We consider a distributed computing setting wherein a central entity seeks power from computational providers by offering a certain reward in return. The computational providers are classified into long-term stakeholders that invest a constant amount of power over time and players that can strategize on their computational investment. In this paper, we model and analyze a stochastic game in such a distributed computing setting, wherein players arrive and depart over time. While our model is formulated with a focus on volunteer computing, it equally applies to certain other distributed computing applications such as mining in blockchain. We prove that, in Markov perfect equilibrium, only players with cost parameters in a relatively low range which collectively satisfy a certain constraint in a given state, invest. We infer that players need not have knowledge about the system state and other players' parameters, if the total power that is being received by the central entity is communicated to the players as part of the system's protocol. If players are homogeneous and the system consists of a reasonably large number of players, we observe that the total power received by the central entity is proportional to the offered reward and does not vary significantly despite the players' arrivals and departures, thus resulting in a robust and reliable system. We then study by way of simulations and mean field approximation, how the players' utilities are influenced by their arrival and departure rates as well as the system parameters such as the reward's amount and dispensing rate. We observe that the players' expected utilities are maximized when their arrival and departure rates are such that the average number of players present in the system is typically between 1 and 2, since this leads to the system being in the condition of least competition with high probability. Further, their expected utilities increase almost linearly with the offered reward and converge to a constant value with respect to its dispensing rate. We conclude by studying a Stackelberg game, where the central entity decides the amount of reward to offer, and the computational providers decide how much power to invest based on the offered reward.

We study a stochastic game with a dynamic set of players, for modeling and analyzing their computational investment strategies in distributed computing. Players obtain a certain reward for solving a problem, while incurring a certain cost based on the invested time and computational power. We present our framework while considering a contemporary application of blockchain mining, and show that the framework is applicable to certain other distributed computing settings as well. For an in-depth analysis, we consider a particular yet natural scenario where the rate of solving the problem is proportional to the total computational power invested by the players. We show that, in Markov perfect equilibrium, players with cost parameters exceeding a certain threshold, do not invest; while those with cost parameters less than this threshold, invest maximal power. We arrive at an interesting conclusion that the players need not have information about the system state as well as each others' parameters, namely, cost parameters and arrival/departure rates. With extensive simulations and insights through mean field approximation, we study the effects of players' arrival/departure rates and the system parameters on the players' utilities.

We propose a setting for two-phase opinion dynamics in social networks, where a node’s final opinion in the first phase acts as its initial biased opinion in the second phase. In this setting, we study the problem of two camps aiming to maximize adoption of their respective opinions, by strategically investing on nodes in the two phases. A node’s initial opinion in the second phase naturally plays a key role in determining the final opinion of that node, and hence also of other nodes in the network due to its influence on them. However, more importantly, this bias also determines the effectiveness of a camp’s investment on that node in the second phase. In order to formalize this two-phase investment setting, we propose an extension of Friedkin–Johnsen model, and hence formulate the utility functions of the camps. We arrive at a decision parameter which can be interpreted as two-phase Katz centrality. There is a natural tradeoff while splitting the available budget between the two phases. A lower investment in the first phase results in worse initial biases in the network for the second phase. On the other hand, a higher investment in the first phase spares a lower available budget for the second phase, resulting in an inability to fully harness the influenced biases. We first analyze the non-competitive case where only one camp invests, for which we present a polynomial time algorithm for determining an optimal way to split the camp’s budget between the two phases. We then analyze the case of competing camps, where we show the existence of Nash equilibrium and that it can be computed in polynomial time under reasonable assumptions. We conclude our study with simulations on real-world network datasets, in order to quantify the effects of the initial biases and the weightage attributed by nodes to their initial biases, as well as that of a camp deviating from its equilibrium strategy. Our main conclusion is that, if nodes attribute high weightage to their initial biases, it is advantageous to have a high investment in the first phase, so as to effectively influence the biases to be harnessed in the second phase.

We study the problem of optimally investing in nodes of a social network in a competitive setting, wherein two camps aim to drive the average opinion of the population in their own favor. Using a well-established model of opinion dynamics, we formulate the problem as a zero-sum game with its players being the two camps. We derive optimal investment strategies for both camps, and show that a random investment strategy is optimal when the underlying network follows a popular class of weight distributions. We study a broad framework, where we consider various well-motivated settings of the problem, namely, when the influence of a camp on a node is a concave function of its investment on that node, when a camp aims at maximizing competitors investment or deviation from its desired investment, and when one of the camps has uncertain information about the values of the model parameters. We also study a Stackelberg variant of this game under common coupled constraints on the combined investments by the camps and derive their equilibrium strategies, and hence quantify the first-mover advantage. For a quantitative and illustrative study, we conduct simulations on real-world datasets and provide results and insights.

We study a stochastic game framework with dynamic set of players, for modeling and analyzing their computational investment strategies in distributed computing. Players obtain a certain reward for solving the problem or for providing their computational resources, while incur a certain cost based on the invested time and computational power. We first study a scenario where the reward is offered for solving the problem, such as in blockchain mining. We show that, in Markov perfect equilibrium, players with cost parameters exceeding a certain threshold, do not invest; while those with cost parameters less than this threshold, invest maximal power. Here, players need not know the system state. We then consider a scenario where the reward is offered for contributing to the computational power of a common central entity, such as in volunteer computing. Here, in Markov perfect equilibrium, only players with cost parameters in a relatively low range in a given state, invest. For the case where players are homogeneous, they invest proportionally to the 'reward to cost' ratio. For both the scenarios, we study the effects of players' arrival and departure rates on their utilities using simulations and provide additional insights.

We study the problem of optimally investing in nodes of a social network in a competitive setting, where two camps aim to maximize adoption of their opinions by the population. In particular, we consider the possibility of campaigning in multiple phases, where the final opinion of a node in a phase acts as its initial biased opinion for the following phase. Using an extension of the popular DeGroot-Friedkin model, we formulate the utility functions of the camps, and show that they involve what can be interpreted as multiphase Katz centrality. Focusing on two phases, we analytically derive Nash equilibrium investment strategies, and the extent of loss that a camp would incur if it acted myopically. Our simulation study affirms that nodes attributing higher weightage to initial biases necessitate higher investment in the first phase, so as to influence these biases for the terminal phase. We then study the setting in which a camp's influence on a node depends on its initial bias. For single camp, we present a polynomial time algorithm for determining an optimal way to split the budget between the two phases. For competing camps, we show the existence of Nash equilibria under reasonable assumptions, and that they can be computed in polynomial time.

We consider a two-player resource allocation polytope game, in which the strategy of a player is restricted by the strategy of the other player, with common coupled constraints. With respect to such a game, we formally introduce the notions of independent optimal strategy profile, which is the profile when players play optimally in the absence of the other player; and common contiguous set, which is the set of top nodes in the preference orderings of both the players that are exhaustively invested on in the independent optimal strategy profile. We show that for the game to have a unique PSNE, it is a necessary and sufficient condition that the independent optimal strategies of the players do not conflict, and either the common contiguous set consists of at most one node or all the nodes in the common contiguous set are invested on by only one player in the independent optimal strategy profile. We further derive a socially optimal strategy profile, and show that the price of anarchy cannot be bound by a common universal constant. We hence present an efficient algorithm to compute the price of anarchy and the price of stability, given an instance of the game. Under reasonable conditions, we show that the price of stability is 1. We encounter a paradox in this game that higher budgets may lead to worse outcomes.