Here I assume that no self-intersections mean no repeating vertex. Answer: Yes. Solution. Business's strategy consists of three phases: 1. Business chooses a random city $c_0$. On each turn, he randomly chooses a city $c_{k+1}$ that is previously unselected so that the road $c_kc_{k+1}$ is not destroyed, then he buys that road. Do this $72$ times to get a (bought) path $c_0c_1\ldots c_{72}$. 2. Business spends the next $50$ days choosing new cities $\ell_1,\ell_2,\ldots,\ell_{50}$ and buying the roads $c_0\ell_1,c_0\ell_2,\ldots,c_0\ell_{50}$, then the next $50$ choosing $r_1,r_2,\ldots,r_{50}$ and buying the roads $c_{72}r_1,c_{72}r_2,\ldots,c_{72}r_{50}$. (All $173$ chosen cities so far must be distinct.) This is always possible because given a city $c$, at most $1710$ roads from $c$ are destroyed and at most $172$ cities are already chosen, but there are $2008 > 1710+172$ possible roads from $c$. 3. Now it's the morning of Day $173$. Consider the roads $\ell_i r_j$ for $i,j\in\{1,2,\ldots,50\}$. There are $2500$ such roads, but at most $1720$ are destroyed. Now Business simply picks a non-destroyed road $\ell_x r_y$, getting the cyclic route $\ell_x c_0 c_1 \ldots c_{72} r_y$ which passes through exactly $75$ different cities. $\blacksquare$