  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. More importantly,
this bias also determines the effectiveness of a camp's investment on that node
in the second phase. To formalize this two-phase investment setting, we propose
an extension of Friedkin-Johnsen model, and hence formulate the utility
functions of the camps. There is a tradeoff while splitting the budget between
the two phases. A lower investment in the first phase results in worse initial
biases for the second phase, while a higher investment spares a lower available
budget for the second phase. 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.

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