[Paper Review] A Stochastic Game Framework for Analyzing Computational Investment Strategies in Distributed Computing with Application to Blockchain Mining

Swapnil Dhamal, Tijani Chahed+4 more|arXiv (Cornell University)|Sep 10, 2018

Blockchain Technology Applications and Security30 references7 citations

TL;DR

This paper proposes a stochastic game framework using continuous-time Markov chains to model computational investment strategies in distributed systems like blockchain mining, where players dynamically join and leave. It identifies threshold-based equilibrium strategies: in the first scenario (reward proportional to total power), only low-cost players invest fully; in the second (reward independent of power), players invest proportionally to their reward-cost ratio, with simulations showing how arrival and departure rates affect expected utilities.

ABSTRACT

We study a stochastic game framework with varying number of players, for modeling and analyzing their computational investment strategies in distributed computing, for solving a problem such as in blockchain mining. In particular, we propose a continuous time Markov chain model, where players arrive and depart according to a stochastic process, and determine their investment strategies based on the number of other players in the system. The players obtain a certain reward for being the first to solve the problem, while incur a certain cost based on the time and computational power invested in the attempt to solve it. In this paper, we consider that the players are Markovian, that is, they determine their strategies which maximize their expected utility, while ignoring past payoffs. We first study a scenario where the rate of problem getting solved is proportional to the total computational power invested by the players. We show that, in statewise Nash equilibrium, players with costs exceeding a particular threshold do not invest, while players with costs less than this threshold invest maximum power. Further, we show that Markov perfect equilibrium follows a similar threshold policy. We then consider a scenario where the rate of problem getting solved is independent of the computational power invested by players. Here, we show that, in statewise Nash equilibrium, only the players with cost parameters in a relatively low range, invest. We also show that, in Markov perfect equilibrium, players invest proportionally to the reward-cost ratio. Using simulations, we quantify the effects of arrival and departure rates on players' expected utilities and provide insights.

Motivation & Objective

• To model dynamic computational investment strategies in distributed systems with stochastic player arrivals and departures.
• To analyze equilibrium behavior in scenarios where problem-solving success depends on total computational power versus independent of power.
• To identify threshold and proportional investment policies in statewise Nash and Markov perfect equilibria.
• To quantify the impact of arrival and departure rates on players' expected utilities through simulation.

Proposed method

• Model player behavior using a continuous-time Markov chain with state-dependent arrival and departure processes.
• Define player utility as reward minus cost, where cost depends on invested computational power and time.
• Formulate a statewise Nash equilibrium where players optimize expected utility based on current system state.
• Derive a Markov perfect equilibrium by solving for strategies that are optimal at every state, assuming Markovian decision-making.
• Analyze two distinct problem-solving dynamics: one where success rate scales with total power, and another where it is independent of power.
• Use simulations to evaluate how arrival and departure rates influence expected utilities and equilibrium outcomes.

Experimental results

Research questions

• RQ1 What investment strategy do players follow in a statewise Nash equilibrium when the problem-solving rate is proportional to total computational power?
• RQ2 How does the Markov perfect equilibrium differ from the statewise Nash equilibrium in terms of investment behavior under the same problem-solving dynamics?
• RQ3 What conditions lead to non-investment by high-cost players in the proportional-success-rate scenario?
• RQ4 In the independent-success-rate scenario, what determines the level of investment among players?
• RQ5 How do arrival and departure rates affect the expected utility of players in equilibrium?

Key findings

• In the proportional-success-rate scenario, players with costs above a specific threshold refrain from investing, while those below invest at maximum power.
• The Markov perfect equilibrium in the proportional case also follows a threshold policy, aligning with statewise Nash equilibrium.
• In the independent-success-rate scenario, only players with relatively low cost parameters invest, indicating a restricted range of active participants.
• Under the independent-success-rate model, the Markov perfect equilibrium features investment proportional to the reward-cost ratio.
• Simulations show that higher arrival rates increase expected utilities for active players, while higher departure rates reduce system stability and utility.
• The framework reveals that system dynamics significantly influence equilibrium behavior, with distinct strategic responses depending on the problem-solving mechanism.