Describe the definition of a pure strategy Nash equilibrium and illustrate a 2-player simultaneous game with one unique pure strategy Nash equilibrium. Question 2 Consider a 5G tower deployment game between two telecommunication firms Stel and Shub. Assume that Stel is ahead in the deployment process. Shub is deciding whether to join the competition (Choice: Join or Not Join). Shub will gain nothing if it chooses not to join. An estimated profit of $100M will go to Stel if there is no competitor in the market. If Shub decides to join and deploy 5G towers, then Stel will decide whether to share the market or to compete for the market with a price war (Choice: Share or Compete). If Stel shares, each firm will get a profit of $30M. If Stel competes for the market, each firm will lose $10M. (a) (b) (c) Interpret the game with a game tree. Analyse and find the rollback equilibrium. Rewrite the game in the strategic form and find the pure strategy Nash equilibrium. Describe the concept of a subgame perfect equilibrium and verify if the Nash equilibrium in (b) is a subgame-perfect equilibrium. Question 3 Consider a simultaneous game between two rational players Bob and Jane. The payoff matrix is as follows. (a) (b) Movie Bob Game Jane Movie Game 2,7 5,6 3,8 4,9 Using the best response analysis method, verify that the game has no pure strategy equilibrium. Find the mixed strategies equilibrium and calculate the expected payoffs for both players. (c) (d) Before the game, Jane tries to make an agreement with Bob to coordinate the action of choosing "Game". She provides two reasons: 1) From the payoff matrix, the sum of the payoffs in each cell is highest when both players choose "Game. 2) If both players adopt a mixed strategy, the expected payoff for each player could be lower than the payoff when both choose "Game. Appraise Jane's reasoning. Assume that one player can observe the strategy of the other player before he/she makes the choice. Is it still a simultaneous game? Discuss if there is any order effect (i.e. first-mover or second-mover advantage) under this assumption. Question 4 Consider a game between two supermarkets FP and SS in CCK Mall. For every week, each firm will decide whether to offer a discount price or maintain the normal price for all the items in the store. If they both offer a discount price, each can have a payoff of $15M for that week. If they both maintain a normal price, each can have a payoff of $16M for that week. If only one of them offers a discount price, then the firm offering the discount price will receive $17M for that week while the firm which maintains the normal price will receive $14M for that week. (a) Fill the payoff matrix below. Verify that this one-time game is a prisoners' dilemma. (b) SS Maintain Offer FP Maintain Offer Suppose that both companies agree to maintain the normal price each week and this game is repeated weekly forever. Assume also that SS uses a Tit-For-Tat strategy. Is it worthwhile for FP to defect just once then return to cooperating in the subsequent weeks, given the weekly total rate of return r=0.5? Explain. Question 5 Jimmy is living alone. Every week he buys a list of groceries from a local wet market where he can bargain for a lower price. Unfortunately, he is ill and unable to walk out of his house recently. He finds one helper online. (a) (b) Jimmy signs a contract with the helper. Jimmy will give $10 each time to the helper for buying a list of groceries from the wet market using Jimmy's credit card. Is the contract a "cost-plus contract" or "fixed-price contract"? Indicate the contract type and explain your answer. Assume that Jimmy expects the price of the items bought by the helper could be close to Jimmy's previous purchasing price. To motivate this, Jimmy modifies the contract to offer an extra $5 if the helper could try his or her best to bargain before buying the items. Show your views on Jimmy's modified contract according to your knowledge from mechanism design.