As before, one of our concerns is with problem identification. Accordingly, we consider some cases in which the game in extensive form helps us to specify a problem of interactive decision-making that may be relevant for public policy.

Ulysses and the Sirens

We recall that the entrenched monopolist in Game 2.2 was unable to prevent the entry of new competition, because the threat of a price war was not subgame perfect. Yet the entrenched monopolist might not be entirely helpless. The monopolist might consider a large-scale investment that would increase both its production capacity and its costs, creating a situation in which the restricted output corresponding to accommodation of the new entering firm would be less profitable, and the increased production incident on a price war more profitable. If it anticipates competitive entry (for example, following deregulation) the firm might undertake such an investment, in the hope that the entry would be prevented. This is strategic investment to deter entry, and Game 6.1 is an example. (See Figure 6.1; the numbers in parentheses will be explained below.) As before, the first payoff is to Firm B.

In Game 6.1, if Firm A decides not to invest, then we have Game 2.2, but if Firm A decides to invest, we have a different subgame. To solve this more complex game, we again identify the basic subgames, and they are A’s second round of decisions. Both are basic. For the lower one, we already know that the perfect behavior strategy is “accommodate.” For the upper subgame, however, “retaliate” is the perfect response. Thus, Firm B can anticipate that the behavior strategy “enter” will pay 1 in the lower subgame but -1 in the upper. “Enter” is a perfect behavior strategy in the lower subgame but not the upper. Anticipating all this, Firm A expects that investing will lead to profits of 4 while not investing will lead to profits of 1. Thus, the subgame perfect sequence of behavior strategies is “invest,” “don’t enter.”

The investment may or may not be efficient. To know the answer to that, we need to know more about the benefits to customers. In economics, the consumers’ surplus measures the net benefit to the buyer from buying at a particular price. The consumers’ surplus plus the total profits of the two firms measures the net social benefit for this industry, in the absence of externalities. For this example, the numbers in parentheses represent consumers’ surpluses corresponding to the different degrees of price competition and output capacity in the various possible outcomes of the game. The largest total, 8 on our arbitrary scale of measurement, occurs if Firm A does not make the investment, Firm B does enter, and entry is accommodated. In that case, the consumers have the benefits of both expanded production capacity and increased price competition. By contrast, in the

Game 6.1

Figure 6.1 Game 6.1: strategic investment to deter entry

subgame perfect equilibrium, consumers benefit from increased capacity but not from increased price competition, so the total, 7, is less. Efficiency is improved - without either entry or new investment, at the bottom of the diagram, total benefit is 6 - but the resulting outcome is not fully efficient because of the entry-limiting investment.

The key point is that the investment has been successful in deterring entry. In Game 2.2, Firm A was unable to deter entry because the threat of a price war was incredible: Firm B could anticipate that Firm A would not undertake such an unprofitable step. If Firm A were able to do so, it would choose the best response behavior strategy of accommodation. However, by making the strategic investment, Firm A has deprived itself of the opportunity to accommodate profitably.

This illustrates a more general point that emerges from the study of the game in extensive form. In some cases, an agent may be better off with fewer opportunities, fewer options. Schelling (1960) stressed that in order to bind others, we might have to find a way to bind ourselves. This is what Firm A has done in Game 6.1. Elster (1977) drew the analogy to Ulysses having himself bound to the mast so that he could hear the song of the sirens - and not succumb to it.

By considering a strategic investment to deter entry, Firm A has nested Game 2.2 within a larger game. Since Game 2.2 is a subgame of Game 6.1, it is an imbedded game. That is important, since it means that the subgame perfect solution of Game 2.2 is perfect also in the context of the nesting Game 6.1. That is to say, (for purposes of noncooperative game theory) we can analyze the imbedded game as a game in its own right, expecting that its equilibrium will be equilibrial also within the larger game - while that might not be true for a game that is nested but not imbedded in the larger game.

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