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Social choice theory is the framework used to analyze collective decision processes and voting systems. At its core, this theory seeks to understand how groups can make decisions that reflect the preferences and welfare of its individual members. It deals with the aggregation of preferences, dealing with notions such as fairness, utility, and efficiency.

Consider a classroom choosing between watching a documentary or a movie; social choice theory would study the mechanisms the class could use to make this decision. An ideal mechanism would take into account each student's preference to arrive at a decision that best represents the group's welfare. However, as Arrow's Impossibility Theorem points out, finding such a mechanism that is fair in all respects is a complex, if not impossible, endeavor.

Voting systems are the methods by which groups make decisions based on the preferences of individual members. These systems range from simple majority vote to more complex mechanisms like rank-order voting. Each system tries to translate personal preferences into a collective decision in the fairest possible way. However, fairness can be subjective and depends on the criteria used to evaluate it.

When a classroom votes by a show of hands on whether to have a pop quiz or not, they're using a simple majority voting system. If they were to rank their preference of potential quiz dates, they would be using a rank-order voting system. The choice of system can significantly impact the outcome and the perceived fairness of the decision, highlighting the uncertainties and complexities inherent in social choice.

Fairness criteria in voting are foundational principles designed to ensure that a voting system operates justly and equitably. Kenneth Arrow identified several such criteria, and these include non-dictatorship, unanimity or Pareto efficiency, and independence of irrelevant alternatives (IIA).

A non-dictatorship criterion ensures one person doesn't unilaterally control the outcome. Unanimity or Pareto efficiency dictates that if every member of a group prefers one alternative over another, the group’s choice should reflect that preference. IIA requires that the collective ranking of any two options is unaffected by individual rankings of irrelevant alternatives. These criteria represent the pillars of a fair voting system, yet Arrow's Impossibility Theorem demonstrates the challenge in satisfying all simultaneously.

Decision theory is a field that grapples with the logic and philosophy behind making choices. It brings together probability, statistics, and game theory to help individuals and organizations choose among alternatives to maximize outcomes. Decisions can range from the mundane, such as choosing what to eat for dinner, to the high-stakes, such as choosing a country's political leadership.

Decision theory lays the groundwork for understanding the implications of different voting systems. Arrow's Impossibility Theorem is a landmark result within this field, shedding light on the limitations of collective decision-making systems. By analyzing the systemic constraints that impact voting outcomes, decision theory informs the design of more effective and fairer decision-making processes within groups.