Five pirates encounter a mission to capture a treasure ad each pirate decides si
ID: 1137587 • Letter: F
Question
Five pirates encounter a mission to capture a treasure ad each pirate decides simultaneously whether to go on the mission or notGoing out on the mission costs 7. If the pirates successfully capture the treasure, each pirate gets a payoff of 20. Thus, the payoff from going on a successful mission is 13 and the payoff from going on an unsuccessful mission is-7. The mission is only sucessful if all five pirates go. Pirates who don't go on the mission get a payoff of 0. a. Explain why it is a Nash Equilibrium for all the pirates to go on the mission. b. Explain why it is a Nash Equilibrium for none of the pirates to go on the mission. Find a symmetric, mixed-strategy Nash Equilibrium where all five pirates go on the mission with probability p c.Explanation / Answer
a) We can model this situation as a two mission strategic game wherein mission one is going on a mission for pirates and mission two is not going on a mission for pirates. To find the Nash Equillibria we examine each action profile in turn. 1) Niether pirate can increase his payoff by choosing an action different from the other. Thus this action profile is a Nash Equillibrium. 2)Niether pirate can increase his payoff by by not capturing the treasure, which means in order to increase their payoff the pirates have to capture the treasure. This action profile is a Nash Equillibrium. 3) None of the pirates can succeed, if the other one decides not go on a mission i.e. if pirates aligned alternately not to go, then they cannot go on a mission which means they cannot increase their payoffs. This action profile is a Nash equillibrium. We conclude that the action of going on a mission is a Nash Equillibrium.
b) To examine the pirates not going on a mission is a Nash equillibrium we consider the following actions. 1) By choosing not to go on a mission when other pirates go on a mission, the other pirates get a payoff, given the action of pirate one. Thus in turn is an action of Nash Equillibrium. 2) By choosing to go on a mission when other pirates do not go on the mission, pirate one gets a payoff, given the action of other pirates. This action is not a Nash equillibrium. 3) While other pirates successfully capture the treasure and pirate one does not capture the treaure, the other pirates get a payoff, despite the action of pirate one. This action is not a Nash equillibrium.4) If none of the pirates go on a mission, they do not get any payoff. This is a Nash equillibrium. Thus we conclude that it is Nash equillibrium if none of the pirates goes on a mission.
c) In a matrix form (p,-p) (-p,p) (-p,p) (p,-p). If no equillibrium exists in pure strategies, at least one should exist in mixed strategy. A mixed strategy is probability distribution over two or more pure strategies. That is the the pirates choose randomly among their options in equillibrium. In equillibrium, each players probability distribution makes others indifferent between their pure strategies.
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