Sheldon and Leonard had a million-dollar idea. In order to make it happen, they have to...

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Sheldon and Leonard had a million-dollar idea. In order to make it happen, they have to do special research first. Only Kripke can help them in this matter. But Kripke is known to be the first-class snake of the whole university. He will help them, but only if their research benefits his own research ideas. If their research turns out to be useless for him, he will steal their idea and sell it as his own. If the S&L idea does not help Kripke's own project and he helps them, he would go against his natural nastiness and get -8. If they trust him and he betrays them, he gets +8 because he can make fun of them and sell the idea as his own, but it won't be as good as when S&L were included. If he wants to help them but they do not ask for it, he suffers a loss of -4 on his ego utility. If S&L do not trust Kripke while he wants to help them, they miss their chance and will never finish their paper (utility -5). If they trust him and get his help, they finish the project (+10) but if they trust and are betrayed, they lose their idea and will be humiliated (-10). If S&L do not trust Kripke and he would betray them, the outcome is 0 for him while they are happy as they saved their idea, even though they cannot finish it (+3). a) write down the payoff matrix and find all the pure-strategy Nash equilibria if Sheldon's and Leonard's research will not help Kripke's research. (0.1p) Help Sheldon, Leonard / Kripke Trust Not trust Betray b) Now imagine that the research of S+L will actually help Kripke's research. In such case, payoffs from his actions change: betrayal means -4 in any case, helping them while they trust him leads to +8, and wanting to help them while they do not trust him leads to -3 of his ego utility. Write down the new payoff matrix and find all pure-strategy Nash equilibria. (0.1p) Sheldon, Leonard / Kripke Help Betray Trust Not trust c) Suppose that Sheldon and Leonard are not sure if Kripke is working on something meaningful (i.e., that their research will help him). They think it is a 30% chance that he is, while 70% chance is not. Kripke is sure of what he is doing. Find all pure-strategy Bayesian equilibria of this game. (0.4p) d) Suppose that the probability is now switched because Kripke invested his time into their own research (i.e., they think there is 70% chance, their research will benefit his). Find all pure- strategy Bayesian equilibria of this game. (0.4p) Sheldon and Leonard had a million-dollar idea. In order to make it happen, they have to do special research first. Only Kripke can help them in this matter. But Kripke is known to be the first-class snake of the whole university. He will help them, but only if their research benefits his own research ideas. If their research turns out to be useless for him, he will steal their idea and sell it as his own. If the S&L idea does not help Kripke's own project and he helps them, he would go against his natural nastiness and get -8. If they trust him and he betrays them, he gets +8 because he can make fun of them and sell the idea as his own, but it won't be as good as when S&L were included. If he wants to help them but they do not ask for it, he suffers a loss of -4 on his ego utility. If S&L do not trust Kripke while he wants to help them, they miss their chance and will never finish their paper (utility -5). If they trust him and get his help, they finish the project (+10) but if they trust and are betrayed, they lose their idea and will be humiliated (-10). If S&L do not trust Kripke and he would betray them, the outcome is 0 for him while they are happy as they saved their idea, even though they cannot finish it (+3). a) write down the payoff matrix and find all the pure-strategy Nash equilibria if Sheldon's and Leonard's research will not help Kripke's research. (0.1p) Help Sheldon, Leonard / Kripke Trust Not trust Betray b) Now imagine that the research of S+L will actually help Kripke's research. In such case, payoffs from his actions change: betrayal means -4 in any case, helping them while they trust him leads to +8, and wanting to help them while they do not trust him leads to -3 of his ego utility. Write down the new payoff matrix and find all pure-strategy Nash equilibria. (0.1p) Sheldon, Leonard / Kripke Help Betray Trust Not trust c) Suppose that Sheldon and Leonard are not sure if Kripke is working on something meaningful (i.e., that their research will help him). They think it is a 30% chance that he is, while 70% chance is not. Kripke is sure of what he is doing. Find all pure-strategy Bayesian equilibria of this game. (0.4p) d) Suppose that the probability is now switched because Kripke invested his time into their own research (i.e., they think there is 70% chance, their research will benefit his). Find all pure- strategy Bayesian equilibria of this game. (0.4p)