Game Theory with Applications to Finance and Marketing

Step 2: Fix any x ∈ (c, 1). Then neither F nor G can have a jump atx.To see this, suppose instead that x ∈ (c, 1) is a point of jump of F .Thus firm 1 may randomize on the price x with a strictly positiveprobability. In this case, there exists ɛ > 0 small enough such thatall prices contained in the interval [x, x + ɛ) are dominated for firm 2by some price q < x with q sufficiently close to x. Since (F, G) is byassumption an NE, it must be that G assigns zero probability to theinterval [x, x + ɛ). Again, since F is the best response against G, thismust imply that x is not a best response for firm 1 (the price x + ɛ, for 2example, is strictly better than x), and firm 1 should not have assigneda strictly positive probability to x, a contradiction. We conclude thatF and G are continuous except possibly at p and P (when p = c orP = 1).Step 3: If P > p then every point in (p, P ) is contained in both S Fand S G .Suppose instead that, say, F is flat on an interval [a, b] ∈ (p, P ). Sincep is the infimum of S F , F (b) > 0. Moreover, since b < P , F (b) < 1.(If instead F (b) = 1 then P ≠ sup S F , a contradiction.) There existsa smallest x ∈ [c, 1] such that F (x) = F (b) so that 0 < F (x) < 1: Bythe right continuity of F , we have x = inf{y ∈ (p, P ) : F (y) = F (b)}.We claim that x is a best response for firm 1. Either x = p so thatF (x) > 0 implies that it is a point of jump or x > p but for all ɛ > 0,F (x−ɛ) < F (x) (by definition of x) so that there exists a best responsey n in each term of the sequence of intervals {(x − 1 , x]; n ∈ Z n +} andlim n→∞ y n = x. In the latter case, note that each best response y ngives rise to the same expected profit for firm 1 and by Step 2, firm1’s expected profit is continuous in (p, P ). We conclude that x attainsthe equilibrium expected profit of firm 1, and hence a best response offirm 1. On the other hand, if this were an equilibrium, then G wouldnever randomize over (x, b), but then x could not be a best responsefor firm 1: it is weakly dominated by the price x + b−x . Hence we have2a contradiction. Together with Step 2, we conclude that F and G arestrictly increasing and continuous on (p, P ). Moreover, each point in(p, P ) is a best response for both firms.We just reached the conclusion that S F = S G = [p, P ]. We next showthat p = P .24