Few questions about game theory

Q1. Consider the strategic games described below. In each case, state how you would classify the game according to the six dimensions outlined in the text.

Are moves sequential or simultaneous?

Is the game zero-sum or not?

Is the game repeated?

Is there imperfect information, and if so, is there incomplete (asymmetric) information?

Are the rules fixed or not?

Are cooperative agreements possible or not?

If you do not have enough information to classify a game in a particular dimension, explain why not.

(a) Rock-Paper-Scissors: On the count of three, each player makes the shape of one of the three items with his hand. Rock beats Scissors, Scissors beats Paper, and Paper beats Rock.

(b) Roll-call voting: Voters cast their votes orally as their names are called. The choice with the most votes wins.

(c) Sealed-bid auction: Bidders on a bottle of wine seal their bids in envelopes. The highest bidder wins the item and pays the amount of his bid.

Q2. You and a rival are engaged in a game in which there are three possible outcomes: you win, your rival wins (you lose), or the two of you tie. You get a payoff of 50 if you win, a payoff of 20 if you tie, and a payoff of 0 if you lose. What is your expected payoff in each of the following situations?

(a) There is a 50% chance that the game ends in a tie, but only a 10% chance that you win. (There is thus a 40% chance that you lose.)

(b) There is a 50–50 chance that you win or lose. There are no ties.

(c) There is an 80% chance that you lose, a 10% chance that you win, and a 10% chance that you tie.

Q3. Find out equilibrium of the game by using backward induction