1. Consider set of alternatives and set of voters , that have the following preferences on set A:

1. Consider set of alternatives and set of voters , that have the following preferences on set A:а) Find the majority graph and Condorcet winners.b) What alternative should be chosen according to the Bordarule?c) Find the social decision if the federation rule will be used. Here means that alternative is less preferable than alternative for voter .2. Three friends decides whether to go to the night club on Friday night or not every week. The data of several votings is presented in the table.AndreyBobConstantineFinal decision1yesnonono2yesyesnoyes3noyesnono4yesyesyesyes5noyesyesnoThere is no more information about the voting system, the results of other polls are also unknown. Is their rule: 1) anonymous, 2) neutral, 3) monotone? Explain your answer.3. a) Check using definitions whether voting rule , where is a number of voters, satisfy:a. non-imposition condition,b. unanimity condition,c. monotonicity condition,d. neutrality condition,e. anonymity condition.Prove it or provide a counterexample.b) Check these properties using list representation of this rule for the case N = {1,2,3,4}.c) Does this rule always construct a linear order as social decision? A partial order?4. . The local rule has the following list representation:a) ,b) ,c) let А = {a, b, c} be the set of alternatives: , , .Check all normative properties using the list representation.5. Consider the Coombs rule that belongs to the class . Is this rule local, anonymous, neutral, monotone?

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