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Monopolist processor
The monopoly situation can be obtained by setting I2 = 0. I1 = 1 if the processor is trusted and
zero otherwise. This yields an inverse demand function for good 1 given by p1(y1)=aI1-by1,
and hence per-period profits (after substitution of the equilibrium output levels) are
π11* = (a - s1 )2/4b . Plugging this back into the monopolist problem, we obtain
max
s1∈S 4b
(a - s1 )2___________
(1 - β (1 - ω + ωλ( s1 )))
Duopoly situation when reputation is a public good
This case is obtained by letting I1 and I2 equal one if no failure has been detected and zero
otherwise. The inverse demand for processor i is given by pi(y1,y2) =aIi-b(y1+y2). Then,
equilibrium quantities for the second stage and per-period profits are easily found to be
y* =( a - 2 si + s- i )/3 b and π2 * ( si, s - i ) = ( a - 2 si + s - i )2/9 b, i = 1,2. Plugging this into the
first-stage problem, we find that processor’s 1 objective is given by
max
s1∈S
(a-2 s1+s2)
9 b (1 - β (1 - ω + ωλ( s1 ))(1 - ω + ωλ( s 2 )))
Duopoly situation when reputation is a private good
For this case, I1 equals one in states 1 and 2 and zero otherwise. Also, I2 equals one in states
1 and 3, and zero otherwise. As long as the stochastic process stays in the initial state (both
processors have good reputations), per-period profits are as in the duopoly situation previously
presented. When the system reaches states 2 or 3, the monopolist’s per-period profit (also
previously presented) becomes relevant. Here, the problem of processor 1 is