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Equilibria of the Colonel Blotto Games with Costs

Stanisław Kaźmierowski

TL;DR

This work extends the Colonel Blotto framework by incorporating global obtainment costs and battlefield-specific costs, formalizing a discrete-resource, multi-battlefield model. It establishes a strategic equivalence between the Blotto game with costs and a zero-sum Blotto variant with one extra battlefield, enabling a polynomial-time computation of Nash equilibria via a linear program based on the BDDHS approach. Key results include the interchangeability of Nash equilibria and the convexity of equilibrium sets, plus a constructive reduction to the sunk-cost Blotto problem with a proven LP size bound of $\Theta((D^{\max})^2 \cdot (n+1))$. Experimental results corroborate the theory, showing how equilibrium resource allocation responds to cost structures and demonstrating practical runtimes. Overall, the paper provides a principled, scalable method for equilibrium analysis in cost-augmented Blotto games with broad applicability to strategic resource planning under diverse cost regimes.

Abstract

This paper studies a generalized variant of the Colonel Blotto game, referred to as the Colonel Blotto game with costs. Unlike the classic Colonel Blotto game, which imposes the use-it-or-lose-it budget assumption, the Colonel Blotto game with costs captures the strategic importance of costs related both to obtaining resources and assigning them across battlefields. We show that every instance of the Colonel Blotto game with costs is strategically equivalent to an instance of the zero-sum Colonel Blotto game with one additional battlefield. This enables the computation of Nash equilibria of the Colonel Blotto game with costs in polynomial time with respect to the game parameters: the number of battlefields and the number of resources available to the players.

Equilibria of the Colonel Blotto Games with Costs

TL;DR

This work extends the Colonel Blotto framework by incorporating global obtainment costs and battlefield-specific costs, formalizing a discrete-resource, multi-battlefield model. It establishes a strategic equivalence between the Blotto game with costs and a zero-sum Blotto variant with one extra battlefield, enabling a polynomial-time computation of Nash equilibria via a linear program based on the BDDHS approach. Key results include the interchangeability of Nash equilibria and the convexity of equilibrium sets, plus a constructive reduction to the sunk-cost Blotto problem with a proven LP size bound of . Experimental results corroborate the theory, showing how equilibrium resource allocation responds to cost structures and demonstrating practical runtimes. Overall, the paper provides a principled, scalable method for equilibrium analysis in cost-augmented Blotto games with broad applicability to strategic resource planning under diverse cost regimes.

Abstract

This paper studies a generalized variant of the Colonel Blotto game, referred to as the Colonel Blotto game with costs. Unlike the classic Colonel Blotto game, which imposes the use-it-or-lose-it budget assumption, the Colonel Blotto game with costs captures the strategic importance of costs related both to obtaining resources and assigning them across battlefields. We show that every instance of the Colonel Blotto game with costs is strategically equivalent to an instance of the zero-sum Colonel Blotto game with one additional battlefield. This enables the computation of Nash equilibria of the Colonel Blotto game with costs in polynomial time with respect to the game parameters: the number of battlefields and the number of resources available to the players.

Paper Structure

This paper contains 13 sections, 6 theorems, 19 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Colonel Blotto Game
  3. Our contribution:
  4. Model
  5. Analysis
  6. Equilibria Properties
  7. Equilibria Computation
  8. Colonel Blotto Game with Sunk Costs
  9. Expressing $\mathcal{B}_{0}$ as $\mathcal{B}$
  10. Computational Complexity
  11. Experimental Results
  12. Conclusions
  13. Acknowledgments

Key Result

Proposition 1

For the same set of parameters $(D^\mathsf{A}, D^\mathsf{B}, n, \bm{v}, \bm{c}^{\mathsf{A}},\bm{c}^{\mathsf{B}}, g^{\mathsf{A}}, g^{\mathsf{B}})$, games $\mathcal{B}_{\$}$ and $\mathcal{B}_{0}$ have the same set of Nash equilibria.

Figures (5)

  • Figure 1: The payoff matrix of game $\mathcal{B}_{\$}^*$.
  • Figure 2: The range of parameters considered in the experiments aimed to test Hypothesis \ref{['obs:structure']}. The min (max) row corresponds to the lower (upper) bound on the considered parameter, and the interval row describes the difference between the two consecutive values of the parameter.
  • Figure 3: The $y$-axis is the number of resources used in equilibrium and the $x$-axis is the inverse $c_0^{\shortminus 1}$ of the unit costs, $c_0$. The considered number of available resources of both players is $D^{\mathsf{A}} = 40$ and players compete on $n=4$ battlefields.
  • Figure 4: The $y$-axis is the cost carried in an equilibrium and the $x$-axis is the inverse $c_0^{\shortminus 1}$ of the unit costs, $c_0$.
  • Figure 5: The averages of 10 running times of our implementation of Algorithm \ref{['alg:lp_0']}. The fourth column ($\hat{\pi}$) presents the running times for models with linear obtainment costs, while the fifth column ($\bar{\pi}$) presents the running times for models with squared assignment costs.

Theorems & Definitions (12)

  • Definition 1: Colonel Blotto game with costs
  • Proposition 1
  • proof
  • Corollary 1
  • Corollary 2
  • Example 1
  • Definition 2
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 2 more