EXERCISES IN GAME THEORY 1. Bob and Alice have strategies "Run" and "Stop" in a game. If both of them run, they both get the payoff 1. If Bob runs and Alice stops then the payoffs are -1 and -3 respectively. But if Bob stops and Alice runs, the payoffs are 2 and -1 respectively. If both Bod and Anna stop then the payoffs are 2 and 3 respectively. a) Construct the normal form of the game. b) Determine if there is a dominant strategy in the given normal form of a game. 2. What should be the value of a so that Player 1 will have a dominant strategy? PLAYER 2 C D PLAYER A (3,4) (0,5) 1 B (a, 0) (1.1)3. Determine the Nash Equilibrium for the following normal form of the game. A) HARLEY UP DOWN UP (10,10) (8,4) MARY DOWN (4.8) (5,5) B) PLAYER 2 LEFT CENTER RIGHT UP (3.0) (2.1) (0.0) PLAYER MIDDLE (1. 1) (1,1) (5.0 1 DOWN (0.1) (4,2) (0.1) C) Apply the iterative elimination. PLAYER 2 D E F A (3,1) (0,1) (0.0) PLAYER B (1,1 ) (1,1) (5.0) C (0,1) (4.2) (0.1)D) Apply the iterative elimination. PLAYER 2 W X Y Z A (6.3) (6.2) (6,4) (6,7) PLAYER B (1,1) (1.99) (1.4) (1,0) 1 C (3.7) '99,2) (3,99) (0,1) D (5.1) (0.2) 99.4) 5.5) 4. Draw an extensive form of the corresponding normal form in #4 (B, C, & D) then apply the backward induction method to determine the Nash Equilibrium.5. Determine the best decision for Company A using backward induction, Consider the following: 13t Branch: Company A will join () the market or not (NJ ) and 2 0 Branch: If A joins the market, Company B will make negative ads (P), positive ads (Ne), or do neither (Nothing - No) 3"d Branch: If B will make the ads whether positive or negative, Company A will either retaliate (R) or leave (L) the market. Re (2,4) A Ne L (0, 1) B Re (5,5) A A L (1, 0) N (4, 3) (4,4 )