Suppose there are three individuals in society trying to rank three social states \((A, B,\) and C). For each of the methods of social choice indicated, develop an example to show how (at least) one of the Arrow axioms will be violated. a. Majority rule without vote trading. b. Majority rule with vote trading. c. Point voting where each voter can give \(1,2,\) or 3 points to each alternative and the al ternative with the highest point total is selected.

Question: Give an example that demonstrates each of the following social choice methods failing to satisfy one of Arrow's axioms: (a) Majority rule without vote trading, (b) Majority rule with vote trading, and (c) Point voting. Answer: (a) Majority rule without vote trading violates the Independence of Irrelevant Alternatives axiom, as shown in the scenario where removal of alternative C changes the ranking between A and B. (b) Majority rule with vote trading also violates the Independence of Irrelevant Alternatives axiom, as the same scenario from example a applies even with vote trading. (c) Point voting, with individuals allocating 1, 2, or 3 points per alternative, violates the Independence of Irrelevant Alternatives axiom when the removal of alternative B changes the ranking between alternatives A and C.