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Multiplayer General Lotto game

Yan Liu, Bonan Ni, Weiran Shen, Zihe Wang, Jie Zhang

TL;DR

This work extends General Lotto to multiplayer contexts with asymmetric budgets and heterogeneous battlefields, establishing Nash equilibrium existence via a discretization-and-limiting framework and delivering a detailed single-battlefield characterization with uniqueness under a two-player-maximum-budget condition. It then analyzes multi-battlefield scenarios, showing that players concentrate on few battlefields as the number of players grows and that equilibria need not be unique in general, while providing symmetric-case closed-form equilibria. The methods combine discretized games, Kakutani fixed-point arguments, and Helly selection to bridge finite-grid equilibria to the continuous, unbounded GL setting. Overall, the paper significantly broadens the GL literature by addressing multiplayer dynamics, structural properties, and symmetric solutions in complex resource-allocation games with diverse budgets and values.

Abstract

In this paper, we investigate the multiplayer General Lotto game across multiple battlefields, a significant variant of the Colonel Blotto game. In this version, each player employs a probability distribution for resource allocation, ensuring that their expected expenditure does not exceed their budget. We first establish the existence of the Nash equilibrium in a general setting, where players' budgets are asymmetric and the values of the battlefields are heterogeneous and asymmetric among players. Next, we provide a detailed characterization of the Nash equilibrium for multiple players on a single battlefield. In this characterization, we observe that the upper endpoints of the supports of players' equilibrium strategies coincide, and that the minimum value of a player's support above zero inversely correlates with his budget. We demonstrate the uniqueness of Nash equilibrium over a single battlefield in some scenarios. In the multi-battlefield setting, we prove that there is an upper bound on the average number of battlefields each player participates in. Additionally, we provide an example demonstrating the non-uniqueness of the Nash equilibrium in the context of multiple battlefields with multiple players. Finally, we present a solution for the Nash equilibrium in a symmetric case.

Multiplayer General Lotto game

TL;DR

This work extends General Lotto to multiplayer contexts with asymmetric budgets and heterogeneous battlefields, establishing Nash equilibrium existence via a discretization-and-limiting framework and delivering a detailed single-battlefield characterization with uniqueness under a two-player-maximum-budget condition. It then analyzes multi-battlefield scenarios, showing that players concentrate on few battlefields as the number of players grows and that equilibria need not be unique in general, while providing symmetric-case closed-form equilibria. The methods combine discretized games, Kakutani fixed-point arguments, and Helly selection to bridge finite-grid equilibria to the continuous, unbounded GL setting. Overall, the paper significantly broadens the GL literature by addressing multiplayer dynamics, structural properties, and symmetric solutions in complex resource-allocation games with diverse budgets and values.

Abstract

In this paper, we investigate the multiplayer General Lotto game across multiple battlefields, a significant variant of the Colonel Blotto game. In this version, each player employs a probability distribution for resource allocation, ensuring that their expected expenditure does not exceed their budget. We first establish the existence of the Nash equilibrium in a general setting, where players' budgets are asymmetric and the values of the battlefields are heterogeneous and asymmetric among players. Next, we provide a detailed characterization of the Nash equilibrium for multiple players on a single battlefield. In this characterization, we observe that the upper endpoints of the supports of players' equilibrium strategies coincide, and that the minimum value of a player's support above zero inversely correlates with his budget. We demonstrate the uniqueness of Nash equilibrium over a single battlefield in some scenarios. In the multi-battlefield setting, we prove that there is an upper bound on the average number of battlefields each player participates in. Additionally, we provide an example demonstrating the non-uniqueness of the Nash equilibrium in the context of multiple battlefields with multiple players. Finally, we present a solution for the Nash equilibrium in a symmetric case.
Paper Structure (31 sections, 23 theorems, 56 equations, 1 figure)

This paper contains 31 sections, 23 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Preliminaries
  5. Existence of Nash Equilibrium in the General Lotto game
  6. Remark on the proof approach.
  7. Nash Equilibrium Characterization of A Single Battlefield
  8. Characterization of Nash Equilibrium
  9. The Uniqueness of the Nash Equilibrium
  10. Analysis of Nash Equilibrium with Multiple Battlefields
  11. Example.
  12. Conclusion
  13. Proofs for Section 2
  14. Proof of Lemma \ref{['lem-cont-linear']}
  15. Proof of Lemma \ref{['lem-one-pos-b']}
  16. ...and 16 more sections

Key Result

Lemma 1

If $(F_{i, j})_{i \in [n], j \in [m]}$ is a Nash equilibrium, then there exist constants $a_i > 0$ for every $i \in [n]$ and $b_{i,j} \ge 0$ for every $i \in [n], j \in [m]$ such that and for all $x \geq 0$, it holds that for every $i \in [n]$ and $j \in [m]$.

Figures (1)

  • Figure 1: The support of Nash equilibrium strategies, blue dot indicates mass point of distribution, black line represents support of distribution. The only difference between the left and right figures is that, in the left figure, player 2 has a mass point at $0$, whereas in the right figure, player 2 does not have a mass point at $0$.

Theorems & Definitions (44)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • ...and 34 more