Question 1

Suppose that we have a firm with an owner \(O\), a manager \(M\), and a potential worker \(W\). The owner first decides whether to hire the worker, to refuse to hire the worker, or to let the manager decide. If the owner lets the manager decide, then the manager decides whether to hire the worker or not. If the worker is hired then they choose to work diligently or to shirk. They do not know who was responsible for hiring them.

If the worker is not hired then everyone gets payoff \(0\). If the worker shirks then both the owner and manager get payoff \(-1\) and the worker gets payoff \(1\). If the worker is diligent then both the worker and the person who hired them get payoff \(1\), and the other person getspayoff \(0\). Represent this game in extensive form.

Question 2

Suppose that Anita, Brendan each start with two playing cards, each having a king \(K\) and queen \(Q\). Anita places one of her cards into an envelope and gives it to Brendan, who can look inside and then place one of his own cards into the envelope as well.

The envelope is then given to Chak who has not seen the moves of Anita and Brendan. after observing the contents of the envelope Chak selects yes \(Y\) or no \(N\) in answer to the question “Did Anita put a king in the envelope?”. If Chak is correct then he gets payoff \(1\) and the others each get payoff \(0\). Otherwise Chak gets payoff \(0\) and the others each get payoff \(1\). Represent this game in extensive form.

Question 3

Suppose that Idris and Juliet are traders of some commodity which they each secretly believe has value \(v_I\) and \(v_J\) respectively. First Idris decides whether they want to buy or sell the commodity. Then both traders simultaneously decide on the price which they are prepared to exchange the commodity for.

If the buyer’s price is higher than the seller’s then the buyer pays their bid to the seller; the payoff to the buyer is their value for the commodity minus the price paid, while the payoff to the seller is the price paid minus their value for the commodity.

If the buyer’s price is not higher than the seller’s then no trade occurs and the payoff to each player is zero.

Represent this game in extensive form.

Question 4

Can any of the games in questions 1-3 be represented in normal form as matrix games?