Pre-trial settlement A contractor is being sued for damages by a municipality that hired him to construct a bridge, because the bridge has collapsed. The contractor knows whether or not the collapse of the bridge is due to negligence on his part, or due to an act of nature beyond his control, but the municipality does not know which of these two alternatives is the true one.

Both sides know that if the matter is settled by a court trial, the truth will eventually be uncovered. The contractor can try to arrive at a pre-trial settlement with the municipality. He has two alternatives: to make a low settlement offer, under which he pays the municipality $300,000, or a high offer, under which he pays the municipality $500,000.

After the contractor has submitted a settlement offer, the municipality must decide whether or not to accept it. Both parties know that if the suit goes to trial, the contractor will pay lawyer fees of $600,000, and that, in addition to this expense, if the court finds him guilty of negligence, he will be required to pay the municipality $500,000 in damages. Assume that the municipality has no lawyer fees to pay.

Answer the following questions:

(a) Describe this situation as an extensive-form game, where the root of the game is a chance move that determines with equal probability whether the contractor was negligent or not.

(b) Explain the significance of the above assumption, that a chance move determines with equal probability whether the contractor was negligent or not.

(c) Find all the Nash equilibria of this game.

(d) Find all the sequential equilibria of this game.

(e) Repeat items (c) and (d) when the chance move selects whether the contractor was negligent or not with probabilities p and 1 - p respectively.