resources), where player $\mathcal{X}$ has budget $X_1\geq 0$ of type 1 resources, $X_2 \geq 0$ of type 2 resources, and so on. The success on each contest is now determined by a winning rule that depends on the combined allocation of resource types from both players. Our main contributions characterize equilibrium payoffs and strategies for two types of winning rules. Theorem \ref{['thm:WL']} considers the weakest-link rule, wherein player $\mathcal{X}$ needs to allocate more of every resource type to win a contest, whereas player $\mathcal{Y}$ only needs to allocate more of only a single type of resource. Proposition \ref{['thm:WC']} considers a setting where each resource type has an associated weight, or effectiveness. A player wins the contest if the aggregate weighted amount of resources exceeds that of the opponent.
• Figure 2: (Left) This plot shows the equilibrium payoff to player $\mathcal{X}$ in the classic single-resource General Lotto game (Theorem \ref{['thm:GL']}), as a function of its resource budget $X \geq 0$. In this plot, we fix player $\mathcal{Y}$'s budget $Y=1$. Note that for any given performance level, e.g. a payoff of 0.1, there is a unique budget for $\mathcal{X}$ that achieves this performance level. (Center) This plot shows the equilibrium payoff to player $\mathcal{X}$ in the two-resource General Lotto game with the weakest-link winning rule (Theorem \ref{['thm:WL']}). For a fixed performance level, e.g. 0.1, there is now a contour of budget pairs $(X_1,X_2)$ that achieves this payoff. Here, we fix budgets $Y_1 = Y_2 = 1$. (Right) Depiction of the two-resource game from the center figure.
• Figure 3: These plots illustrate the equilibrium properties of the ML-C game established in Theorem \ref{['thm:investment']} through a simulation example with three resource types. We fix costs $\boldsymbol{\sigma} = (3, 2, 1.8)$ for player $\mathcal{Y}$, and vary the cost of only resource 1 for player $\mathcal{X}$: $\boldsymbol{\sigma} = (\kappa_1, 0.2, 0.3)$, with $\kappa_1 \geq 0$ as the x-axis for all plots above. (Top Left) The players' total resource investment in equilibrium, denoting $X_{\text{tot}}^* = \sum_{t=1}^3 X_t^*$ and $Y_{\text{tot}}^* = \sum_{t=1}^3 Y_t^*$. Interestingly, player $\mathcal{X}$ does not change its total resource investment $X_{\text{tot}}^*$ for low costs, $\kappa < 2.2$. The dashed line $M^*=\sum_{t=1}^3 \kappa_t X_t^* = \sum_{t=1}^3 \sigma_t Y_t^*$ denotes the money spent by each player, which is the same for both. (Top Right) Equilibrium payoffs to both players. Increasing $\kappa_1$ linearly increases the ratio $r=\sum_{t\in\mathcal{T}} \frac{\kappa_t}{\sigma_t}$, which determines the equilibrium payoff. In this example $r = 1$ when $\kappa_1 = 2.2$. (Bottom Left) Player $\mathcal{Y}$'s investment fractions in the three resource types. As resource 1 becomes more expensive for player $\mathcal{X}$, player $\mathcal{Y}$ takes advantage by investing more into resource 1. Here, we denote $\bar{Y}_i^* = Y_i^*/Y^*_{tot}$. (Bottom Right) Player $\mathcal{Y}$'s investment fractions in the three resource types. These fractions remain constant since they only depend on the cost parameters $\boldsymbol{\sigma}$ of player $\mathcal{Y}$.

Theorems & Definitions (21)

• Definition 1
• Theorem 2.1: Adapted from Hart_2008
• Theorem 3.1
• Theorem 4.1
• Lemma 4.1
• proof : Proof of Theorem \ref{['thm:investment']}
• Proposition 5.1
• Lemma 6.1
• Definition 2
• Definition 3