Claim $16\mathrm{.}2$ In any separating perfect Bayesian equilibrium, in the first period a highcost incumbent must produce q$1$H

$1=1\mathrm{.}5.$

Proof From claim $16\mathrm{.}1,$ in any separating perfect Bayesian equilibrium it must be

that following q$1$H

$1$ firm $2$ will enter, and following q$1$L

$1$ firm $2$ will stay out and let firm $1$

be a monopolist $($otherwise it would not be a separating perfect Bayesian equilibrium$).$

To see that in a separating perfect Bayesian equilibrium it must be that q$1$H

$1=1\mathrm{.}5,$

assume in negation that this is not the case. Then in period $1$ firm $1$ is making less than

monopoly profits when its marginal costs are c$1=2,$ and in period $2$ firm $1$ is making

Cournot profits. Now consider a deviation of firm $1,$ when c$1=2,$ to the monopoly

quantity of q$1$H

$1=1\mathrm{.}5.$ In period $1$ profits will be higher, which means that for this

deviation not to be profitable it must be that firm $1$ gets less than Cournot profits in

the second period. But this cannot happen because either: $(1)$ firm $2$s beliefs after

the deviation remain Pr$\{$c$1=2\}=1,$ in which case they will play the same Cournot

game, or $(2)$ firm $2$ changes its beliefs to Pr$\{$c$1=2\}<1,$ in which case firm $1$ will

make higher$-$than$-$Cournot profits $($either firm $2$ will stay out or it will play Cournot

against an unknown rival and produce less than $1,$ depending on its beliefs$).$ Thus we

conclude that if q$1$H

$1=1\mathrm{.}5$ then firm $1$ has a profitable deviation to q$1$H

$1=1\mathrm{.}5.$

We have therefore established from our analysis that if a separating perfect

Bayesian equilibrium exists then it must satisfy q$2$L

$1=2,$ q$2$H

$1=1,$ q$2$

$2($q$1$L

$1)=0,$

q$2$

$2($q$1$H

$1)=1,$ $($q$1$L

$1)=0,$ and $($q$1$H

$1)=1.$ From claim $16\mathrm{.}2$ we established that it

must satisfy q$1$H

$1=1\mathrm{.}5.$ We are left to find two more elements: we must define beliefs

for all other quantities q$1$

$1$ in $\{$q$1$H

$1,$ q$1$L

$1\},$ and we have to find q$1$L

$1.$ If we find these

values in a way for which strategies and beliefs satisfy requirements $15\mathrm{.}115\mathrm{.}4$ from

Section $15\mathrm{.}2$ then we have found a separating perfect Bayesian equilibrium.

Step $4$: Setting off$-$the$-$equilibrium$-$path beliefs.

To set off$-$the$-$equilibrium$-$path beliefs that will support behavior on the equilibrium path we will use a $$trick$$ that is common for games with continuous strategy

sets and is similar to what we did for the MBA game. Recall that we want the separating perfect Bayesian equilibrium to work in such a way that each type of firm $1$

in $\{$L$,$ H$\}$ will stick to his strategy q$1$

$1,$ rather than deviating to some other quantity

q$1$

$1.$ To do this, we can make the continuation game following any deviation from either

q$1$L

$1$ or q$1$H

$1$ to be as undesirable as possible for firm $1.$How can we achieve this? Precisely by causing firm $2$ to enter following any such deviation, which is guaranteed to

happen when firm $2$ believes that firm $1$ has high costs. Indeed when firm $2$ acts in this

way then firm $1$ faces the most severe second$-$period competition, and this is the most

$16\mathrm{.}2$ Limit Pricing and Entry Deterrence $.327$

undesirable outcome for firm $1.$ Hence the easiest way to prevent deviations and keep

firm $1$ on the equilibrium path is by setting beliefs that make off$-$the$-$equilibrium$-$path

behavior very unattractive to firm $1.$ This will be satisfied if

$($q$1$

$1)=$

$1$ if q$1$

$1=$ q$1$L

$1$

$0$ if q$1$

$1=$ q$1$L

$1.$

These beliefs cause a unique best response for firm $2,$ which is not to enter if q$1$

$1=$ q$1$L

$1$

and to enter and produce q$2$

$2=1$ if q$1$

$1=$ q$1$L

$1.$

Step $5$: What should q$1$L

$1$ be$?$

Once we have calculated all of the equilibrium components, for q$1$L

$1$ to satisfy the

missing piece of the puzzle it must satisfy the following two important conditions:

$1.$ When firm $1$ is an L type, it prefers to choose q$1$L

$1$ over any other quantity, in

particular over q$1$H

$1.$

$2.$ When firm $1$ is an H type, it prefers to choose q$1$H

$1$ over q$1$L

$1.$

The first condition is that type L is playing a best response. The second condition

just says that an H type does not want to imitate an L type. What about other

quantities? Using the belief system we defined earlier, claim $16\mathrm{.}2$ already implies

that an H type prefers q$1$H

$1=1\mathrm{.}5$ to any other nonmonopoly profit that induces entry.

The reason we have to care about imitating an L type is because q$1$L

$1$ prevents entry,

allowing firm $1$ to remain a monopolist in the second period.

We call these two conditions$$that each type prefers choosing his designated

action rather than imitating some other type$$incentive compatibility constraints.

The meaning is precisely that in equilibrium each type has an incentive to choose

his prescribed strategy and not choose the prescribed strategies for the other types.

Similar incentive compatibility constraints held for the separating perfect Bayesian