# Negotiations under expected utility maximization

A buyer is negotiating with a used car salesperson. The value of the car to the seller is uniformly distributed between 0 and 5000. Value to the buyer is 50 percent more than that of the seller (i.e. 1.5 times of the seller's). Both are rik neutral, neither party has a private signal, and all of the previous is common knowledge. One party will offer a price. The other party can accept the offer, in which case the buyer will pay the price to the salesperson in exchange for the car, or make a counteroffer. The counteroffer is take-it-or-leave-it; the first party can either accept the counteroffer, or there is no deal. If both parties are expected utility maximizers, what would you expect to happen when the salesperson makes the first offer? What about when the buyer makes the first offer? If you were one of the parties to the negotiation, would you prefer to make the first move, or the second?

My Solution:

The expected utility of the car for the seller is $$\frac{(0+5000)}{2} = 2500$$ and the expected utility of the buyer is $$1.5 \times \frac{(0+5000)}{2} = 3750$$.

Seller makes the 1st offer: The seller would ask a price greater than or equal to 2500. Trade occurs only if the offer is less than or equal to 3750. The seller wants to get the maximum payoff so he will ask for the maximum price which is equal to the expected value of the buyer i.e. 3750. The buyer than makes a counteroffer equal to the expected value of the seller i.e. 2500. Seller accepts that offer as it is within his domain of acceptance.

Buyer makes the 1st offer: The buyer would give a price less than or equal to 3750. Trade occurs only if the offer is greater than or equal to 2500. The buyer wants to get the maximum payoff so he will ask for the minimum price which is equal to the expected value of the seller i.e. 2500. The seller then makes a counteroffer equal to the expected value of the buyer i.e. 3750. Buyer accepts that offer as it is within his domain of acceptance.

What would I do: I would want to move second as that would allow me to give a take-it-orleave-it offer which provides me a higher bargaining power.

I would like your valuable feedback on my solution of the question.