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Game theory and starategy; Problem 6 (20 Points) [From Practice HW]. Consider the scene from The Princess Pride where Prince Humperdinck discovers Wesley alive in

Game theory and starategy;

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Problem 6 (20 Points) [From Practice HW]. Consider the scene from "The Princess Pride" where Prince Humperdinck discovers Wesley alive in a bedroom with Princess Buttercup after having killed him earlier that day. Prince Humperdinck suspects that Wesley has no strength (is Weak) but isn't sure. He also knows that if Wesley is strong there is no way he could take him. Wesley therefore has two types: Strong and Weak. Prince Humperdinck believes the probability that he is strong is only 1/4 (after all Wesley has been mostly-dead all day). Wesley has two strategies, he can get out of bed O (act tough, bluff, etc.) or he can stay in bed, B. After observing Wesley's action, Prince Humperdinck can choose to surrender S or fight F. Wesley's payoffs are listed first. (1, 0) S (1, 0) B Wesley O (0, -2).* Strong (0, -2) Humperdinck Humperdinck O Nature (1, 0) Weak :. 0) S 1- 1 1 -7 B' Wesley (-1, 1). 1) Find all Perfect Bayes Nash equilibria (PBE) of the game.[From Practice HW]. Consider the signaling game below. There are 2 players, Pl and P2. Player 1 has 2 types, N and S. Nature chooses / and probability 1/2 and S at probability 1/2. After learning their type, P1 chooses either Left L or right R. Player 2 observes player 1's strategy and then updates beliefs about player 1's type before choosing a strategy up U or down D. (3.2) (2. 0).+ L R D N (1,0) Nature C (1,0) (2.0).+ L P1 R P2 D (1, 1) The separating pure strategy profiles for this game are (LN / RS, U) (RN / LS, D) (RN / LS, U) (RN/ LS, D) The pooling pure strategy profiles for this game are (LN/ LS, U) (LN/ LS, D) (RN/ RS, U) (RN / RS, D) (A) (10 points) Can any of the separating strategy profiles above be sustained as a Perfect Bayes Equilibria (PBE)? If so, fully describe which ones and any required conditions on beliefs. (B) (10 points) Can any of the pooling strategy profiles above be sustained as a Perfect Bayes Equilibria (PBE)? If so, fully describe which ones and any required conditions on beliefs.Nature Gunslinger Cowpoke Stranger Stranger D. W D W Wyatt Earp D W D W D W D W Stranger 00 19 Wyatt Earp 2 09 There are 8 pure strategy profiles for the simultaneous game of incomplete information above. The stranger's strategies are listed first, then Wyatt Earps. (D" / DC, D) (DG/WG, D) (WG/ DC, D) (WG/WC, D) (DG /DC, W) (DG/WC. W) (WG/DC, W) (WG/WC, W) (A) (5 points) What is the Gunslinger type's dominant strategy? Which of the pure strategy profile's above does this eliminate from being possible BNEs? (B) (15 points) Find the 2 pure strategy Bayes-Nash equilibria of the game

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