Suppose that you are bargaining against a robot that is programmed to choose a strategy that maximizes its own profit against rational opponents. If two alternative options are equally profitable for the robot, it will toss a fair coin to decide what to do. You don’t care at all how much money the robot makes, but you do want to maximize your own profit.

You are trying to buy an object from the robot that is worth $0 to the robot. It is worth $100 to you if you get it on your first offer. Each offer or counteroffer that is made uses up one period of time. Each period of time that passes before the object is sold causes the value of the object to you to fall to 80 percent of its value in the previous period. For example, if your first offer is rejected and the robot makes a counteroffer that you accept, the object will be worth $80 to you when you get it. If your first offer is rejected and you reject the robots counteroffer, and it accepts your second offer, the object will be worth $64, and so on. Neither you nor the robot can make change for a dollar, so you and the robot have to make offers and counteroffers in dollar units. If at the end of the time allotted for offers and counteroffers, no agreement is reached, then you and the robot both get $0.

a. If you can make only one offer, and the robot must either accept your offer or reject it, how much should you offer the robot for the object?

b. Suppose that the robot is allowed to make a counteroffer to your first offer, and you must either accept or reject the counteroffer. What price should you offer the robot when you make your initial offer?

c. Suppose that the robot can make a counteroffer to your first offer, and if you reject the counteroffer, you can make a second offer that the robot must accept or reject. What is the lowest price that you can offer initially and be sure that the robot will accept your initial offer?