There are two firms (i = 1,2).In period 1, both firms are in the market, Only firm 1 (the “incumbent”) fakes an action a1. The action space has two elements: “prey” and “accommodate.” Firm 2 (the “entrant”) has potential profit D2, if firm 1 accommodates and P2, if firm 1 preys, with D2 > 0 > P2. Firm 1 has one of two potential types: “sane” and “crazy.” A sane firm 1 makes D1, when it accommodates and P1 when it preys, where D1 > P1. Thus, a sane firm prefers to accommodate rather than to prey. However, it would prefer to be a monopoly, in which case it would make M1 > D1 per period. When crazy, firm 1 enjoys predation and thus preys (its utility function is such that it is always worth preying). Let p (respectively, 1 — p) denote the prior probability that firm 1 is sane (respectively, crazy).
In period 2, only firm 2 chooses an action a2. This action can take two values: “stay” and “exit.” If firm 2 stays, it obtains payoff D2, if firm 1 is actually sane and P2, if it is crazy; if firm 2 exits, it obtains payoff 0. The idea is that, unless it is crazy, firm 1 will not prey in the second period, because there is no point to building or keeping a reputation at the end. (This assumption can be derived more formally from the description of the second-period competition.) The sane firm gets D1, if firm 2 stays and M1, if firm 2 exits.
We presumed that the crazy type always preys. The interesting thing to study is thus the sane type’s behavior. From a static point of view, it would want to accommodate in the first period; however, by preying it might convince firm 2 that it is of the crazy type, and thus induce exit (as 0 > P2) and increase its second period profit.