Hey guys, i have to solve these questions. I already came up with a solution. Can anyone look over it and state their opinion? Many thanks in advance 
CHALLENGE I
1) The number of Nash Equilibria of the game presented in the first question is one. The first thing that comes into the mind of the players should be that the “winning number” is lower than 33 (or in extreme case 33 itself) since all of the higher numbers than 1/3 of 100 are “loosing numbers” for the simple reason that they are higher than 1/3 of the highest number. So now they will have the same game but with a smaller range: 1-33, and if all of the player are thinking in the same way then the new “winning number” would be 11, if they keep going on this reasoning (strong and necessary assumption) the only available solution as Nash Equilibrium is the number 1 since all the others number will be eliminated one by one.
(1) Is this a Nash equilibrium? Why or why not?
(2) Recall that Leonard did start the auction with an opening bid of €0.01. In retrospect, was it wise of Penny to respond with a bid of €0.02? Explain.
2.1) This is not a Nash Equilibrium and never it will.
The reasoning behind it is that both of the player keep raising their bid since they don’t want to lose their money and this procedure continues, as we have already seen, until their bidding arrives to 0,99$, in favour of Leonard. Now Penny obviously does not want to lose its 0,98$ and would prefer to pay 1$ for 1$, that will lead to an offer of 1$ from Penny. The situation then becomes more complicated.
In this way Leonard would lose 0,99$ if he does not raise the bid, and he would prefer to pay 1.01$ and lose just 0,01$ winning the bidding than obviously lose 0,99$, and then it comes his offer. This reasoning can go on until the end of the day and for them would always be better do another offer and “lose less” than stop the bidding.
2.2) It wasn’t wise at all, since it led to the situation discussed in the point 2.1.
It would have been better for Penny to bid directly 0,99$, in this way she would have not arrived at the uncomfortable situation discussed before, and she would have gained at least 0,01$ (coming out with a slight prize, but still far better off than the previous situation).
However, if possible an even better decision would have been to not bid and share the profits, in that case it would have been 1-0,01= 0,99 0,99/2 = 0,495 for both. In addition, they should request Sheldon to play this game repeatedly but only If they agree on not biding. Because from the point onwards if the lower bid reaches 0,50 $ Sheldon makes profit, whereas penny and Leonard are less likely to make profits if they try to outbid each other.
CHALLENGE I
- Consider the following game between Sheldon, Leonard, Rajesh, Howard and Penny who all pay €1 to participate. Each player must simultaneously choose a number (integers) between 1 and 100. All aim to win the game by outguessing one another. The person that is closest to 1/3 of the average of all guesses wins €100. The others get nothing. In case of ties, the €100 is split equally between the winners. This game has a unique Nash equilibrium; what is it and why?
1) The number of Nash Equilibria of the game presented in the first question is one. The first thing that comes into the mind of the players should be that the “winning number” is lower than 33 (or in extreme case 33 itself) since all of the higher numbers than 1/3 of 100 are “loosing numbers” for the simple reason that they are higher than 1/3 of the highest number. So now they will have the same game but with a smaller range: 1-33, and if all of the player are thinking in the same way then the new “winning number” would be 11, if they keep going on this reasoning (strong and necessary assumption) the only available solution as Nash Equilibrium is the number 1 since all the others number will be eliminated one by one.
- Sheldon, who won the previous game, feels a bit sorry for the “losers”. To get rid of this uncomfortable mood, he wants to offer the others an opportunity to win back part of their submission fee. Towards that end, he decides to auction €1. The rules are as follows: (1) The highest bidder wins the €1 and (2) the two highest bidders both have to pay their bids to Sheldon. Leonard and Penny are both eager to win back part of their money and Leonard places a bid of €0.01. Penny responds with a bid of €0.02. This process of outbidding each other continues until at some point Penny bids €0.98 and Leonard responds with a bid of €0.99.
(1) Is this a Nash equilibrium? Why or why not?
(2) Recall that Leonard did start the auction with an opening bid of €0.01. In retrospect, was it wise of Penny to respond with a bid of €0.02? Explain.
2.1) This is not a Nash Equilibrium and never it will.
The reasoning behind it is that both of the player keep raising their bid since they don’t want to lose their money and this procedure continues, as we have already seen, until their bidding arrives to 0,99$, in favour of Leonard. Now Penny obviously does not want to lose its 0,98$ and would prefer to pay 1$ for 1$, that will lead to an offer of 1$ from Penny. The situation then becomes more complicated.
In this way Leonard would lose 0,99$ if he does not raise the bid, and he would prefer to pay 1.01$ and lose just 0,01$ winning the bidding than obviously lose 0,99$, and then it comes his offer. This reasoning can go on until the end of the day and for them would always be better do another offer and “lose less” than stop the bidding.
2.2) It wasn’t wise at all, since it led to the situation discussed in the point 2.1.
It would have been better for Penny to bid directly 0,99$, in this way she would have not arrived at the uncomfortable situation discussed before, and she would have gained at least 0,01$ (coming out with a slight prize, but still far better off than the previous situation).
However, if possible an even better decision would have been to not bid and share the profits, in that case it would have been 1-0,01= 0,99 0,99/2 = 0,495 for both. In addition, they should request Sheldon to play this game repeatedly but only If they agree on not biding. Because from the point onwards if the lower bid reaches 0,50 $ Sheldon makes profit, whereas penny and Leonard are less likely to make profits if they try to outbid each other.