  We introduce a model of estimation in the presence of strategic,
self-interested sensors. We employ a game-theoretic setup to model the
interaction between the sensors and the receiver. The cost function of the
receiver is equal to the estimation error variance while the cost function of
the sensor contains an extra term which is determined by its private
information. We start by the single sensor case in which the receiver has
access to a noisy but honest side information in addition to the message
transmitted by a strategic sensor. We study both static and dynamic estimation
problems. For both these problems, we characterize a family of equilibria in
which the sensor and the receiver employ simple strategies. Interestingly, for
the dynamic estimation problem, we find an equilibrium for which the strategic
sensor uses a memory-less policy. We generalize the static estimation setup to
multiple sensors with synchronous communication structure (i.e., all the
sensors transmit their messages simultaneously). We prove the maybe surprising
fact that, for the constructed equilibrium in affine strategies, the estimation
quality degrades as the number of sensors increases. However, if the sensors
are herding (i.e., copying each other policies), the quality of the receiver's
estimation improves as the number of sensors increases. Finally, we consider
the asynchronous communication structure (i.e., the sensors transmit their
messages sequentially).
