We consider certain mixtures, Γ, of classes of stochastic games and provide sufficient conditions for these mixtures to possess the orderfield property. For 2-player zero-sum and non-zero sum stochastic games, we prove that if we mix a set of states S1 where the transitions are controlled by one player with a set of states S2 constituting a sub-game having the orderfield property (where S1 ∩ S2 = ∅), the resulting mixture Γ with states S = S1 ∪ S2 has the orderfield property if there are no transitions from S2 to S1. This is true for discounted as well as undiscounted games. This condition on the transitions is sufficient when S1 is perfect information or SC (Switching Control) or ARAT (Additive Reward Additive Transition). In the zero-sum case, S1 can be a mixture of SC and ARAT as well. On the other hand,when S1 is SER-SIT (Separable Reward -State Independent Transition), we provide a counter example to show that this condition is not sufficient for the mixture Γ to possess the orderfield property. In addition to the condition that there are no transitions from S2 to S1, if the sum of all transition probabilities from S1 to S2 is independent of the actions of the players, then Γ has the orderfield property even when S1 is SER-SIT. When S1 and S2 are both SER-SIT, their mixture Γ has the orderfield property even if we allow transitions from S2 to S1. We also extend these results to some multi-player games namely, mixtures with one player control Polystochastic games. In all the above cases, we can inductively mix many such games and continue to retain the orderfield property.
Social Clouds have been gaining importance because of their potential for efficient and stable resource sharing without any (monetary) cost implications . There is a need, however, to look at how a social structure or relationship evolves to build a Social Cloud (by identifying factors that affect the social structure) and how social structure impacts individual resource sharing behavior. This paper presents a pairwise resource (or pairwise service) sharing social network model to explore the interdependence between social structure and resource (service) availability for an individual user or player. The paper also investigates effects of social structure on individual resource availability. Further, the paper analyzes positive and negative externalities, and aims to characterize stable social clouds.
In this paper, we study the formation of endogenous social storage cloud in a dynamic setting, where rational agents build their data backup connections strategically. We propose a degree-distance-based utility model, which is a combination of benefit and cost functions. The benefit function of an agent captures the expected benefit that the agent obtains by placing its data on others’ storage devices, given the prevailing data loss rate in the network. The cost function of an agent captures the cost that the agent incurs to maintain links in the network. With this utility function, we analyze what network is likely to evolve when agents themselves decide with whom they want to form links and with whom they do not. Further, we analyze which networks are pairwise stable and efficient. We show that for the proposed utility function, there always exists a pairwise stable network, which is also efficient. We show that all pairwise stable networks are efficient, and hence, the price of anarchy is the best that is possible. We also study the effect of link addition and deletion between a pair of agents on their, and others’, closeness and storage availability.
Social storage systems are a good alternative to existing data backup systems of local, centralized, and P2P backup. Till date, researchers have mostly focussed on either building such systems by using existing underlying social networks (exogenously built) or on studying Quality of Service (QoS) related issues. In this paper, we look at two untouched aspects of social storage systems. One aspect involves modelling social storage as an endogenous social network, where agents themselves decide with whom they want to build data backup relation, which is more intuitive than exogenous social networks. The second aspect involves studying the stability of social storage systems, which would help reduce maintenance costs and further, help build efficient as well as contented networks.We have a four fold contribution that covers the above two aspects. We, first, model the social storage system as a strategic network formation game. We define the utility of each agent in the network under two different frameworks, one where the cost to add and maintain links is considered in the utility function and the other where budget constraints are considered. In the context of social storage and social cloud computing, these utility functions are the first of its kind, and we use them to define and analyse the social storage network game. Second, we propose the concept of bilateral stability which refines the pairwise stability concept defined by Jackson and Wolinsky (1996), by requiring mutual consent for both addition and deletion of links, as compared to mutual consent just for link addition. Mutual consent for link deletion is especially important in the social storage setting. The notion of bilateral stability subsumes the bilateral equilibrium definition of Goyal and Vega-Redondo (2007). Third, we prove necessary and the sufficient conditions for bilateral stability
Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard Deng, 2006 [5], Chen, Deng and Teng 2009 [6]). The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant, 2005 [1]). However, there do exist polynomial time tractable classes of win-lose bimatrix games -such as, very sparse games (Codenotti, Leoncini and Resta, 2006 [8]) and planar games (Addario-Berry, Olver and Vetta, 2007 [2]).We extend the results in the latter work to K 3,3 minor-free games and a subclass of K 5 minorfree games. Both these classes of games strictly contain planar games. Further, we sharpen the upper bound to unambiguous logspace, a small complexity class contained well within polynomial time. Apart from these classes of games, our results also extend to a class of games that contain both K 3,3 and K 5 as minors, thereby covering a large and non-trivial class of win-lose bimatrix games. For this class, we prove an upper bound of nondeterministic logspace, again a small complexity class within polynomial time. Our techniques are primarily graph theoretic and use structural characterizations of the considered minor-closed families.
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