file:///var/mobile/Library/SMS/Attachments/a6/06/D3E6A9BA-7B5F-4246-A794-B53AED0B24EE/hw5_24spring.pdf 1. (10pts) For the following game (Player 1 chooses T or B; Player 2 chosses C, or R): payoffs L overline C R T overline 2.4 0,3 1, 2 B 3, 0 1,1 0, 3 Note: (b) can be solved without solving (a) ( a) (5pts) Show that for player 2, some mixed strategy involving only L and R qL+(1-; q)R, strictly dominates pure strategy C. (You should find some value of for it to be true.) ( b) (5pts) Now, eliminate strategy C. Then find all mixed- strategy and pure strategy equilibria in the simplified game. (Hint : no need to use mixed- strategy best response functions.) 2. (10pts) Consider again Example 1 in Lecture Notes Sequential -Move Games. This time assume the players reverse their order of moves: Player B goes first and Player A goes second ( a) ( 4pts) Draw the game tree. Make sure to assign players to correct nodes , label edges with actions, and write payoffs next to appropriate terminal nodes ( b) (3 pts) Use backward induction to solve the game: an arrow on each edge that each player will choose when the player gets the chance ( ) ( 3pts) What is the subgame perfect equilibrium (SPE )? What is the SPE outcome What are the SPE payoffs? 3. (16pts) (Stackelberg Duopoly ) Two firms produce identical products. Each has the same constant average cost of $10 per unit of output. Market price P = 110 - 2(q_{1} + q_{2}) Both firms choose outputs to compete. ( ) (5pts) Find the subgame perfect equilibrium outcome of the Stackelberg Duopoly game with Firm 1 moving first. First, solve for the follower's (Firm 2's) best response function. Then solve for the leader's optimal strategy. ( b) (4pts) Find the Nash equilibrium of the Cournot duopoly under the same assumptions on costs and demand . ( ) (3pts) Compare the two equilibria . Discuss the differences as to which firm earns more profit in which case. ( d) (4pts) Suppose again firm 1 chooses output first and firm 2 chooses output second. But now before firm makes a decision, firm 2 makes a claim that it would produce the Cournot equilibrium level of output regardless of what output q firm chooses first. If firm 1 believes firm 2's claim, then what is the best output q_{1} for firm 1 to choose first Compared with standard Stackelberge equilibrium in part (a), verify if firm 2 will earn more profit in this case. What might be the challenge for firm 2 to achieve this outcome