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We model this situation as a potentially infinitely lived duopoly game, where the
termination time is random. The uncertainty comes again from the fact that processors cannot
exert absolute control on the quality of the good they procure and/or produce. Rob and
Sekiguchi model a similar situation. However, in their model, firms compete in price in the
second period, and only one firm is able to make sales until it loses its reputation and the other
occupies its place. When the second firm loses the market, consumers switch back to the first
firm, and so on.
In the first stage, the two processors decide once and for all, independently and in a
non-cooperative fashion, what QAS to implement. After observing each other’s choice of
QAS, processors compete a la Cournot. Production and QAS technologies are known by both
firms. Consumers buy the product at the end of the first period and update their beliefs about
the quality of the industry’s output. In the periods to follow, firms keep competing in output,
and consumers keep updating their beliefs, until a failure occurs and is detected by consumers.
In that period, confidence is lost by the entire industry forever.
We work backwards, first finding the equilibrium level of output after technologies
have been chosen, and then solving the first-stage problem applying the second-period
equilibrium rules. The second-stage problem is a standard Cournot game. Following the same
argument used to construct the monopolist’s objective function in the previous section, we find
per-period profits of player i in state 1, and y= (yi, y-i) represents a vector of output.
that firm i ’s
expected profit is E(∏i,1 (y,s))=
π ,1 ( y1, y- i, si )
1 - βmi ( si ) m - i ( s - i )
where πi,1 denotes
Because per-period profits are zero if the firm has lost its reputation, we drop the superscript
that indexes the state of the world. Since both si and s-i are predetermined, the problem is to