# Sequential care in the bilateral care model

Up to this point we have assumed that the injurer and the victim move simultaneously. But this may not be a realistic assumption in all situations. In many cases, one party may choose their level of care first, and this may be observed by the other, who then chooses his level of care. This situation can give rise to different incentives than in the simultaneous move case.

Suppose again that there is a victim and an injurer. They can take care of *x _{t}* and

*x*The marginal costs of care are

_{v}.*w*and

_{t}*w*The expected harm is

_{v}.*H (x*

_{t},

*x*

_{v}).

## If the injurer moves first

We first consider a situation in which the injurer moves first at the victim moves second. We assume that the victim can observe the injurer's choice of care.

The model is solved sequentially, by assuming that the first-mover has chosen some level of care, working out what the second mover will do for each possibility, and then rolling back to the first period to examine what the first mover will do, given that he can anticipate how the second mover will react.

*5.3.3.1.1 Strict liability.* First, consider a rule of strict liability for the injurer. Suppose that the injurer has chosen *x.* The victim observes this and solves:

which gives *x _{v} =* 0. Anticipating this, the injurer chooses:

We have *x _{t}*(0) >

*x* = x*(x*). A rule of strict liability has the same outcome as if the game was played simultaneously. This is because the victim has a dominant strategy, and so irrespective of whether he can observe what the injurer does or not, under this legal rule he is always fully compensated and will always choose

_{t}*x*0.

_{v}=- 5.3.3.1.2
*No liability.*Now consider a rule of no liability. It is easy to see that because the injurer has a dominant strategy (namely, to choose*x*0), the fact that there is sequential care has no bearing on the outcome under this legal rule. The victim therefore chooses_{i}=*x*(0) > x* =_{v}= x_{v}*x*(x*)._{v} - 5.3.3.1.3
*Negligence rule.*Now consider a negligence rule for injurers, where the due standard is set at*z*x*. We find the subgame perfect equilibrium by first assuming that the injurer has chosen some level of care_{i}=*x,*and then examine the incentives that this creates for the victim.

In the case of a negligence rule, we split the injurer's possible actions up into two possible classes of outcomes, which depend on whether or not the injurer has met the due standard.

First, suppose that the injurer has chosen *x _{{} >* x*. The injurer would never choose

*x*x*, since taking care that exceeds the due standard does not reduce his expected costs. So we must have

_{t}>*x**. Then the costs of the accident now full upon the victim, and so the victim's expected costs are:

_{{}= x

This is minimised at the point *x _{i} =* x*.

On the other hand, suppose that the injurer has chosen *x _{{} <* x*. Then the injurer has behaved negligently, and so will bear the full costs of any accident. Hence the victim's expected costs are:

Hence, the victim would choose *x _{v} =* 0 in response to this.

*Figure 5.3.2* The injurer's expected costs

Now roll back to the first period. The injurer can anticipate the behaviour of the victim, and so therefore expects his costs to be:

As Figure 5.3.2 shows, these costs are minimised at the point *x _{i} = x*. *Therefore, it cannot happen that

*x*< x* and

_{t}*x*0. We have therefore found a unique subgame perfect equilibrium outcome: in the sequential situation, the negligence rule induces both victims and injurers to behave efficiently. The subgame perfect equilibrium payoffs in this situation are w

_{v}=_{i}x* for the injurer and w

_{v}x* +

*H*(x*, x*) for the victim.