Question

Problem 5. Game Theory

Two firms use a common resource (a river or a forest, for example) to produce output. As the total amount of the resource used by the two firms increases, each firm can produce less output. Denote by xi the amount of the resource used by firm i. Assume that firm i’s output is xi(1- x1 - x2) if x1+x2 <= 1, and zero otherwise. Each firm i chooses xi to maximize its output. Find the Nash equilibria of the game. Find an action profile (x1, x2) at which each firm’s output is higher than it is at any Nash equilibrium.

<= refers to "less than or equal to"

Answer 1

player attaches no value to the time spent waiting for the other player to concede, but the
object in dispute loses value as time passes. (Think of a rotting animal carcass or a melting
ice cream cone.) Assume that the value of the object to each player i after t units of time
is vi – t (and the value of a 50% chance of obtaining the Specify the
strategic game that models this situation (take care with the payoff functions). Find the players’ best response functions, and hence
find the Nash equilibrium of the game.