Consider a game of voluntary contribution to a public radio, that is, two individuals, A and...

 

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Consider a game of voluntary contribution to a public radio, that is, two individuals, A and B, contribute to a public radio and once the radio broadcasts anybody can listen to it. In particular, we have two listeners: A and B; and a public radio G. Both Listener A and B has an income 20. Each listener chooses how much to contribute to support public radio, G₁ in [0; 20] and GB in [0; 20]. Listener A's preferences are given by the payoff function U₁(GA; GB) = In (20-G₁) + 3 In (G); and Listener B's are by UB(GA GB) In (20 GB) + 3 In (G) where G=G₁; + GB. a. Find the Nash equilibrium and the payoffs at the Nash equilibrium. b. Find the cooperative solution, that is, contributions of the two players that maximize the total payoff of the two players. c. Assuming that at the cooperative solution the two players contributes at the same amount, find payoffs of the two players and compare them with payoffs at the Nash equilibrium. d. Assume that Listener A has enjoys the public radio a lot. So that, his payoff function is UA(GA; GB) = In (20-GA) = ln (20-GA) + 6 - In (G). Find the Nash equilibrium of the game when Listener A's payoff function is given as above and compare the payoffs of the two listeners when Listener A's payoff is given as in part (a) and in part (d). Consider a game of voluntary contribution to a public radio, that is, two individuals, A and B, contribute to a public radio and once the radio broadcasts anybody can listen to it. In particular, we have two listeners: A and B; and a public radio G. Both Listener A and B has an income 20. Each listener chooses how much to contribute to support public radio, G₁ in [0; 20] and GB in [0; 20]. Listener A's preferences are given by the payoff function U₁(GA; GB) = In (20-G₁) + 3 In (G); and Listener B's are by UB(GA GB) In (20 GB) + 3 In (G) where G=G₁; + GB. a. Find the Nash equilibrium and the payoffs at the Nash equilibrium. b. Find the cooperative solution, that is, contributions of the two players that maximize the total payoff of the two players. c. Assuming that at the cooperative solution the two players contributes at the same amount, find payoffs of the two players and compare them with payoffs at the Nash equilibrium. d. Assume that Listener A has enjoys the public radio a lot. So that, his payoff function is UA(GA; GB) = In (20-GA) = ln (20-GA) + 6 - In (G). Find the Nash equilibrium of the game when Listener A's payoff function is given as above and compare the payoffs of the two listeners when Listener A's payoff is given as in part (a) and in part (d).

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a Find the Nash equilibrium and the payoffs at the Nash equilibrium The Nash equilibrium is given by ... View the full answer