Suppose Bills and Brothers Co. and Sally and Sisters ltd are bargaining over the wages W...

  

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Suppose Bills and Brothers Co. and Sally and Sisters ltd are bargaining over the wages W at which Sally will perform a one-week computer programming service for Bill who is about to start an internet business. It is common knowledge to both parties that the maximum wage Bill will pay is $300 and the minimum wage Sally will accept is $200. As such, if they agree at some wage W, then Bill's payoff is $(300-W) and Sally's payoff is $(W-200). Suppose both players discount future income by 10% when they fail to reach an agreement in the first round and bargaining rolls over to the second round, which is all known to both players. (This 10% payoff loss can be interpreted as a lawyer fee you should pay for each round of bargaining). So, if they reached an agreement in the first round, , then they would receive the payoffs shown above. But if they reached agreement in the second round then their payoffs would be the discounted value (i.e. 90%) of that described above. (1) Bill first offers a wage W, to Sally, who can either accept or reject. If she accepts Bill's first offer, the game ends. If Sally rejects the offer, the game goes into the second round when she can ask for an alternative wage W, which he can accept or reject. If Bill accepts then each earns the discounted payoff but if he rejects then each receives a payoff of $0. The game does not go into a third round. It is assumed that each accepts an offer if (s)he is indifferent between accepting and rejecting it. Draw a game tree and solve for the subgame perfect Nash equilibrium. (2) How would your answer be different if Sally asks for a wage first, which Bill can accept or reject, and make another offer to Sally if he rejects Sally's first request? (The rest of the game rules are the same as in (1) and payoffs for each outcome are similarly determined.) Suppose Bills and Brothers Co. and Sally and Sisters ltd are bargaining over the wages W at which Sally will perform a one-week computer programming service for Bill who is about to start an internet business. It is common knowledge to both parties that the maximum wage Bill will pay is $300 and the minimum wage Sally will accept is $200. As such, if they agree at some wage W, then Bill's payoff is $(300-W) and Sally's payoff is $(W-200). Suppose both players discount future income by 10% when they fail to reach an agreement in the first round and bargaining rolls over to the second round, which is all known to both players. (This 10% payoff loss can be interpreted as a lawyer fee you should pay for each round of bargaining). So, if they reached an agreement in the first round, , then they would receive the payoffs shown above. But if they reached agreement in the second round then their payoffs would be the discounted value (i.e. 90%) of that described above. (1) Bill first offers a wage W, to Sally, who can either accept or reject. If she accepts Bill's first offer, the game ends. If Sally rejects the offer, the game goes into the second round when she can ask for an alternative wage W, which he can accept or reject. If Bill accepts then each earns the discounted payoff but if he rejects then each receives a payoff of $0. The game does not go into a third round. It is assumed that each accepts an offer if (s)he is indifferent between accepting and rejecting it. Draw a game tree and solve for the subgame perfect Nash equilibrium. (2) How would your answer be different if Sally asks for a wage first, which Bill can accept or reject, and make another offer to Sally if he rejects Sally's first request? (The rest of the game rules are the same as in (1) and payoffs for each outcome are similarly determined.)

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