Consider a world withkmen andhwomen. Mani’s height isxiand womanj’sheight isyjfor alliandj. Assume that no two individuals in this world havethe same height. Also assume that both men and women prefer taller partners toshorter ones. Finally, both men and women prefer to be married to being single.(a) How many stable matchings are possible in this world? Find them.(b) Are there any unstable Pareto efficient matchings? Explain your answer.(c) Suppose all women prefer shorter partners to taller ones. Is there more thanone stable matching in this case?4. Consider the government that wants to hire a contractor to build a bridge. ThereareNpotential contractors to choose from. Each contractor is characterized bya costciof building the bridge. The cost is drawn independently from a uniformdistribution with the support [α,β], whereβ > α >0. Each potential contractorknows its own cost, but does not know the costs of others. All contractors arerisk-neutral expected profit maximizers.(a) Suppose a government decides to run a second-price sealed bid tender to select acontractor. All potential contractors simultaneously and independently submittheir quotes to the auctioneer. A contractor who submits thelowestquote, getsthe contract. The government pays this contractor the second lowest quote.This tender is similar to the second-price sealed bid auction, but since thegovernment tries to minimize the expenditure, it selects the lowest price, ratherthen the highest price. Argue that submitting contractori’s own private costcias a quote is a weakly dominant strategy for contractori.(b) Suppose a government decides to run a first-price sealed bid tender to select acontractor. All potential contractors simultaneously and independently submittheir quotes to the auctioneer. A contractor who submits thelowestquote, getsECON0027 Game Theory, HA52CONTINUED
the contract. The government pays this contractor his quote. This tender issimilar to the first-price sealed bid auction, but since the government tries tominimize the expenditure, it selects the lowest price, rather then the highestprice. Write down the payoff maximization problem for a contractori. What isthe expected probability of winning the tender for a contractoriif he submitsa quotebi.(c) Find a symmetric Bayesian Nash Equilibrium in the first-price sealed bid tenderdescribed above. You can assume that in this equilibrium a contractor with thehighest type possible submits a quote that is equal to his costs:ci=β=⇒bi=β(Hint:it may be useful to make the following change of variables:θi=β−ciβ−αandpi=β−biβ−α. If you do so, give the intuition about what these newvariables represent.