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Blotto on the Ballot: A Ballot Stuffing Blotto Game

Harsh Shah, Jayakrishnan Nair, D Manjunath, Narayan Mandayam

TL;DR

The paper tackles a zero-sum resource-allocation problem where a contestant may stuff ballots across $J$ stations, incurring convex costs $g_j(z_j)$ and facing $K$ inspectors deployed by an election-oversight agent. It models the Ballot Stuffing Game $\mathcal{B}(g,G,K)$, analyzes a Stackelberg variant to derive a unique stuffing vector $z^*$ with a structured partition of battlefields and a key structure slope $\theta_z$, and then characterizes a Nash equilibrium via a mixed defender strategy $q^*$ supported on $K$-subsets of the critical set $A_{z^*}$. The paper also provides an efficient convex-optimization-based algorithm to compute $z^*$ and extends the approach to a parliamentary setting through a Gaussian-tail approximation, yielding practical insights for security and auditing in plebiscite-like and parliamentary contexts. The results illuminate how inspector deployment reshapes optimal stuffing and offer scalable methods for computing equilibria in multi-battlefield elections with adversarial stuffing dynamics.

Abstract

We consider the following Colonel Blotto game between parties $P_1$ and $P_A.$ $P_1$ deploys a non negative number of troops across $J$ battlefields, while $P_A$ chooses $K,$ $K < J,$ battlefields to remove all of $P_1$'s troops from the chosen battlefields. $P_1$ has the objective of maximizing the number of surviving troops while $P_A$ wants to minimize it. Drawing an analogy with ballot stuffing by a party contesting an election and the countermeasures by the Election Commission to negate that, we call this the Ballot Stuffing Game. For this zero-sum resource allocation game, we obtain the set of Nash equilibria as a solution to a convex combinatorial optimization problem. We analyze this optimization problem and obtain insights into the several non trivial features of the equilibrium behavior. These features in turn allows to describe the structure of the solutions and efficient algorithms to obtain then. The model is described as ballot stuffing game in a plebiscite but has applications in security and auditing games. The results are extended to a parliamentary election model. Numerical examples illustrate applications of the game.

Blotto on the Ballot: A Ballot Stuffing Blotto Game

TL;DR

The paper tackles a zero-sum resource-allocation problem where a contestant may stuff ballots across stations, incurring convex costs and facing inspectors deployed by an election-oversight agent. It models the Ballot Stuffing Game , analyzes a Stackelberg variant to derive a unique stuffing vector with a structured partition of battlefields and a key structure slope , and then characterizes a Nash equilibrium via a mixed defender strategy supported on -subsets of the critical set . The paper also provides an efficient convex-optimization-based algorithm to compute and extends the approach to a parliamentary setting through a Gaussian-tail approximation, yielding practical insights for security and auditing in plebiscite-like and parliamentary contexts. The results illuminate how inspector deployment reshapes optimal stuffing and offer scalable methods for computing equilibria in multi-battlefield elections with adversarial stuffing dynamics.

Abstract

We consider the following Colonel Blotto game between parties and deploys a non negative number of troops across battlefields, while chooses battlefields to remove all of 's troops from the chosen battlefields. has the objective of maximizing the number of surviving troops while wants to minimize it. Drawing an analogy with ballot stuffing by a party contesting an election and the countermeasures by the Election Commission to negate that, we call this the Ballot Stuffing Game. For this zero-sum resource allocation game, we obtain the set of Nash equilibria as a solution to a convex combinatorial optimization problem. We analyze this optimization problem and obtain insights into the several non trivial features of the equilibrium behavior. These features in turn allows to describe the structure of the solutions and efficient algorithms to obtain then. The model is described as ballot stuffing game in a plebiscite but has applications in security and auditing games. The results are extended to a parliamentary election model. Numerical examples illustrate applications of the game.

Paper Structure

This paper contains 15 sections, 13 theorems, 45 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model and Notation
  3. The Ballot Stuffing Game
  4. Plebiscite Model
  5. A Stackelberg variation
  6. Nash equilibrium under the Plebiscite model
  7. Algorithm for computing $z^*$
  8. Parliamentary Model
  9. Numerical Experiments
  10. Discussion and Conclusion
  11. Technical Appendix
  12. Proof of Lemma \ref{['lemma:Stackelber1']}
  13. Proof of Lemma \ref{['lemma:q_existence']}
  14. Proof of Theorem \ref{['thm:Stackelberg-structure']}
  15. Proof of Lemma \ref{['lemma: PQerrors']}

Key Result

Lemma 1

The optimization eq:Stackelberg1 admits a unique optimal solution $z^*.$ Moreover, i.e., the set of indices of $z^*$ that have the maximum value has cardinality at least $K+1$ and cost of constraints hold with equality.

Figures (4)

  • Figure 1: Optimum stuffed votes $z^*$ as a function of budget $G$ for $K=0$ inspectors.
  • Figure 2: Equilibrium stuffed votes $z^*$ as a function of budget $G$ for $K=1$ inspector.
  • Figure 3: Equilibrium stuffed votes $z^*$ as a function of budget $G$ for $K=2$ inspectors.
  • Figure 4: Probability of $P_1$ winning the nationwide plebiscite as a function of $K$ when $G=8,000,000.$

Theorems & Definitions (23)

  • Definition 1
  • Lemma 1
  • Theorem 1: Structure of $z^*$
  • Lemma 2
  • Theorem 2: Nash Equilibrium Characterization
  • proof : Proof of Theorem \ref{['thm:ballot_NE']}
  • Lemma 3
  • Remark 1
  • Lemma 4
  • Lemma 5
  • ...and 13 more