In the projective plane PG(2, q), upper bounds on the smallest size t 2(2, q) of a complete arc are considered. For a wide region of values of q, the results of computer search obtained and collected in the previous works of the authors and in the present paper are investigated. For q ≤ 301813, the search is complete in the sense that all prime powers are considered. This proves new upper bounds on t 2(2, q) valid in this region, in particular $$\begin{array}{ll}t_{2}(2, q) \;\; < 0.998 \sqrt{3q {\rm ln}\,q} \quad\;\;\,\,{\rm for} \;\; \qquad \quad \;\;7 \leq q \leq 160001;\\ t_{2}(2, q) \;\; < 1.05 \sqrt{3q {\rm ln}\, q}\qquad\,\,{\rm for}\;\; \qquad \quad \;\;7 \leq q \leq 301813;\\ t_{2}(2,q)\;\; < \sqrt{q}{\rm ln}^{0.7295}\,q \qquad \,\,\,\,\,{\rm for} \;\; \quad \quad \,\,\,109 \leq q \leq 160001;\\ t_{2}(2,q) \;\; < \sqrt{q}{\rm ln}^{0.7404}\,q \qquad \,\,\,\,\,{\rm for }\;\;\, \quad 160001 < q \leq 301813.\end{array}$$ The new upper bounds are obtained by finding new small complete arcs in PG(2,q) with the help of a computer search using randomized greedy algorithms and algorithms with fixed (lexicographical) order of points (FOP). Also, a number of sporadic q’s with q ≤ 430007 is considered. Our investigations and results allow to conjecture that the 2-nd and 3-rd bounds above hold for all q ≥ 109. Finally, random complete arcs in PG(2, q), q ≤ 46337, q prime, are considered. The random complete arcs and complete arcs obtained by the algorithm FOP have the same region of sizes; this says on the common nature of these arcs.