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An Equilibrium Analysis of the Arad-Rubinstein Game

Christian Ewerhart, Stanisław Kaźmierowski

Abstract

Colonel Blotto games with discrete strategy spaces effectively illustrate the intricate nature of multidimensional strategic reasoning. This paper studies the equilibrium set of such games where, in line with prior experimental work, the tie-breaking rule is allowed to be flexible. We begin by pointing out that equilibrium constructions known from the literature extend to our class of games. However, we also note that irrespective of the tie-breaking rule, the equilibrium set is excessively large. Specifically, any pure strategy that allocates at most twice the fair share to each battlefield is used with positive probability in some equilibrium. Furthermore, refinements based on the elimination of weakly dominated strategies prove ineffective. To derive specific predictions amid this multiplicity, we compute strategies resulting from long-run adaptive learning.

An Equilibrium Analysis of the Arad-Rubinstein Game

Abstract

Colonel Blotto games with discrete strategy spaces effectively illustrate the intricate nature of multidimensional strategic reasoning. This paper studies the equilibrium set of such games where, in line with prior experimental work, the tie-breaking rule is allowed to be flexible. We begin by pointing out that equilibrium constructions known from the literature extend to our class of games. However, we also note that irrespective of the tie-breaking rule, the equilibrium set is excessively large. Specifically, any pure strategy that allocates at most twice the fair share to each battlefield is used with positive probability in some equilibrium. Furthermore, refinements based on the elimination of weakly dominated strategies prove ineffective. To derive specific predictions amid this multiplicity, we compute strategies resulting from long-run adaptive learning.
Paper Structure (11 theorems, 17 equations, 1 figure)

This paper contains 11 theorems, 17 equations, 1 figure.

Key Result

Proposition 1

Impose Assumptions 1 through 3. Then, a symmetric equilibrium strategy of $\mathcal{B}_{\alpha}(N,K)$ is given by uniform randomization over the set of pure strategies In the resulting equilibrium, players' expected payoffs amount to $\pi^* = K\cdot\frac{m+ \frac{\alpha}{2}}{2m+ 1}$.

Figures (1)

  • Figure 1: Rank-order analysis

Theorems & Definitions (16)

  • Example 1: Hart, hart
  • Example 2: Arad and Rubinstein, arad_rubinstein
  • Proposition 1
  • Corollary 1: Equilibrium in the Arad-Rubinstein game
  • Proposition 2
  • Proposition 3
  • Corollary 2
  • Proposition 4
  • Corollary 3
  • Example 3
  • ...and 6 more