In this problem, it does not matter how much money you make, just how long you can make it. The firm that can hang on the longest can force out its more profitable rival as soon as duopoly profits begin to turn negative.

As suggested in the hint, if David can hold on until the third quarter of 1989 he is home free. From then on, the worst possibility is that Goliath stays in the market through the fourth quarter of 1989. This will cost David $2.25 in duopoly losses. But when 1990 comes, Goliath must exit, since he suffers losses either as a duopolist or as a monopolist. Thus, David can count on making $2.50 in monopoly profits during the 1990s, which is enough to tide him over any possible losses during the final two quarters of 1989.

Now, the power of backward reasoning picks up steam. Given that David is committed to staying upon reaching July 1989 (exiting is a dominated strategy), Goliath can expect to earn only losses from July 1989 onward. Thus, he will exit immediately if he ever finds himself as a duopolist on that date. That means that David can expect to make the $2.50 as a monopolist in 1990 and $2.75 as a monopolist in the final two quarters of 1989. This windfall of $5.25 more than covers the maximum duopoly losses up until that date ($1.50), and therefore David should never exit before January 1991. Given that David is committed to staying, Goliath should leave as soon as duopoly profits turn negative, July 1988.

Note that Goliath cannot make the same commitment to stay in the market for the same length of time. That commitment breaks down first in January 1990, and then the guaranteed exit by January 1990 translates into a forced exit by July 1989. The slippery slope for Goliath brings him back all the way to October 1988, the first instance when the market isn’t big enough for the two of them.

This simple story of fighting for market share in declining industries may help explain the observation that large firms are often the first to exit. Charles Baden Fuller, an expert in the decline of British markets, reports that when the demand for U.K. steel casing fell by 42 percent over the period 1975-1981, executives of the two largest firms, F. H. Lloyd and the Weir Group, “felt that they had borne the brunt of the costs of rationalization; they accounted for 41 percent of the industry output in 1975, but for 63% of the capacity that was withdrawn over the 1975-1981 period, reducing their combined market share to 24 percent.”

Remember that size is not always an advantage: in judo and here in exit strategies, the trick is to use your rival’s bigger size and consequently inflexibility against him.