Fast, optimal, and dynamic electoral campaign budgeting by a generalized Colonel Blotto game

TL;DR

It is demonstrated that the Colonel Blotto game can be a practical model for competitive allocation environments by implementing the multiplicative weights update algorithm from Beaglehole et al. (2023), and strategies for a more realistic model of political campaigning are studied, which are term Electoral Colonel Blotto.

Abstract

The Colonel Blotto game is a deeply studied theoretical model for competitive allocation environments including elections, advertising, and ecology. However, the original formulation of Colonel Blotto has had few practical implications due to the lack of fast algorithms to compute its optimal strategies and the limited applicability of its winner-take-all reward distribution. We demonstrate that the Colonel Blotto game can be a practical model for competitive allocation environments by implementing the multiplicative weights update algorithm from Beaglehole et al. (2023). In particular, using that this algorithm allows for arbitrary winning-rules, we study strategies for a more realistic model of political campaigning we term Electoral Colonel Blotto. Contrary to existing theory and the implemented allocation strategies from U.S. presidential elections, we find that the optimal response to Democratic and Republican strategies in the 2008 and 2020 presidential elections was to focus allocations on a subset of states and sacrifice winning probability on others. We also found that campaigners should compete for undecided voters even in states where the opponent has significantly many more decided voters.

Figures (14)

• Figure 1: Allocations from MWU for battles with symmetric values, and equal resource capacities for the two players. Columns indicate the choice of warm-start strategy for the historical loss matrix, and $I_0$ indicates the number of warm-start rounds. Bar heights represent the average number of soldiers allocated by the one such player under the $0/1$, Popular Vote (PV), and Electoral Vote (EV) winning rules. For the Popular Vote (PV) and Electoral Vote (EV) winning rules, the gray bars indicate the the theoretical equilibria proposed in three_halves. Battle values were chosen uniformly at random from $\{1,\ldots,100\}$.
• Figure 2: Allocations from MWU for battles with symmetric values, and equal resource capacities for the two players. Columns indicate the choice of warm-start strategy for the historical loss matrix, and $I_0$ indicates the number of warm-start rounds. Bar heights represent the average number of soldiers allocated by the one such player. For the Popular Vote (PV) and Electoral Vote (EV) winning rules, the gray bars indicate the the theoretical equilibria proposed in three_halves.
• Figure 3: Comparison of regret and average allocation when one player plays an integer approximation of the Three-Halves Allocation while the opponent plays the MWU allocation. We use the Electoral Vote winning rule with $I_0 = 0$ for the MWU allocation.
• Figure 4: Average allocations returned from the MWU algorithm using data from the 2008 presidential candidates. States presented are those which were listed as being viewed as battleground states by at least one of the two parties according to 2008_strategies. The fixed strategies represent the actual number of visits made by the ticket 2008_strategies. Electoral college votes for each state at the time of the election are in parenthesis.
• Figure 5: Resource allocations using data from the 2020 presidential candidates. Biden and Harris had $45$ total visits, while Trump and Pence had $61$. States presented are those which were visited at least twice by each ticket 2020_strategies. Electoral college votes for each state at the time of the election are in parenthesis.

Theorems & Definitions (4)

• Definition 1: Electoral Colonel Blotto
• Proposition 2
• Proposition 3: Fast MWU in Blotto gamesbeaglehole2022sampling
• proof : Proof of Proposition \ref{['prop:regret_opt']}