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Stackelberg vs. Nash in the Lottery Colonel Blotto Game

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

TL;DR

This paper studies sequential resource competition in the Lottery Colonel Blotto game by modeling it as a Stackelberg game where the leader commits first and the follower best-responds. It develops a water-filling interpretation of the follower's BR, a battlefield-splitting/merging framework that reduces the BR support to at most $n$ options, and a univariate alpha-based method to compute the leader's optimal commitment in polynomial time. It provides necessary and sufficient conditions for SE to coincide with NE and characterizes how the leader's advantage depends on the budget ratio, including cases where the leader's gain can be arbitrarily large. The results offer practical and theoretical insights into first-mover advantages in Blotto-type contests and yield tractable algorithms for computing Stackelberg commitments under asymmetric valuations and budgets.

Abstract

Resource competition problems are often modeled using Colonel Blotto games, where players take simultaneous actions. However, many real-world scenarios involve sequential decision-making rather than simultaneous moves. To model these dynamics, we represent the Lottery Colonel Blotto game as a Stackelberg game, in which one player, the leader, commits to a strategy first, and the other player, the follower, responds. We derive the Stackelberg equilibrium for this game, formulating the leader's strategy as a bi-level optimization problem. To solve this, we develop a constructive method based on iterative game reductions, which allows us to efficiently compute the leader's optimal commitment strategy in polynomial time. Additionally, we identify the conditions under which the Stackelberg equilibrium coincides with the Nash equilibrium. Specifically, this occurs when the budget ratio between the leader and the follower equals a certain threshold, which we can calculate in closed form. In some instances, we observe that when the leader's budget exceeds this threshold, both players achieve higher utilities in the Stackelberg equilibrium compared to the Nash equilibrium. Lastly, we show that, in the best case, the leader can achieve an infinite utility improvement by making an optimal first move compared to the Nash equilibrium.

Stackelberg vs. Nash in the Lottery Colonel Blotto Game

TL;DR

This paper studies sequential resource competition in the Lottery Colonel Blotto game by modeling it as a Stackelberg game where the leader commits first and the follower best-responds. It develops a water-filling interpretation of the follower's BR, a battlefield-splitting/merging framework that reduces the BR support to at most options, and a univariate alpha-based method to compute the leader's optimal commitment in polynomial time. It provides necessary and sufficient conditions for SE to coincide with NE and characterizes how the leader's advantage depends on the budget ratio, including cases where the leader's gain can be arbitrarily large. The results offer practical and theoretical insights into first-mover advantages in Blotto-type contests and yield tractable algorithms for computing Stackelberg commitments under asymmetric valuations and budgets.

Abstract

Resource competition problems are often modeled using Colonel Blotto games, where players take simultaneous actions. However, many real-world scenarios involve sequential decision-making rather than simultaneous moves. To model these dynamics, we represent the Lottery Colonel Blotto game as a Stackelberg game, in which one player, the leader, commits to a strategy first, and the other player, the follower, responds. We derive the Stackelberg equilibrium for this game, formulating the leader's strategy as a bi-level optimization problem. To solve this, we develop a constructive method based on iterative game reductions, which allows us to efficiently compute the leader's optimal commitment strategy in polynomial time. Additionally, we identify the conditions under which the Stackelberg equilibrium coincides with the Nash equilibrium. Specifically, this occurs when the budget ratio between the leader and the follower equals a certain threshold, which we can calculate in closed form. In some instances, we observe that when the leader's budget exceeds this threshold, both players achieve higher utilities in the Stackelberg equilibrium compared to the Nash equilibrium. Lastly, we show that, in the best case, the leader can achieve an infinite utility improvement by making an optimal first move compared to the Nash equilibrium.

Paper Structure

This paper contains 13 sections, 17 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Preliminaries
  5. The Support of the Follower's Best Response Strategy
  6. The Leader's Optimal Commitment Strategy
  7. Stackelberg vs. Nash
  8. Leader's Advantage
  9. Conclusion and Future Work
  10. OMITTED PROOFS FROM SECTION 3
  11. OMITTED PROOFS FROM SECTION 4
  12. OMITTED PROOFS FROM SECTION 5
  13. OMITTED PROOFS FROM SECTION 6

Key Result

Lemma 1

Kovenock-2019 Given the leader's strategy $\bm{x}_a = (x_{aj})_{j = 1}^{n}$, where $x_{aj} > 0$, $\forall j \in [n]$, assume without loss of generality that all battlefields are ordered such that $\frac{v_{b1}}{x_{a1}} \geq \frac{v_{b2}}{x_{a2}} \geq \cdots \geq \frac{v_{bn}}{x_{an}}$. The unique op where $K(\bm{x}_a) = \{ 1, \cdots, k^* \}$ is such that

Figures (1)

  • Figure 1: Utility Curves in Two Equilibria.

Theorems & Definitions (33)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Lemma 4
  • Lemma 5
  • Theorem 3
  • Theorem 4
  • proof
  • ...and 23 more