# If the victim moves first

Now suppose that the victim moves first. The outcomes under strict liability and no liability are the same as before, so we again consider a negligence rule for injurers.

Under a negligence rule for injurers with a due standard of *z _{{} =* x*, the injurer's expected costs are:

We again solve the game by backward induction. Suppose that the victim has chosen x_{v}. We split the victim's choices into two possible classes. Suppose the victim has chosen a suboptimally low level of care,

*Figure 5.3.3* The injurer's expected costs - injurer prefers an inefficiently low level of care

so that *x _{v} < x*.* Then this means that the injurer will face the efficient level of expected damages if

*x*x*, or if

_{v}=*x*x*, this will increase the injurer's costs and he will face an even greater incentive to avoid them. Hence, in this case, the injurer will behave efficiently.

_{v}<The victim, knowing this, would then face the full costs of his actions, and so would choose *x _{v} =* x*. Hence, there is a subgame perfect equilibrium in which both parties behave efficiently.

On the other hand, suppose that the victim chooses an inefficiently high level of care, so that *x _{v} >* x*. Then this reduces the costs that the injurer faces, and there are two possibilities. First, the victim's care may reduce costs by so much that the injurer would rather choose an inefficiently low level of care than meet the due standard. This situation is illustrated in Figure 5.3.3.

But if this is the case, the injurer would be found negligent, and the victim would face no costs. Therefore, he would choose *x _{v} =* 0 in the first period, which is inconsistent with our original assumption that

*x*x*.

_{v}>On the other hand, the victim's additional care may reduce costs by only a little, making it optimal for the injurer to still meet the due standard of care. But if this is the case, the injurer would not be found negligent, and the victim would face the full costs of the accident. Therefore, the victim would not choose *x _{v} > x** in the first period, which is again inconsistent with our original assumption.

We therefore conclude that if the victim moves first, there is a unique subgame perfect equilibrium in which both parties behave efficiently, with the injurer choosing the efficient level of care, and the victim always bearing the accident costs.

In both cases, then - irrespective of who moves first - we find that a negligence rule for injurers induces a unique subgame perfect equilibrium in which each party behaves efficiently. With a negligence rule, there is no first-mover advantage for either the injurer or the victim.