Question

In: Economics

Consider a game in which there are 3 people. Each player can either participate or not...

Consider a game in which there are 3 people. Each player can either participate or not participate in a revolution. If at least 2 people participate, the revolution succeeds. If the revolution succeeds, each player gets a benefit of B. If the revolution fails, each player gets a benefit of 0.

Each player who participates bears a cost c < B, whether or not the revolution succeeds.

(a) Write down each player’s best response correspondence.

(b) Is it a Nash equilibrium for no players to participate? Why or why not?

(c) Is there a Nash equilibrium in which only one player participates? Why or why not?

(d) Is there a Nash equilibrium in which only two players participate? Why or why not?

(e) Is it a Nash equilibrium for all three players to participate? Why or why not?

Solutions

Expert Solution

(a) Best response of a player

  • If no other player participate
    In the case, by participating he gets the payoff of -c as revolution will fail. But if he does not participate he gets 0. So, the best response is not to participate.
  • If one other player participate
    In this case, by participating there will be at least 2 players are part of revolution. Thus he gets B - C which is positive. And, if he doesn't participate he gets 0. Hence, his best response is to participate.
  • If other two players participate
    In this case, revolution will succeed even without participating. But he has to inccur a cost for participating. Thus his best response is not to participate.

(b) Yes, Given no other player is participation the best response is not to participate. Hence, everybody is playing there best response, and it is an equilibrium.

(c) No, Consider a player who is not participating. Given that one other player is participating, his best response will be to participate. Since he is not playing his best response this is not an equilibrium.

(d) Yes, Consider a player who is not participating. Given that two other player is participating, his best response will be not to participate. So, he is not playing his best response. Now consider a player who is participating, if he decide to not to participate his payoff will decrease from B-c to 0. Hence, he is playing his best responce. Thus this is an equilibrium.

(e) No, Consider any player his best response will be not to participate. Hence this is not an equilibrium

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Rahul Sunny answered 1 year ago
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