Game theory. Show all the strategies

1. (a) What are N, S and u in the definition of a finite normal-form (or, equivalently, strategic-form) game G = (N. S, u)? What is the mixed-strategy extension, G = (N.0(5), #) of such a game G? [1 pt] (b) In terms of the mixed-strategy extension G of an arbitrary finite game G: Give the definitions of a strictly dominated strategy, a weakly dominated strategy, a Nash equilibrium, and a perfect equilibrium. [2 pts] (c) Consider the following finite two-player game G, representing price competition in a market where all consumers buy from the seller(s) with the lowest price. Both sellers have to simultaneously choose a price, p and p2, where p; e P = {0, 1, 2,3, 4). The profits to each seller are given in the payoff bi-matrix below, where seller 1 chooses row and seller 2 column. Find all strictly dominated pure strategies, all weakly dominated pure strategies, all pure-strategy Nash equilibria, and all pure-strategy perfect equilibria. [2 pts] PI\\pz 0 1 2 3 4 C -5, -5 -10,0 -10.0 -10,0 -10,0 1 0. -10 0,0 0.0 0.0 0,0 2 0. -10 0,0 3.3 6.0 6.0 3 0. -10 0.0 0.6 4.4 8,0 0. -10 0,0 0.6 0.8 3,3 Solution (a) See book or lecture notes.2. There are n 2 1 partners who together own a firm. Each partner i chooses an effort level r; 2 0, resulting in total profit g(y) for their firm, where y is the sum of all partners' efforts. The profit function g : R - R+ satisfies g(0) = 0 and it is twice differentiable with g' > 0, and g"