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General Lotto Games with Scouts: Information versus Strength

Jan-Tino Brethouwer, Bart van Ginkel, Roy Lindelauf

TL;DR

The paper studies General Lotto games with Scouts (GL-S), a Blotto-style resource-allocation problem where Blue may learn Red's allocation with probability $u\in[0,1]$, and derives exact equilibria for a single field across three regimes defined by the budget ratio $B/R$. It then extends to a multistage, multi-field setting (GL-MS), establishing upper and lower bounds on the game's value via convex-concave envelope techniques and showing conditions under which these bounds coincide. The authors introduce quantitative measures of information versus strength, including an influence ratio and contour-based budget guidelines, to address the weapons-mix problem of optimally trading off scouting information against combat power. The results yield practical insights into when information is valuable, how to structure efficient scout-based strategies, and how to allocate budgets between information and assets in strategic, military-like settings.

Abstract

We introduce General Lotto games with Scouts: a General Lotto game with asymmetric information. There are two players, Red and Blue, who both allocate resources to a field. However, scouting capabilities afford Blue to gain information, with some probability, on the number of Red's resources before allocating his own. We derive optimal strategies for this game in the case of a single field. In addition we provide upper and lower bounds of the value of the game in a multi-stage case with multiple battlefields. We devise several ways to characterise the influence of information versus strength. We conclude by drawing qualitative insights from these characterisations and the game values, and draw parallels with military practice.

General Lotto Games with Scouts: Information versus Strength

TL;DR

The paper studies General Lotto games with Scouts (GL-S), a Blotto-style resource-allocation problem where Blue may learn Red's allocation with probability , and derives exact equilibria for a single field across three regimes defined by the budget ratio . It then extends to a multistage, multi-field setting (GL-MS), establishing upper and lower bounds on the game's value via convex-concave envelope techniques and showing conditions under which these bounds coincide. The authors introduce quantitative measures of information versus strength, including an influence ratio and contour-based budget guidelines, to address the weapons-mix problem of optimally trading off scouting information against combat power. The results yield practical insights into when information is valuable, how to structure efficient scout-based strategies, and how to allocate budgets between information and assets in strategic, military-like settings.

Abstract

We introduce General Lotto games with Scouts: a General Lotto game with asymmetric information. There are two players, Red and Blue, who both allocate resources to a field. However, scouting capabilities afford Blue to gain information, with some probability, on the number of Red's resources before allocating his own. We derive optimal strategies for this game in the case of a single field. In addition we provide upper and lower bounds of the value of the game in a multi-stage case with multiple battlefields. We devise several ways to characterise the influence of information versus strength. We conclude by drawing qualitative insights from these characterisations and the game values, and draw parallels with military practice.
Paper Structure (27 sections, 7 theorems, 77 equations, 9 figures, 2 tables)

This paper contains 27 sections, 7 theorems, 77 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model, notation and assumptions
  3. General Lotto
  4. General Lotto with Scouts
  5. General Lotto with Scouts: Single Field
  6. Case 1: GL-S with 1 < B/R
  7. Case 2: GL-S with u < B/R < 1
  8. Case 3: GL-S with B/R < u
  9. Solution of GL-S
  10. General Lotto with Scouts: Multistage
  11. The multistage game
  12. Existence of Game Value
  13. Upper bound on the game value
  14. Lower bound on the game value
  15. New payoff function
  16. ...and 12 more sections

Key Result

Theorem 1

Let $R \geq B > 0$. The value $V$ of the General Lotto game is and the unique optimal strategies are $X^*$ for Red and $Y^*$ for Blue, where

Figures (9)

  • Figure 1: The value of $GL\text{-}S$ for fixed detection probability $u$ as a function of $B/R$.
  • Figure 2: The value of $GL\text{-}S$ for fixed $B/R$ ratios as a function of the detection probability $u$.
  • Figure 3: A heatmap of the value of $GL\text{-}S$, with the resource ratio $B/R$ on the x-axis and the detection probability $u$ on the y-axis.
  • Figure 4: The function $\psi_i$ and its convex envelope $\psi_i^\dag$ for two different $u_i$, both with $w_i = 1$.
  • Figure 5: The function $\phi_i$ and the transformed convex envelope $\phi_i^\dag$ for two different $u_i$, both with $w_i = 1$.
  • ...and 4 more figures

Theorems & Definitions (15)

  • Theorem 1: Hart
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • Remark 1: Comparison with known results when $u=0$
  • Remark 2: Comparison with known results when $u=1$
  • ...and 5 more