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QUESTION 1 There are two customer service workers; worker i (i{1, 2}) chooses an effort level [0, ) i e to exert in helping a customer. The cost of effort is 2 1 1 1 ( ) 15 e c e for worker 1 and 2 2 2 2 ( ) 10 e c e for worker 2. The customer's satisfaction level equals the total effort that the workers exert, up to a maximum satisfaction level of 10, that is, it is equal to min{ ,10} 1 2 e e . Each worker's payoff is the difference between the customer's satisfaction level and his/her own effort cost. Consider first the game G in which the two workers choose their efforts simultaneously. (a) Write down each worker's payoff function in G. (b) Describe each worker's best response (or reaction) function in G. (c) Draw the two best response functions in the same diagram and find all the pure-strategy Nash equilibria of G. Now consider sequential versions of the interaction. In game 2 , first Player 1 (= worker 1) selects an effort level, then Player 2 (= worker 2) after observing Player 1's choice chooses her own effort level. (d) Completely describe all the pure-strategy subgame perfect equilibria of 2 . In game 3 , the sequence of events starts with those in 2 , but after Player 2 moves, Player 1 observes Player 2's choice and decides whether to exert some additional effort 1 [0, ), so that his own total effort is 1 1 e . (e) (e.1) What is a pure strategy for Player 1 in 3 ? (e.2) What is a pure strategy for Player 2? [You are free to state your answers in words, rather than in mathematical notation; but, in either case, your answers should be explicit and complete.] (e.3) Give an example of a pure strategy for Player 1 and a pure strategy for Player 2 in the special case where each effort level must be chosen from the set {3,4}, that is, when 1 2 1 e e, , {3,4}. (f) Completely describe all pure strategy subgame perfect equilibria of 3 when 1 2 e e, [0,10] (instead of [0, )) and 1 1 [0, 10 ] (instead of [0, )).

Consider a simultaneous two-player second-price auction concerning a single indivisible good. The game-frame is as follows: 1 2 S S B where 1 2 { , ,..., } B p p p m is a finite subset of [0, ) with k k 1 p p for every k m {1,..., 1}, the set of outcomes is the set of pairs ( , ) i p where i{1, 2} is the winner of the auction and p B is the price that winner has to pay and the outcome function is as follows ( i b denotes the bid of player i): 2 1 2 1 2 1 (1, ) if ( , ) (2, ) otherwise b b b f b b b . Let i v be the value of the object to Player i (that is, Player i views getting the object as equivalent to getting $ i v ). We shall consider three kinds of preferences. We state them in terms of Player 1, but the same definitions apply to Player 2. The following apply to all three preferences (this is the "selfish" part): for every p < 1 v and for every p, 1 (1, ) (2, ) p p ; for every p and p, 1 (1, ) (1, ) p p if and only if p p . Player 1 is selfish and uncaring if, in addition, his preferences are as follows: for every p and p, 1 (2, ) (2, ) p p ; for every p, 1 1 (2, ) (1, ) p v ; and everything that follows from the above by transitivity. Player 1 is selfish and benevolent if, in addition, his preferences are as follows: for every p and p, 1 (2, ) (2, ) p p if and only if p p ; 1 1 (2, ) (1, ) mp v ; and everything that follows from the above by transitivity. Player 1 is selfish and spiteful if his preferences are as follows: for every p and p, 1 (2, ) (2, ) p p if and only if p p ; 1 1 1 (2, ) (1, ) p v ; and everything that follows from the above by transitivity. In what follows assume that m > 3, 1 2 v v B , , 1 1 m p v p and 1 2 m p v p . (a) Suppose that Player 1 is selfish and uncaring. Does she have a weakly or strictly dominant strategy? If your answer is Yes, state whether it is weak or strict dominance; if your answer is No, prove it. (b) Suppose that Player 1 is selfish and benevolent. Is bidding 1 v a weakly dominant strategy? Prove your claim.

For questions (d), (e) and (f) assume that B {1,2,3,4,5}, 1 v 3 and 2 v 5 (d) Suppose that both players are selfish and uncaring. Find all the pure-strategy Nash equilibria. (e) Suppose that both players are selfish and benevolent. Find all the pure-strategy Nash equilibria. (f) Suppose that both players are selfish and spiteful. Find all the pure-strategy Nash equilibria. Now suppose that we have a situation of incomplete information. As a matter of fact, both players are selfish and benevolent (B), but this is not common knowledge. The following is common knowledge: 1. Each player knows her own preferences. 2. Player 1 is either selfish and benevolent (B) or selfish and uncaring (U). If Player 1 is B then she attaches probability 1 to Player 2 being B, while if Player 1 is U then she attaches probability 1 4 to Player 2 being B and probability 3 4 to Player 2 being S (= selfish and spiteful). 3. Player 2 is either B or S. If Player 2 is B then she attaches probability 1 6 to Player 1 being B and probability 5 6 to Player 1 being U, while if Player 2 is S then she attaches probability 1 to Player 1 being U. (g) Use an interactive knowledge-belief structure to represent this situation. (h) For this question assume that 1 2 B p p { , }, 1 2 2 v v p and the auction is not simultaneous: Player 1 bids first and Player 2 bids second after having been informed of Player 1's bid. (h.1) Apply the Harsanyi transformation to the incomplete-information situation of part (g), without writing payoffs but writing the outcomes. (h.2) Suppose that you know that the players have von Neumann-Morgenstern preferences, but all you know is that their von Neumann-Morgenstern preferences induce the ordinal rankings given above. Is it possible to find a pure-strategy weak sequential equilibrium of the game of part (h.1)? If you claim that it is not possible because not enough information is given, state what extra information you would need, otherwise find a pure-strategy weak sequential equilibrium and prove that it is indeed a weak sequential equilibrium.