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- A particular two-player game starts with a pile of diamonds and a pile of rubies. Onyour turn, you can take any number of diamonds, or any number of rubies, or an equalnumber of each. You must take at least one gem on each of your turns. Whoever takesthe last gem wins the game. For example, in a game that starts with 5 diamonds and10 rubies, a game could look like: you take 2 diamonds, then your opponent takes 7rubies, then you take 3 diamonds and 3 rubies to win the game.You get to choose the starting number of diamonds and rubies, and whether you gofirst or second. Find all starting configurations (including who goes first) with 8 gemswhere you are guaranteed to win. If you have to let your opponent go first, what arethe starting configurations of gems where you are guaranteed to win? If you can’t findall such configurations, describe the ones you do find and any patterns you see.Consider a two-player game that is set up with two piles of stones. The two players are taking turns removing stones from one of the two piles. In each turn, a player must choose a pile and remove one stone or two stones from it. The player who removes the last stone (making both piles empty) wins the game. Show that if the two piles contain the same number n ∈ Z+ of stones initially, then the second player can always guarantee a win.Suppose that each player in a two-person zero sum game has four strategies. Ignoring nonnegativity, how many constraints are in the linear programming formulation of this game for the row player? 3 5 2 4
- The Chinese government has created a fund worth more than 20 trillion won to foster the semiconductorindustry. Although there is a large technological difference between memory semiconductors, systemsemiconductors can be developed in a short period of time. The number of companies producinghomogeneous quality products has increased.■Question (a) In the Cournot game, when the number of firms increases from 2 to n, compare the output,total output, and profit of each firm with N.E. in the Cournot model. ■Question (b) If the number of companies participating in the semiconductor market increases to infinity,that is, in a perfectly competitive market, what will be the equilibrium point?Determine whether the following games have saddle points. For games without a saddle point,using the simplex method to find the optimal strategies and the values of the games.A tennis tournament has 85 participants. Players who lose a game are immediately eliminated; players who win a game keep playing. Still, the organizers have a lot of choices to make. They could give a first round bye to some players so that after the first round there are 26 = 64 players and no more bye are needed. Or they could give even a second round bye to the best players, or possibly even a third round bye to the very best ones. What is the best strategy for the organizers if they want to choose the winner of the tournament using as few games as possible?
- Two players, A and B, are playing an asymmetrical game. There are n points on the game board. Each turn player A targets a pair of points and player B says whether those two points are connected or unconnected. A can target each pair only once and the game ends when all pairs have been targeted. Player B wins iff a point is connected with all other points on the very last turn, while player A wins if any point is connected with all other points on any turn but the very last one OR if no point is connected to all other points after the last turn. For what values of n does either player have a winning strategy?The game of Chomp is played by two players. In this game, cookies are laid out on a rectangular grid. The cookie in the top left position is poisoned. The two players take turns making moves; at each move, a player is required to eat a remaining cookie, together with all cookies to the right and/or below (that is all the remaining cookies in the rectangle, in which the first cookie eaten is the top left corner). The loser is the player who has no choice but to eat the poisoned cookie. Prove that if the board is square (and bigger than 1 × 1) then the first player has a winning strategy.A fast food company is offering a prize promotion game. The company claims that in 0.5 % of all orders will receive $100 cash1% of all orders will receive $ 10 cash and 10\% of orders will receive a coupon for a free soft drink. The remaining orders will receive no prizes. A group of long- time customers were excited about the new promotion, and over the course of the promotion, they placed 651 ordersOf these orders, 2 ended up winning $100,6 won $10 and 52 won a free soft drink. If this group wanted to do a chi-squared goodness of fit test to test the advertised odds, what would be the appropriate null and alternative hypotheses?
- Suppose that there is a negotiation between two players over a painting. Person 1,the seller ,has no interest in the painting. On the other hand the painting is worth $100 to the buyer. If the painting is sold at a price in between 0 and 100, both are better off. Player 1 proposes a price p to player 2. Then after observing player 1's offer, player decides whether to accept it or to reject it. If the offer is rejected both zero and the game ends.Find the unique SPNE of this game.At a college basketball game, one lucky ticket holder will win an all-expenses-paid trip to the conference championship game. There are 4,000 ticket holders at the game. The organizers want to place chips, each labeled with the seat number of a ticket holder, in a bin and draw the winner from the bin. Unfortunately, the bin will hold only 400 chips. Which method assures both that the organizers can use the bin for the drawing and that each ticket holder will have a fair chance of winning? A. Place chips labeled with the seat numbers of each ticket holder into 10 groups based on age. Then randomly select one of the groups and place it in the bin. Randomly select the winner of the trip from the bin. B. Randomly assign the 4,000 ticket holders to 40 equal-sized groups. Then randomly select 4 of the groups and place their seat numbers in the bin. Randomly select the winner of the trip from the bin. C. Place 400 chips labeled with the seat numbers of ticket holders from a…Consider the following game with two dice (red and blue). The rules for the game areas follows:When a red and a blue dice are rolled simultaneously, the result is only acceptable if:• the sum of the numbers on the two dice is not greater than 10• the number value on the red dice is less than or equal to 5• the numerical value on the blue dice is greater than 1;• the number on the blue dice is not more than double the number of the red dice By sketching the feasible region, make a list of all possible throws?