ECON3152/4453/8053 Tutorial 3

1. Formulate the following problems as extensive form games.

(a) There are two players, each drawn independently an integer uniformly between  1 and  100;  the probability that any integer i between 1 and 100 is drawn with probability .01.  Player 1 chooses whether to challenge Player 2. If Player 1 challenges Player 2, Player 2 chooses to accept or concede.

If Player 1 does not challenge Player 2, each player receives payoff 0; if Player 1 challenges and Player 2 concedes, Player 1 receives payoff .5 and Player 2 receives payoff -.5. If Player 1 challenges and Player 2 accepts, the player who draws the bigger integer receives payoff 1 and the other player receives payoff 0; in case the two players draw the same integer, each receives payoff 0.

(b) There are two types of university graduates, S(mart) and N(ormal).  Observing his own type, a university graduate chooses whether to pursue an advanced degree (honour or master depending on how you want to intrepret the model). For simplicity, assume that the student always manages to receive the degree, but the pain it takes to doing so, cθ , depends on his type θ  ∈ {S, N} with cN  > cS . Observing the student’s degree but not type, a high-end employer decides whether to offer a job to him. The student’s payoff is w if he does not receive a job offer from the high-end employer; receiving a high-end job offer (which he always accepts) increases his payoff by w − w  > 0, while pursuing an advanced degree reduces his payoff by cθ  independently. The employer’s payoff is zero is she does not offer a job and Rθ  − w if she does, where RS  > RN  > 0 is the value added by a new employee depending on his type.

(c) There is an investor and an advisor. The advisor observes a signal s ∈ {G, B} about how profitable a particular investment is and then sends a message m ∈ M to the investor, where M is a given set. Observing m but not s, the investor decides whether to make a fixed investment. The advisor receives share α ∈ (0, 1) of the proceeds from the investment no matter how profitable it is.  The investor’s payoff is zero if she does not invest and (1 − α)rs  − c if he does where c is a commission payment from the investor to the advisor and rB   < 0 < (1 − α) − 1 c < rG .  (The signal may not be fully accurate, but rB  and rG  are interpreted as the expected return of the investment based on the