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- Solve for all mixed Nash equilibria for the three-person game below. X Y X Y х 3, —2,4 | 0, —3, 2 Y 0, -3,2 0, –3, 2 х 0, —3,2 | 0, —3, 2 Ү 0, —3, 2 | 12, 1, 10 6. XP1 L W P1 P1 R U J P2 P2 P2 F M H T P1 5, 4 -1,1 7,0 2,6 7,4 2,1 7,3 K V 4,1 2,1 a)Find the subgame perfect equilibria of this game b)Would IDSDS eliminate any equilibria you found in (a)? If so, which ones? If not, why not?Computer Science Use graph to answer the questions a) Which pairs of variables will be made independent by conditioning on A? b) Is it possible to switch the direction of a single edge to make ?⊥⊥? | ? , and still maintain a valid DAG? If so, list the two variables connected by the edge that should be switched. c) Give the expression for the factorized joint probability distribution of all the variables (A, B, C, D, E) that is specifically implied by this graphical model.
- 4. The matching game is played in a bipartite graph G = (V1, V2, E) in which edges are connect only vertices Vị to vertices in V2. The players are the vertices in the graph that is Vi U V2. Each player has to select one of its neighbors. Player i gets utility 1 when the selection is mutual (player i selects j and player j selects i) otherwise he gets 0. Provide a formal characterization of the strategy profiles that are pure Nash equilibrium of the matching game. Analyze the complexity of the problems related to pure Nash equilibria for this family of games.A seller has an indivisible asset to sell. Her reservation value for the asset is s, which she knows privately. A potential buyer thinks that the assetís value to him is b, which he privately knows. Assume that s and b are independently and uniformly drawn from [0, 1]. If the seller sells the asset to the buyer for a price of p, the seller's payoff is p-s and the buyer's payoffis b-p. Suppose simultaneously the buyer makes an offer p1 and the seller makes an offer p2. A transaction occurs if p1>=p2, and the transaction price is 1/2 (p1 + p2). Suppose the buyer uses a strategy p1(b) = 1/12 + (2/3)b and the seller uses a strategy p2(s) = 1/4 + (2/3)s. Suppose the buyer's value is 3/4 and the seller's valuation is 1/4. Will there be a transaction? Explain. Is this strategy profile a Bayesian Nash equilibrium? Explain.A two-person zero-sum game with an nn rewardmatrix A is a symmetric game if A AT.a Explain why a game having A AT is called asymmetric game.b Show that a symmetric game must have a value ofzero.c Show that if (x1, x2,..., xn) is an optimal strategyfor the row player, then (x1, x2,..., xn) is also an opti-mal strategy for the column player.d What examples discussed in this chapter are sym-metric games? How could the results of this problemmake it easier to solve for the value and optimal strate-gies of a symmetric game?
- 1. Consider the generalization of stable matching problem where a certain man-woman pairs are forbidden. The set of forbidden pairs is denoted as F. Each man m ranks all the woman w for which (m, w) ¢ F and each woman w ranks all the man m for which (m, w) ¢ F. Consider the following algorithm for finding a stable matching that consists of only unforbidden pairs: Initially all mE M and w e W are free; S = Ø While there is a man m who is free and hasn't proposed to every woman w for which (m, w) ¢ F Choose such a man m Let w be the highest-ranked woman in m's preference list to which m has not yet proposed If w is free: Add (m, w) to solution S Else (w is current matched with m'): If w prefers m' to m m remains free Else Replace (m', w) by (m, w) in S Return solution S Answer true or false for the following questions: (a) Any woman w remains engaged from the point at which she receives her first proposal, and the se- quence of partners to which she is engaged get better and better. (b)…Correct answer will be upvoted else downvoted. Computer science. Each digger should mine precisely one jewel mine. Each excavator has a snare, which can be utilized to mine a jewel mine. On the off chance that an excavator at the point (a,b) utilizes his snare to mine a precious stone mine at the point (c,d), he will burn through (a−c)2+(b−d)2−−−−−−−−−−−−−−−√ energy to mine it (the distance between these focuses). The diggers can't move or help one another. The object of this game is to limit the amount of the energy that diggers spend. Would you be able to track down this base? Input The input comprises of different experiments. The main line contains a solitary integer t (1≤t≤10) — the number of experiments. The portrayal of the experiments follows. The main line of each experiment contains a solitary integer n (1≤n≤105) — the number of diggers and mines. Every one of the following 2n lines contains two space-isolated integers x (−108≤x≤108) and y (−108≤y≤108),…1. Consider the following set P of men's and women's preferences. Preferences Men Women Preferences T U V V U T U T V V T S А S D АВ C В S T ADCВ C D A C D C S U B A D U V В (a) Find the stable matching for set P, that would be produced by running the Gale- Shapley algorithm as presented in class, in which the men do the proposing. (b) Set P has k different stable matchings. What is k? If you cannot determine k exactly, give the best upper bound and the best lower bound that you can. Justify your statement.
- Correct answer will be upvoted else downvoted. Computer science. way from block u to impede v is an arrangement u=x0→x1→x2→⋯→xk=v, where there is a street from block xi−1 to obstruct xi for each 1≤i≤k. The length of a way is the amount of lengths over all streets in the way. Two ways x0→x1→⋯→xk and y0→y1→⋯→yl are unique, if k≠l or xi≠yi for some 0≤i≤min{k,l}. Subsequent to moving to another city, Homer just recollects the two unique numbers L and R yet fails to remember the numbers n and m of squares and streets, separately, and how squares are associated by streets. Be that as it may, he accepts the number of squares ought to be no bigger than 32 (on the grounds that the city was little). As the dearest companion of Homer, if it's not too much trouble, let him know whether it is feasible to view as a (L,R)- constant city or not. Input The single line contains two integers L and R (1≤L≤R≤106). Output In case it is difficult to track down a (L,R)- ceaseless city…Algorithm for Match in a pyramid tournament.in: set P of players in the pyramid structure; players p and q (p, q ∈ P ∧((rank(p) = rank(q) ∧ ¬peerWinner(q)) ∨ (rank(p) = rank(q) − 1 ∧peerWinner(q))))out: set R of players after p and q have had a matchlocal: match outcome mn = 9. sign and Analysis of Algorithms (3rd ed.) [Levitin 2011-10-09].pdf ✓ 1 2 3 zoom O 4 O 5. How would you modify the dynamic programming algorithm for the coin- collecting problem if some cells on the board are inaccessible for the robot? Apply your algorithm to the board below, where the inaccessible cells are shown by X's. How many optimal paths are there for this board? 1 3 4 5 6 8.1 Three Basic Examples 2 X O 5 XXX 2 O Xo X O Search Fri May 12 291 6. Rod-cutting problem Design a dynamic programming algorithm for the fol- lowing problem. Find the maximum total sale price that can be obtained by cutting a rod of n units long into integer-length pieces if the sale price of a piece i units long is p; for i = 1, 2, ..., n. What are the time and space efficiencies of your algorithm? 7. Shortest-path counting A chess rook can move horizontally or vertically to any square in the same row or in the same column of a chessboard. Find the number of shortest paths by which a rook can move…