Characterizing the interplay between information and strength in Blotto games

TL;DR

This work studies how informational asymmetry interacts with resource strength in competitive Blotto/Lotto games with stochastic battlefield valuations. Using a Bayesian framework, it derives unique equilibrium payoffs in representative two- and three-battlefield settings and quantifies the value of information as the payoff gap relative to complete information, leveraging a reduction to all-pay auctions via the Siegel_2014 algorithm. A key finding is that information strictly benefits the informed player in the Lotto setting, with the magnitude depending on budget ratio $\gamma$ and valuation dispersion (e.g., parameters $\alpha,\beta$); in the two-battlefield Blotto case, information fails to overturn the inherent disadvantage, though it increases the informed player's ex-ante payoff. The results illuminate the delicate balance between information and strength in adversarial environments and connect General Lotto equilibria to all-pay auction theory, offering a tractable path to analyze broader information structures in Blotto/Lotto contests.

Abstract

In this paper, we investigate informational asymmetries in the Colonel Blotto game, a game-theoretic model of competitive resource allocation between two players over a set of battlefields. The battlefield valuations are subject to randomness. One of the two players knows the valuations with certainty. The other knows only a distribution on the battlefield realizations. However, the informed player has fewer resources to allocate. We characterize unique equilibrium payoffs in a two battlefield setup of the Colonel Blotto game. We then focus on a three battlefield setup in the General Lotto game, a popular variant of the Colonel Blotto game. We characterize the unique equilibrium payoffs and mixed equilibrium strategies. We quantify the value of information - the difference in equilibrium payoff between the asymmetric information game and complete information game. We find information strictly improves the informed player's performance guarantee. However, the magnitude of improvement varies with the informed player's strength as well as the game parameters. Our analysis highlights the interplay between strength and information in adversarial environments.

Figures (2)

• Figure 1: (a) A diagram of the Colonel Blotto game with two battlefields and asymmetric budgets and information. There are two possible battlefield valuation sets, each realized with probability $\frac{1}{2}$. Player I has fewer resources than player U, but knows the true realization. (b) The equilibrium payoff \ref{['eq:blotto_payoff_midrange']} to the informed player in the Blotto game $\text{CB}(X_I,X_U,V,\frac{1}{2})$, where $\bar{v}=1$ and $\underline{v}=\alpha\in(0,1)$. It is negative when $\gamma \in (\frac{1}{2},1)$ and $\alpha \in (0,1)$. (c) The value of information, $\pi_I^* - \pi_I^{\text{GW}}$, is non-negative. Information offers the most improvement for low budget ($\gamma$ near $\frac{1}{2}$) and when there is higher priority in the diagonal battlefields ($\alpha$ low).
• Figure 2: (a) The equilibrium payoff $\pi_I^*$ to the informed player in the Lotto game $\text{GL}(X_I,X_U,V_{\alpha\beta},\frac{1}{3}\mathbb{1}_3)$, under the special case $\alpha = \beta$. The dashed line are the points $(\gamma,\alpha)$ for which $\pi_I^*=0$. That is, for parameters below the line, the informed player "wins" the game and "loses" for parameters above the line (see Corollary \ref{['lemma_zerocrossing']}). (b) The value of information, or payoff gain or loss to player $I$ from purchasing information at the cost $c_I = \frac{1}{5}$. The dashed black line in Figure \ref{['fig:lotto_VoI']} indicates the parameters for which $\text{VoI}(\alpha,\gamma) = 0$. (c) The maximal cost \ref{['eq:bVoI']}, or the largest fraction of its resource budget $\gamma$ that player $I$ is willing to give up in exchange for information before it experiences a payoff loss. Information is more valuable in lower $\alpha$ ranges - when the battlefield rewards are concentrated at diverse locations.

Theorems & Definitions (16)

• Proposition 1
• proof
• Theorem 1
• Corollary 1
• proof
• Theorem 2
• Corollary 2
• proof
• Corollary 3
• proof