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1.1 Observe an auction with 5 bidders who have valuations v1 = 10, v2 = 8, v3 = 6, v4 = 4, v5 = 2 for an object. Given a profile of bids b1, , b5, the object is awarded to the agent with the highest bid. In the case of a tie, the object goes to those among the tied bidders with the lowest index (e.g. if Agents 2,3,5 are tied, Agent 2 wins the object). Agents may bid any non- negative amount. The price the winner of the object has to pay is the average between the 1\"t highest price and the 2nd highest price (e.g. if the bids are 5,4,3,2,1, then Agent 1 wins and pays (5+4)/2 = 4.5. Note that the 1St highest and 2nd highest may be the same e.g. if the bids are 4,4,3,2,1, then Agent 1 wins and pays (4+4)/2 = 4). Agents who do not win the object pay nothing. Is the profile of bids in which each agent bids their own valuation an equilibrium? If not, find an equilibrium. (20%) 1.2 For the model in 1.1, is there an equilibrium in which the agent with the lowest valuation (Agent 5) wins the object? (Hint: In such an equilibrium, how high can the winning bid be?) (15%) 1.3 Consider the model from part 1.1 with the following change: All agents who do not win the object have to pay their own bids (e.g. if the bids are 5,4,3,2,1, then Agent 1 wins and pays (5+4)/2 = 4.5, while each other agent pays their own bid: Agent 2 pays 4, Agent 3 pays 3, etc.). Show that there is no equilibrium. (Hint: Show first that if there were an equilibrium, all agents except the winner would bid 0.) (15%)