1. Consider the following normal form game between A and B. Use iterated elimination of dominated strategies, if possible, to find the solution to the three games. B C1 C2 C3 1, Y 7,7 3, 10 A R1 5, 4 R2 Х, 12 9, 10 a. Let X = 5 & Y = 2. Find the solution. b. Let X = 6 & Y = 9. Find the solution. c. Let X = 3 & Y = 11. Find the solution.
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- c) Knowing that we have the equality: 12Ar+3= B2hez determine the value of x and encircle it.(This problem assumes knowledge of the basic rules of basketball.) One controversial topic in basketball (college or any other level) is whether to foul a player deliberately with only a few seconds left in the game. Consider the following scenario. With about 10 sec- onds left in the game, team A is ahead of team B by three points, and team B is just about to inbound the ball. Assume team A has committed enough fouls so that future fouls result in team B going to the free-throw line. If team A purposely commits a foul as soon as possible, team B will shoot two foul shots (a point apiece). The thinking is that this is better than letting team B shoot a three-point shot, which would be their best way to tie the game and send it into over- time. However, there is a downside to fouling. Team B could make the first free throw, purposely miss the second, get the rebound, and score a two-point shot to tie the game, or it even score a three-point shot to win the game. Examine this decision,…Consider the following bridge crossing problem where n people with speeds s1, ··· , sn wish to cross the bridge as quickly as possible. The rules remain: • It is nighttime and you only have one flashlight. • A maximum of two people can cross at any one time • Any party who crosses, either 1 or 2 people must have the flashlight with them. • The flashlight must be walked back and forth, it cannot be thrown, etc. • A pair must walk together at the rate of the slower person’s pace. Give an efficient algorithm to find the fastest way to get a group of people across the bridge. You must have a proof of correctness for your method.
- - The bucket has a weight of 400 N and is being hoisted using three springs, each having an unstretched length of 6 = 0.45 m and stiffness of k = 800 N/m. Determine the vertical distance d from the rim to point A for equilibrium. 400 N D. 120 120° 0.45 m 120° B 3d EF, = 0; 400 – F = 0 d² + (0.45)² 400 N 3d [800 Jd² + (0.45)² -0.45)] = 0 400 d² + (4.5)² |d² + (0.45)² -0.45) = 0.16667 0.45 m d² + (4.5)² d d² + (0.45)² -0.45 d = 0.16667 Jd² + (0.45)² Va? + (0.45)² (d– 0.16667) = 0.45 0.45 m d [d² + (0.45)*] [d° – 2d(0.16667) + (0.16667)°] = (0.45)² d² d* – 0.33334 d + 0.027779 d – 0.0675 d + 0.0056252 = 0 120° 0.45 m d = 0.502 m AnsPlease solve a,b, c, and d.b) What happens inside PC-1(Permuted choice 1) and PC-2(Permuted choice 2)? How they are different from each other? c) Solve the following equations. 3302 mod 13 [Euler's theorem]. 250 mod 17 [Fermat’s theorem]. 1. 2.
- Question 4 A red die, a blue die, and a yellow die (all six-sided) are rolled. We are interested inthe probability that the number appearing on the blue die is less than that appearing on theyellow die which is less than that appearing on the red die. (That is, if B (R) [Y ] is the numberappearing on the blue (red) [yellow] die, then we are interested in P(B < Y < R). )(a) What is the probability that no two of the dice land on the same number?(b) Given that no two of the dice land on the same number, what is theconditional probability that B < Y < R?(c) What is P(B < Y < R)?8. (a) Show that a = Vxa if x is not free in a. (b) Show that (a) need not hold if x is free in a.Problem 3: A day at the beach. A group of n people are lying on the beach. The beach is represented by the real line R and the location of the i-th person is some integer r; e Z. Your task is to prevent people from getting sunburned by covering them with umbrellas. Each umbrella corresponds to a closed interval I = [a, a + L] of length L e N, and the i-th person is covered by that umbrella if æ¡ € I. Design a greedy algorithm for covering all people with the minimum number of umbrellas. The input consists of the integers x1,..., En, and L. The output of your algorithm should be the positions of umbrellas. For example, if the input is r1 = 1, x2 = 3, r3 = 5, and L = 2, then an optimum solution is the set of two umbrellas placed at positions 2 and 5, covering intervals [1,3] and [4,6]. 1 2 3 4 5 6 Prove that your algorithm is correct and that it runs in time polynomial in n.
- Question: Suppose there are n people in a group, each aware of a scandal no one else in the group knows about. These people communicate by telephone; when two people in the group talk, they share information about all scandals each knows about. For example, on the first call, two people share information, so by the end of the call, each of these people knows about two scandals. The gossip problem asks for G(n), the minimum number of telephone calls that are needed for all n people to learn about all the scandals. a). Find G(1), G(2), G(3), and G(4). b). Use mathematical induction to prove that G(n) 4. [Hint: In the inductive step, have a new person call a particular person at the start and at the end.] c). Prove that G(n)= 2n – 4 for n> 4.Please solve the following. Thank you.Solve the linear programming below by (a) graphing and (b)using simplex method. A certain company makes 2 models of motorcycles, model 2A and model 2B. Both models pass through assembling and painting. A model 2A motorcycle needs 4 hours for in assembling and 3 hours in painting. A model 2B motorcycle needs 8 hours in assembling and 12 hours in painting. Machines in assembling are available for 64 hours, while those in painting, 72 hours. The profit for each model 2A motorcycle is P2700 and P3600 for a model 2B motorcycle. How many model 2A and 2B motorcycles must be manufactured to maximize profit? What is the maximum profit? Solve by graphing Correct objective function and constraints Correct Process (conversion-determining points – verifying at point (0,0) Correct graphand correct values and interpretation of x, y, and z Using Simplex method Correct placement of values of the first matrix Correct process and complete solution (determining the pivot row, column, and…