  We introduce a strategic behavior in reinsurance bilateral transactions,
where agents choose the risk preferences they will appear to have in the
transaction. Within a wide class of risk measures, we identify agents'
strategic choices to a range of risk aversion coefficients. It is shown that at
the strictly beneficial Nash equilibria, agents appear homogeneous with respect
to their risk preferences. While the game does not cause any loss of total
welfare gain, its allocation between agents is heavily affected by the agents'
strategic behavior. This allocation is reflected in the reinsurance premium,
while the insurance indemnity remains the same in all strictly beneficial Nash
equilibria. Furthermore, the effect of agents' bargaining power vanishes
through the game procedure and the agent who gets more welfare gain is the one
who has an advantage in choosing the common risk aversion at the equilibrium.
