LOTR (1p) Frodo and Sam need to get to Mordor to destroy the One Ring, but...

  

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LOTR (1p) Frodo and Sam need to get to Mordor to destroy the One Ring, but they don't know the exact way. They met a creature called Gollum who claims to know the best way to Mordor. Gollum is obviously drawn to the One Ring and wants it for himself, but there is still a chance he is harmless and will lead Frodo and Sam safely to Mordor. If Frodo and Sam decide to trust Gollum and he will really help them, they will get utility of +10, but if Gollum betrays them, he will capture the One Ring and the Middle Earth will be doomed, resulting in their utility of -10. If they trust him and Gollum helps them get to Mordor, he will be left only with regret that he didn't take the ring and will get utility of -7, but if he betrays them, he will get the ring and get utility of +6. If Frodo and Sam don't trust him and go alone while he wants to help them, it will take them a lot longer, which leads to the utility of Frodo and Sam of +4. Gollum however cannot acquire the ring and therefore gets again utility of -7. If Gollum wants to betray them but they don't trust him, Gollum gets angry because his plot didn't work and attacks them straight. Frodo and Sam defeat him, resulting in utility of +2 for them and -5 for him, as he flees from the battle but at least tried to get the ring. a) write down the payoff matrix and find all the pure-strategy Nash equilibria. (0.1p) Frodo+Sam/Gollum Trust Help Betray Not trust b) now imagine that Gollum was enchanted by Gandalf and the utility from helping them will result in +7 of Gollum's utility regardless of their trust (and not -7 as before), while the rest of the payoffs remain the same. Write down the new payoff matrix and find all the pure-strategy Nash equilibria. (0.1p) Frodo+Sam/Gollum Trust Not trust Help Betray c) suppose Frodo knows that Gandalf has cast the spell but thinks it is a 40% chance that the magic was successful while 60% chance Gollum resisted the spell (i.e., 60% chance he plays the first game and 40% chance he plays the second game). Gollum knows whether spell worked or not. Find all pure-strategy Bayesian equilibria of this game. (0.4p) d) now suppose that the probability is switched because Frodo remembers that Gandalf did actually some pretty good magic in the past (i.e., he thinks with 60% chance playing the second game). Find all pure-strategy Bayesian equilibria of this game. (0.4p) LOTR (1p) Frodo and Sam need to get to Mordor to destroy the One Ring, but they don't know the exact way. They met a creature called Gollum who claims to know the best way to Mordor. Gollum is obviously drawn to the One Ring and wants it for himself, but there is still a chance he is harmless and will lead Frodo and Sam safely to Mordor. If Frodo and Sam decide to trust Gollum and he will really help them, they will get utility of +10, but if Gollum betrays them, he will capture the One Ring and the Middle Earth will be doomed, resulting in their utility of -10. If they trust him and Gollum helps them get to Mordor, he will be left only with regret that he didn't take the ring and will get utility of -7, but if he betrays them, he will get the ring and get utility of +6. If Frodo and Sam don't trust him and go alone while he wants to help them, it will take them a lot longer, which leads to the utility of Frodo and Sam of +4. Gollum however cannot acquire the ring and therefore gets again utility of -7. If Gollum wants to betray them but they don't trust him, Gollum gets angry because his plot didn't work and attacks them straight. Frodo and Sam defeat him, resulting in utility of +2 for them and -5 for him, as he flees from the battle but at least tried to get the ring. a) write down the payoff matrix and find all the pure-strategy Nash equilibria. (0.1p) Frodo+Sam/Gollum Trust Help Betray Not trust b) now imagine that Gollum was enchanted by Gandalf and the utility from helping them will result in +7 of Gollum's utility regardless of their trust (and not -7 as before), while the rest of the payoffs remain the same. Write down the new payoff matrix and find all the pure-strategy Nash equilibria. (0.1p) Frodo+Sam/Gollum Trust Not trust Help Betray c) suppose Frodo knows that Gandalf has cast the spell but thinks it is a 40% chance that the magic was successful while 60% chance Gollum resisted the spell (i.e., 60% chance he plays the first game and 40% chance he plays the second game). Gollum knows whether spell worked or not. Find all pure-strategy Bayesian equilibria of this game. (0.4p) d) now suppose that the probability is switched because Frodo remembers that Gandalf did actually some pretty good magic in the past (i.e., he thinks with 60% chance playing the second game). Find all pure-strategy Bayesian equilibria of this game. (0.4p)

Answer rating: 100% (QA)

1 Payoff Matrix and PureStrategy Nash Equilibria To create the payoff matrix for Frodo and Sam FS and Gollum G we need to consider their utilities in each of the four possible scenarios FS trust Gollu... View the full answer