Homework 2

1. Analyse the following game:
   TWO-ROUND BARGAINING(5,3): There is a fixed number, say 5, of dollar bills for both players. Ann makes an offer how to share them, which Beth can either accept or reject. If she accepts, the bills are divided as agreed upon. If the rejects, two dollar bills are taken away and only three remain on the desk. Then Beth makes an offer how to share these three dollars, which Ann can accept or reject. If Ann accepts, the bills are divided accordingly, if she rejects, nobody gets anything.
   Draw the extensive form of the game, and explain what will happen if both play optimal.
2. Analyse the 3·3 version of REC THE SQUARE game. Draw the extensive form of the game, and explain what will happen if both play optimal.
   Note that you can describe the game using the following 18 states, where each symbol subsumes the cases obtained by rotation and reflection.
   
   
   
   
   
3. Analyse the 3·3 version of CREATING HOLES, a variant of REC THE SQUARE. Draw the extensive form of the game, and explain what will happen if both play optimal.
4. a) Two cards are selected randomly from a shuffled 52-card deck. Someone tells you that the first card is black. What is the probability that the second card is black too?
   b) Again two cards are selected randomly from a shuffled 52-card deck. Someone tells you that at least one of the two cards is black. What is the probability that both cards are black?
5. Assume you flip a coin and get $3 if the result is heads, and have to pay $2 if the result is tails. What is the expected value?
6. You face 5 envelopes, containing $0, $1000, $2000, $3000, and $4000. You randomly choose one of them. What is the expected value?
7. You have five $1-bills, three $5-bills, and one $10 bills in your pocket. You randomly choose one of them to give the taxi driver a tip. What is the expected value?
8. 5 Envelopes (a challenge)

   $10000 is distributed into 5 envelopes. One envelope, the "bomb", remains empty. In the basic version the player knows how much is put into each envelope, let's say $1000, $2000, $3000, $4000, and 0. Then the envelopes are shuffled and in each round you, the player, can either choose to get one of the remaining envelopes or keep the money you have collected so far and quit. If you ask for a new envelope, randomly one of the remaining envelopes is selected and handed to the player. If this envelope contains nothing (the bomb), the player you lose all the money you collected in the previous envelopes, otherwise you add the money to the money taken so far.

   How much would you "expect" to win when you were playing the game? Would you expect to win more or less if the $10000 were differently distributed on the envelopes, say as $2000, $2000, $2000, $2000, and 0? What if it were distributed as $100, $500, $2600, $7000, and 0? What about other distributions? Would you play different then? And what exactly does "expectation" mean?

   Version b: In another version, you, the player can distribute the money into the envelopes (using the requirement that one of them, the bomb, has to remain empty) before starting to play. How would you distribute the money, and how would you play?

   Version c: In still another version, the distribution of the money on the 5 envelopes is not known to the player. It is just known that exactly one envelope is empty and that the other four contain all the 10000 Dollar. How would expected value and strategy change? Would they change?