Question

Question 3. You are about to appear on a TV game show. To start the competition, the show gives you a “pot” of 1000 TL. You will then be asked a sequence of three questions: Question Payoff Probability of answering correctly 1 1000 TL 0.5 2 2000 TL 0.4 3 3000 TL 0.3 Before being asked each of the three questions, you can choose between quitting the game and trying to answer the question. If you quit, you keep the money in your pot and the game is over. If you choose to try to answer, there are two possibilities. If you answer correctly, the payoff from the question is added to your pot and you may continue to the next round of the game. If you do not answer correctly, you forfeit half the money in your pot and exit the game with the other half. If you answer all three questions correctly, you get to keep all the money in your pot (which would be 7000 TL at that point). a) Represent the problem by using a decision tree. b) Assuming you are risk neutral, determine the best strategy and find the maximum expected payoff. Show your work. (Show your calculations for the expected value of uncertain events, show the eliminated decisions etc.)

DECISION TREE ANALYSIS REQUIRED

Answer 1

The square (or rhombus) nodes represent the decision nodes and the round (or circle) nodes represent the chance (states of nature) nodes. The decision tree is shown below

Folding from the right, we see that the expected payoff is

At the chance node 3, the expected payoff is 0.3*7000 + 0.7*2000 = 3500

This means at the 3rd question’s decision node, we should choose not to answer and quit. This will give us a total payoff of 4000

At the chance node 2, the expected payoff is 0.4*4000 + 0.6*1000 = 2200

The choice is between quitting (2000 payoff) or trying (2200 payoff). We should choose to try. This will give us an expected payoff of 2200.

At the chance node 1, the expected payoff is 0.5*2200 + 0.5*500 = 1350

Given the expected payoff of trying is now 1350 vs quitting is 1000, we should try..

The strategy

Try to answer the first question, try to answer the second question but quit at the third question. The expected payoff is 1350TL