SEQUENTIAL GAMES: COMMITMENT AND BACKWARD INDUCTION c

cSections 4.2 and 4.3 cover relatively more advanced material, which may be skipped in a first reading of the book.

In the previous section, we justified the use of simultaneous-choice games as a realis-tic way of modeling situations where observation lags are so long that it is as if players were choosing strategies simultaneously. When the time between strategy choices is suf-ficiently long, however, the assumption of sequential decision making is more realistic.

Consider the example of an industry that is currently monopolized. A second firm must decide whether or not to enter the industry. Given the decision of whether or not to enter, the incumbent firm must decide whether to price aggressively or not. The incumbent’s de-cision is taken as a function of the entrant’s dede-cision. That is, first the incumbent observes whether or not the entrant enters, and then decides whether or not to price aggressively.

In such a situation, it makes more sense to consider a model with sequential rather than simultaneous choices. Specifically, the model should have the entrant—Player 1—move first and the incumbent—Player 2—move second.

The best way to model games with sequential choices is to use a game tree. A game tree is like a decision tree except that there is more than one decision maker involved.

An example is given in figure4.6, where strategies and payoffs illustrate the case of entrant and incumbent described earlier. In figure 4.6, a circle denotes a decision node.

The game starts with decision node 1. At this node, Player 1 (entrant) makes a choice between e and¯e, which can be interpreted as “enter” and “not enter,” respectively. If the latter is chosen, then the game ends with payoffs1= 0 (entrant’s payoff) and 2= 50 (incumbent’s payoff). If Player 1 chooses e, however, then we move on to decision node 2.

This node corresponds to Player 2 (incumbent) making a choice between r and¯r, which can be interpreted as “retaliate entry” or “not retaliate entry,” respectively. Games which, like figure 4.6, are represented by trees are also referred to as games in extensive form.^d

dFrom this and the previous sections, one might erroneously conclude that games with simulta-neous choices must be represented in the normal form, and games with sequential moves in the extensive form. In fact, both simultaneous and sequential choice games can be represented in both the normal and extensive forms. However, for simple games such as those con-sidered in this chapter, the choice of game representation considered in the text is the more appropriate.

This game has two Nash equilibria:(e, ¯r) and (¯e, r). Let us first check that (e, ¯r) is indeed a Nash equilibrium, that is, that no player has an incentive to change its strategy given what the other player does. First, if Player 1 chooses e, then Player 2’s best choice is to choose¯r (it gets 20, it would get −10 otherwise). Likewise, given that Player 2 chooses

¯r, Player 1’s optimal choice is e (it gets 10, it would get 0 otherwise).

Let us now check that(¯e, r) is an equilibrium. Given that Player 2 chooses r, Player 1 is better off by choosing ¯e: this yields Player 1 a payoff of 0, whereas e would yield

−10. As for Player 2, given that Player 1 chooses ¯e, its payoff is 50, regardless of which strategy it chooses. It follows that r is an optimal strategy (though not the only one).

Figure 4.6 Extensive-Form Representation: The Sequential-Entry Game.

Although the two solutions are indeed two Nash equilibria, the second equilibrium does not make much sense. Player 1 is not entering because of the “threat” that Player 2 will choose to retaliate. But, is this threat credible? If Player 1 were to enter, would Player 2 decide to retaliate? Clearly, the answer is “no”: By retaliating, Player 2 gets−10, compared with 20 from no retaliation. We conclude that(¯e, r), although it is a Nash equilibrium, is not a reasonable prediction of what one might expect to be played.

One way of getting rid of this sort of “unreasonable” equilibria is to solve the game backward, that is, to apply the principle of backward induction. First, we consider node 2, and conclude that the optimal decision is¯r. Then, we solve for the decision in node 1 given the decision previously found for node 2. Given that Player 2 will choose¯r, it is now clear that the optimal decision at node 1 is e. We thus select the first Nash equilibrium as the only one that is intuitively “reasonable.”

Solving a game backward is not always this easy. Suppose that, if Player 1 chooses e at decision node 1 we are led not to a Player 2 decision node but rather to an entire new game, say, a simultaneous-move game as in figures4.1 to 4.5. Because this game is a part of the larger game, we call it a subgame of the larger one. In this setting, solving the game backward would amount to first solving for the Nash equilibrium (or equilibria) of the subgame, and then, given the solution for the subgame, solving for the entire game.

Equilibria that are derived in this way are called subgame-perfect equilibria.⁴⁶

In the game of figure 4.6, the equilibrium(¯e, r) was dismissed on the basis that it requires Player 2 to make the “incredible” commitment of playing r in case Player 1 chooses e. Such threat is not credible because, given that Player 1 has chosen e, Player 2’s best choice is¯r. But suppose that Player 2 writes an enforceable and non renegotiable contract whereby, if Player 1 chooses e, Player 2 chooses r. The contract is such that,

Games and Strategy 57

Figure 4.7 The Value of Commitment.

were Player 2 not to choose r and chose¯r instead, Player 2 would incur a penalty of 40, lowering its total payoff to−20.^e

eThis is a very strong assumption as most contracts are renegotiable.

However, for the purposes of the present argument, what is important is that Player 2 has the option of imposing on itself a cost if it does not choose r. This cost may result from breach of contract or from a different cause.

The situation is illustrated in figure4.7. The first decision now belongs to Player 2, who must choose between writing the bond described above (strategy b) and not doing anything (strategy ¯b). If Player 2 chooses ¯b, then the game in figure 4.6 is played. If instead Player 2 chooses b, then a different game is played, one that takes into account the implications of signing the bond.

Compare the two subgames starting at Player 1’s decision nodes. The one on the right is the same as in figure 4.6. As we then saw, the equilibrium payoff for Player 2 in this game is 20. The subgame on the left-hand side is identical to the one on the right except for Player 2’s payoff following(e, r). The value is now −20 instead of 20. At first, it might seem that this makes Player 2 worse off: Payoffs are the same in every case except one, and in that one case, payoff is actually lower than it was initially. However, as we will see next, Player 2 is better off playing the left-hand-side subgame than the right-hand-side subgame.

Let us solve the left-hand-side subgame backward, as before. When it comes to Player 2 to choose between r and¯r, the optimal choice is r. In fact, this gives Player 2 a payoff of−10, whereas the alternative would yield −20 (Player 2 would have to pay for breaking the bond). Given that Player 2 chooses r, Player 1 finds it optimal to choose¯e:

It is better to receive a payoff of zero than to receive−10, the outcome of e followed by r. In summary, the subgame on the left-hand side gives Player 2 an equilibrium payoff of 50, the result of the combination of ¯e and r.

We can finally move backward one more stage and look at Player 2’s optimal choice between b and ¯b. From what we saw above, Player 2’s optimal choice is to choose b and eventually receive a payoff of 50. The alternative, ¯b, eventually leads to a payoff of 20 only.

This example illustrates two important points. First it shows that

A credible commitment may have significant strategic value.

By signing a bond that imposes a large penalty when playing¯r, Player 2 credibly commits to playing r when the time comes to choose between r and¯r. In so doing, Player 2 induces Player 1 to choose¯e, which in turn works for Player 2’s benefit. Specifically, introducing this credible commitment raises Player 2’s payoff from 20 to 50. The value of commitment is 30 in this example.

The second point illustrated by the example is a methodological one. If we believe that Player 2 is credibly committed to choosing r, then we should model this by changing Player 2’s payoffs or by changing the order of moves. This can be done as in figure4.7, where we model all the moves that lead to Player 2 effectively precommiting to playing r.

Alternatively, this can also be done as in figure 4.8, where we model Player 2 as choosing r or¯r “before” Player 1 chooses its strategy. The actual choice of r or ¯r may occur in time after Player 1 chooses e or¯e. However, if Player 2 precommits to playing r, we can model that by assuming Player 2 moves first. In fact, by solving the game in figure 4.8 backward, we get the same solution as in figure 4.7, namely the second Nash equilibrium of the game initially considered.

To conclude this section, we should mention another instance in which the se-quence of moves plays an important role. This is when the game under consideration

Figure 4.8 Modeling Player 2’s Capacity to Precommit.

Games and Strategy 59

Figure 4.9 A Game with Long-Run and Short-Run Strategy Choices (Timing of Moves).

depicts a long-term situation where players choose both long-run and short-run variables.

For example, capacity decisions are normally a firm’s long-term choice, for production capacity (buildings, machines) typically lasts for a number of years. Pricing, on the other hand, is typically a short-run variable, for firms can change it relatively frequently at a relatively low cost.

When modeling this sort of strategic interaction, we should assume that players choose the long-run variable first and the short-run variable second. Short-run variables are those that players choose given the value of the long-run variables. And this is precisely what we get by placing the short-run choice in the second stage.

This timing of moves is depicted in figure4.9. This figure illustrates yet a third way of representing games: a time line of moves. This is not as complete and rigorous as the normal-form and extensive-form representations we saw before; however, it proves to be useful in analyzing a variety of games.

In a real-world situation, as time moves on, firms alternate between choosing capacity levels and choosing prices, the latter more frequently than the former. If we want to model this in a simple, two-stage game, then the right way of doing it is to place the capacity decision in the first stage and the pricing decision in the second stage. The same principle applies generally when there are long-run and short-run strategic variables. In the following chapters, we encounter examples of this in relation to capacity/pricing decisions (chapter 7), product positioning/pricing decisions (chapter 12), and entry/output decisions (chapter 15).