4 Pirates of the Caribbean (5 points) When Spain colonized the New World, they e

Question

4 Pirates of the Caribbean (5 points) When Spain colonized the New World, they extracted untold riches from there to finance their wars in Europe. The gold and silver had to be transported back to Europe on ships and these were attractive targets for pirates and privateers from other countries. Imagine the fictional Capitano Juan Barcodoro, who is in Cartagena and has to bring his cargo safely back to Sevilla, but he knows that the ruthless French pirate François Dragon is lurking. He can sail one of two routes: following the coast to Trinidad or go for Santo Domingo on Hispaniola. The pirates have to decide where they try to intercept him. Barcodoro is on the left and François Dragon on top, and their payoffs are on the left right. François Trinidad Hispaniola E Trinidad10/6 3/-2 Hispaniola | 4 /-3 | -10 / 5 What is Barcodoro's best response if he believes the pirates are at Trinidad? What if he believes they are at Hispaniola? Also find the pirates' best responses. Is there a Nash equilibrium in pure strategies? What is the Nash Equilibrium in mixed strategies (state probabilities in fractions)?

Explanation / Answer

Barcodoro's Best Response:

If Barcodoro believes that Francois (the pirates) are at Trinidad, then his best response is to sail on Hispaniola. This is because when he believes that the pirates are at Trinidad, he has the option to sail at Trinidad and obtain a payoff of -10 or he can sail at Hispaniola and get a payoff of 4. Sine 4 > -10, his best response to the pirates being at Trinidad is to sail through Hispaniola.

Similarly, if he believes that the pirates are at Hispaniola, he should sail through Trinidad. This is again because when he believes that the pirates are at Hispaniola, he has the option to sail at Trinidad with a payoff of 3 or he can sail at Hispaniola and get a payoff of -10. Since 3 > -10, his best response is to sail through Trinidad.

Pirates Best Response:

If the pirates believe that Barcodoro is at Trinidad, then the Pirates' best response is to go to Trinidad. This is because when he believes that Borcodoro is at Trinidad, he can either go to Trinidad and get a payoff of 6 or go to Hispaniola and get a payoff of -2. Since 6 > -2, the pirates best response is to go to Trinidad, when believes that Borcodoro is at Trinidad.

If the pirates believe that Barcodoro is at Hispaniola, then the Pirates' best response is to go to Hispaniola. This is because when he believes that Borcodoro is at Hispaniola, he can either go to Trinidad and get a payoff of -3 or go to Hispaniola and get a payoff of 5. Since 5 > -3, the pirates best response is to go to Hispaniola, when believes that Borcodoro is at Hispaniola.

There is no Nash Equilibrium in pure strategies. We can see this using the Best Response approach above. None of the players have a pure dominant strategy that would yield to a pure strategy Nash Equilibrium. In this case, we will have to find a Mixed Strategy Nash Equilibrium (MSNE):

We will assume that player Borcodoro plays Trinidad with probability q and plays Hispaniola with probability 1-q. Similarly, player Francois (the pirate) plays Trinidad with probability p and plays Hispaniola with probability 1-p.

Expected payoff for player Borcodoro:

Expected payoff for playing Trinidad = -10p+3(1-p) = -10p+3-3p = 3-13p

Expected payoff for playing Hispaniola = 4p-10(1-p) = 4p-10+10qp= 14p-10

to find p: 3-13p = 14p-10 -------> 13 = 14p+13p = 27p ------> p = 13/27 ; 1-p = 14/27

Expected payogg for player Francois:

Expected payoff for playing Trinidad = 6q-3(1-q) = 6q-3+3q = 9q-3

Expected payoff for playing Hispaniola = -2q+5(1-p) = -2q+5-5q = 5-7q

to find q: 9q-3 = 5-7q -------> 16q = 8 -------> q = 1/2 ; 1-q =1/2

As we can see, player Barcodoro's Best Response depends on p and player Francois's Best Response dependds on q.

Thus, the MSNE = {1/2 Trinidad + 1/2 Hispaniola ; 13/27 Trinidad + 14/27 Hispaniola}

Related Questions

Navigate

• Browse (All)
• Browse 4
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.