### 6.3 Order of moves in the technological tying game

Some of our results flow from the assumption that Firm 1 can impose a technological tie in the second stage of the game, after the firms invest, which makes the threat of a tie an important consideration for firms' investment behavior. This is a reasonable assumption for some competitive environments. For example, a manufacturer of mainframe computers could change the interface protocols that determine how components communicate with the central processor after the manufacturer invests in most of the attributes of the mainframe system. In other environments it is more reasonable to assume that the technological tying decision is made at the same time or before the firm invests to improve the characteristics of its system. For example, the maker of a popular video game console could design the console so that it works with its own games, but not with games supplied by rivals. How would our results change if we reverse stages one and two of the technological tying game? In this modification of the technological tying game, Firm 1 chooses whether to impose the tie in the first stage. In the second stage both firms make their investment decisions, and they choose prices in the third stage. We refer to this as the "modified technological tying game".

The possible equilibria of the modified technological tying game differ according to whether $ r(z^{M}) $ is larger or smaller than $ (\gamma_{2}-\gamma_{1})M $ . Suppose that $ r(z^{M})<(\gamma_{2}-\gamma_{1})M $ . In this case the product improvement game described in Proposition 1 has a unique equilibrium in which the more efficient Firm 2 invests $ z^{M} $ . In the modified technological tying game, if $ w<\gamma_{1}+z^{M}-r(z^{M})/M $ , Firm 1 imposes a tie in the first stage and invests $ z^{M} $ in the second stage, and Firm 2 does not invest. Consequently, Firm 1 sells systems at price $ P_{1}=\gamma_{1}+z^{M} $ and earns $ \pi_{1}=(\gamma_{1}+z^{M})M-r(z^{M}) $ , while Firm 2 sets $ P_{2}=w $ and earns zero. Firm 1 has no incentive to impose a technological tie in the first stage if $ w\geq\gamma_{1}+z^{M}-r(z^{M})/M $ . In this case the equilibrium outcome is the same as in the product improvement game without tying: Firm 2 invests $ z^{M} $ , sells systems at price $ P_{2}=\gamma_{2}+z^{M} $ , and earns $ \pi_{2}=(P_{2}-w)M-r(z^{M}) $ ; Firm 1 earns $ \pi_{1}=wM $ . This is more profitable for Firm 1 than investing itself and foreclosing competition from Firm 2. Note that, given $ r(z^{M})<(\gamma_{2}-\gamma_{1})M $ , the modified technological tying game has a unique equilibrium outcome for every value of w, whereas the technological tying game described in Proposition 2 has multiple equilibria for $ \gamma_{1}+z^{M}-r(z^{M})/M\leq w\leq\gamma_{1}+z^{M} $ . Thus, the modified game avoids the coordination failure of Firm 1 winning the market even though both firms prefer Firm 2 to prevail.

Suppose $ r(z^{M}) \geq (\gamma_{2} - \gamma_{1})M $ . Under this assumption the product improvement game described in Proposition 1 has multiple equilibria. In the modified technological tying game, if $ w < \gamma_{2} $ , it is a dominant strategy for Firm 1 to impose a technological tie in the first stage, as this eliminates any possible competition from Firm 2. Therefore, the unique equilibrium outcome when $ w < \gamma_{2} $ is that Firm 1 imposes a tie in the first stage and invests $ z^{M} $ in the second stage, and Firm 2 does not invest. Firm 1 sells