I am trying to understand egalitarian social welfare in the context of resource allocation. By definition, the aim is to try and maximize the minimum utility value of an agent.
I am not sure if I understand this correctly. Does this mean that we are trying to find an allocation where the utility for the worst agent is as high as possible?
Also how do we find such an allocation? Given a set $A$ of agents $(a_1, a_2,\dots, a_m)$, a set $O$ of items $\{o_1, o_2,\dots, o_n\}$ and the utility function of agents for all the items, is there an algorithm to find an egalitarian maximizing allocation (assuming additive utilities and such allocation exists)?
Let $V_i$ be the utility of agent $i$. What you are describing is a maximin problem:
$$\max_{x_1,...,x_n} \min\{V_1(x_1),...,V_n(x_n)\}.$$
Let's illustrate this in a 3-person, 3-item economy. Suppose the items are $a,b,c$ (with no redundancies) and the agents get a utility of zero with nothing (so we will not leave anyone empty handed), and $$V_1(a)=2,V_1(b)=3,V_1(c)=3\\ V_2(a)=1,V_2(b)=4,V_2(c)=5\\ V_3(a)=5,V_3(b)=3,V_3(c)=1\\.$$
Thera are six possible allocations. We write each as a tuple, where $(x_1,x_2,x_3)$ represents giving $x_i$ to agent $i\quad \forall i\in\{1,2,3\}$.
$$(a,b,c)\implies \min\{V_1(a),V_2(b),V_3(c)\}=\min\{2,4,1\}=1\\ (a,c,b)\implies \min\{V_1(a),V_2(c),V_3(b)\}=\min\{2,5,3\}=2\\ (b,a,c)\implies \min\{V_1(b),V_2(a),V_3(c)\}=\min\{3,1,1\}=1\\ (b,c,a)\implies \min\{V_1(b),V_2(c),V_3(a)\}=\min\{3,5,5\}=3\\ (c,a,b)\implies \min\{V_1(c),V_2(a),V_3(b)\}=\min\{3,1,3\}=1\\ (c,b,a)\implies \min\{V_1(c),V_2(b),V_3(a)\}=\min\{3,4,5\}=3\\ $$
Thus, the maximin (best worst case) utility is $3$ and is achieved by the allocations $(b,c,a)$ and $(c,b,a).$ It is worth noting though that between these two, $(b,c,a)$ is a Pareto improvement. Thus, maximin allocations are not always Pareto efficient.
In general, the problem can be complicated to solve under an infinitude of allocations, but hope this helps clarify the concept. You may also find this and this helpful.