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What are the meanings of a social welfare function and social profile? How are they related?

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In its most general formulation, a social welfare function is just a utility function representing the preferences of "society as a whole" (or the preferences of a hypothetical "benevolent social planner" who makes decisions for the society).

Let $X$ be some space of "social outcomes". (Social outcomes could be anything. But if you want a concrete example, then suppose that social outcomes describe allocations of resources to individuals. If there are $N$ individuals, and there are $M$ distinct types of resources, the an allocation is an ($N \times M$)- dimensional vector, so in this case $X$ would be some subset of $\mathbb{R}^{N\times M}$.) A social welfare function is just a function $W:X \rightarrow\mathbb{R}$; heuristically, if $x$ is some element of $X$ (some "social outcome") then $W(x)$ measures the "total welfare" or "social value" or "overall desirability" (or whatever) of $x$. Thus, the benevolent social planner should aim for policies that maximize the value of $W$.

The key question, of course, is how to define $W$. One way to do this is to somehow build $W$ out of the preferences or utility functions of individuals. Suppose there are $N$ individuals, each of whom has a preference order $\succeq_i$ on $X$. This collection of preference orders $(\succeq_1,\succeq_2,\ldots,\succeq_N)$ is an (ordinal) social profile. (This partly answers your second question.) A big part of the modern theory of social choice and welfare is about defining systematic, principled ways of defining $W$ based on the profile $(\succeq_1,\succeq_2,\ldots,\succeq_N)$.

Typically, purely ordinal preference information about the individuals is not sufficient. (This is one way of reading Arrow's Impossibility Theorem --I won't get into the details here). So we might want to endow each individual with a utility function $u_i:X \rightarrow \mathbb{R}$ (which is more informative than just a preference order). A collection of utility functions $(u_1,u_2,\ldots,u_N)$ is a (utility) social profile. (This is the rest of the answer to your second question.) We then seek to construct $W$ out of $(u_1,u_2,\ldots,u_N)$.

For example, given a social profile of utility functions $(u_1,u_2,\ldots,u_N)$, we could define $W(x):=u_1(x)+u_2(x)+\cdots+u_N(x)$ for all $x\in X$ ---this is the utilitarian social welfare function. Or we could define $W(x):=\min\{u_1(x),u_2(x),\ldots,u_N(x)\}$ for all $x\in X$ ---this is the egalitarian (or Rawlsian) social welfare function. Or (assuming all utilities are strictly positive), we could define $W(x):=u_1(x)\cdot u_2(x) \cdots u_N(x)$ for all $x\in X$ ---this is the Nash social welfare function.

Different social welfare functions have different properties, which may be more or less desirable, depending on your beliefs about moral philosophy and social justice. Much of social welfare theory can be seen as developing a mathematically precise way to think about moral philosophy and social justice, by understanding the properties of different social welfare functions and the tradeoffs between these properties. For example, axiomatic characterization theorems begin with a list of axioms (each encoding a property corresponding to our intuitions about rationality, morality, justice, etc.) and show that there is a unique social welfare function that satisfies all of these axioms.

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