[Paper Review] Approximate Equilibria in Generalized Colonel Blotto and Generalized Lottery Blotto Games

Dong Quan Vu, Patrick Loiseau+1 more|arXiv (Cornell University)|Oct 15, 2019

Experimental Behavioral Economics Studies4 citations

TL;DR

This paper introduces independently uniform (IU) strategies as approximate equilibria in the Generalized Colonel Blotto (GCB) and Generalized Lottery Blotto (GLB) games, where players have asymmetric budgets and battlefields have heterogeneous values. It proves that these strategies yield $ar{ ho}W$-equilibria with approximation error bounded by $\mathcal{O}(nR^{-1}\ln(\varepsilon^{-1}))$, providing a tractable solution for complex, asymmetric resource allocation games.

ABSTRACT

In the Colonel Blotto game, two players with a fixed budget simultaneously allocate their resources across n battlefields to maximize the aggregate value gained from the battlefields where they have the higher allocation. Despite its long-standing history and important applications, the Colonel Blotto game still lacks a complete Nash equilibrium characterization in its most general form where players are asymmetric and battlefields' values are heterogeneous across battlefields and different between the two players---this is called the Generalized Colonel Blotto game. In this work, we propose a simply-constructed class of strategies---the independently uniform strategies---and we prove that they are approximate equilibria of the Generalized Colonel Blotto game; moreover, we characterize the approximation error according to the game's parameters. We also consider an extension called the Generalized Lottery Blotto game, with stochastic winner-determination rules allowing more flexibility in modeling practical contests. We prove that the proposed strategies are also approximate equilibria of the Generalized Lottery Blotto game.

Motivation & Objective

• To address the lack of equilibrium characterization in the Generalized Colonel Blotto (GCB) game, where player budgets and battlefield values are asymmetric and heterogeneous.
• To extend equilibrium analysis to the Generalized Lottery Blotto (GLB) game, which allows stochastic winner-determination rules for greater modeling flexibility.
• To provide a tractable, simply-constructed class of strategies that yield approximate equilibria with quantifiable performance guarantees.
• To characterize the approximation error in terms of game parameters such as budget asymmetry, number of battlefields, and strategy parameters.

Proposed method

• Introduces independently uniform (IU) strategies, where each player's allocation across battlefields is drawn independently from a uniform distribution over a bounded interval.
• Analyzes the expected payoff deviation under unilateral deviation, bounding the maximum gain a player can achieve by deviating from the IU strategy.
• Uses concentration inequalities and tail bounds to quantify the measure of allocations that yield significantly higher payoffs than expected under the IU strategy.
• Applies a coupling argument to relate the joint distribution of allocations to marginal distributions, ensuring budget constraints are approximately satisfied.
• Derives explicit bounds on the approximation error $\delta_{\mu}$ and $\delta_{\nu}$ in terms of $n$, $R$, and $\varepsilon$, using logarithmic and linear scaling in the strategy parameter $R$.
• Establishes that for sufficiently large $n$ and $R$, the IU strategy yields an $\bar{\rho}W$-equilibrium, where $\bar{\rho} = \mathcal{O}(nR^{-1}\ln(\varepsilon^{-1}))$.

Experimental results

Research questions

• RQ1 Can a simple, tractable strategy class serve as an approximate equilibrium in the most general form of the Colonel Blotto game with asymmetric players and heterogeneous battlefield values?
• RQ2 What is the quantifiable approximation error of such strategies in terms of game parameters like budget asymmetry, number of battlefields, and strategy resolution?
• RQ3 Does the same strategy class remain an approximate equilibrium in the generalized Lottery Blotto game with stochastic winner-determination?
• RQ4 How large must the number of battlefields and the strategy parameter $R$ be to ensure a desired level of approximation accuracy?

Key findings

• The independently uniform (IU) strategy is an $\mathcal{O}(nR^{-1}\ln(\varepsilon^{-1}))W$-equilibrium in the Generalized Colonel Blotto game, where $W$ is the maximum possible payoff.
• For any $\varepsilon > 0$, there exists a threshold $n$ and $R$ such that the IU strategy yields an $\bar{\rho}W$-equilibrium with $\bar{\rho} = \mathcal{O}(nR^{-1}\ln(\varepsilon^{-1}))$.
• The approximation error $\delta_{\mu}$ and $\delta_{\nu}$, which bound the maximum payoff gain from unilateral deviation, are both $\mathcal{O}(nR^{-1}\ln(\varepsilon^{-1}))$.
• When $n \geq \tilde{L}\bar{\varepsilon}^{-2}\max\{1,\ln(\bar{\varepsilon}^{-1})\}$ and $R \geq \mathcal{O}(n\bar{\varepsilon}^{-1}\ln(\bar{\varepsilon}^{-1}))$, the IU strategy is an $\bar{\rho}W$-equilibrium with $\bar{\rho} = \bar{\varepsilon}W$.
• The results extend to the Generalized Lottery Blotto game, where the same IU strategy achieves the same approximation guarantee under the same parameter conditions.