Consensus formation in a social network is modeled by a dynamic game of a prescribed duration played by members of the network. Each member independently minimizes a cost function that represents his/her motive. An integral cost function penalizes a member's differences of opinion from the others as well as from his/her own initial opinion, weighted by influence and stubbornness parameters. Each member uses its rate of change of opinion as a control input. This defines a dynamic non-cooperative game that turns out to have a unique Nash equilibrium. Analytic explicit expressions are derived for the opinion trajectory of each member for two representative cases obtained by suitable assumptions on the graph topology of the network. These trajectories are then examined under different assumptions on the relative sizes of the influence and stubbornness parameters that appear in the cost functions.Comment: 7 pages, 9 figure, Pre-print from the Proceedings of the 12th International Conference on Signal Image Technology and Internet-based Systems (SITIS), 201

The question of whether opinions of stubborn agents result in Nash equilibrium under the presence of troll is investigated in this study. The opinion dynamics is modelled as a differential game played by n agents during a finite time horizon. Two types of agents, ordinary agents and troll, are considered in this game. Troll is treated as a malicious stubborn content maker who disagrees with every other agent. On the other hand, ordinary agents maintain cooperative communication with other ordinary agents and they disagree with the troll. Under this scenario, explicit expressions of opinion trajectories are obtained by applying Pontryagin's principle on the cost function. This approach provides insight into the social networks that comprise a troll in addition to ordinary agents.

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The question of whether foraging swarms can form as a result of a noncooperative game played by individuals is shown here to have an affirmative answer. A dynamic game played by N agents in 1-D motion is introduced and models, for instance, a foraging ant colony. Each agent controls its velocity to minimize its total work done in a finite time interval. The game is shown to have a unique Nash equilibrium under two different foraging location specifications, and both equilibria display many features of a foraging swarm behavior observed in biological swarms. Explicit expressions are derived for pairwise distances between individuals of the swarm, swarm size, and swarm center location during foraging.

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