A polling problem I have been thinking about a problem recently twisting my mind. Let me give a brief description of the problem. Imagine $n$ number of people who are about to vote $m$ different objects to chose one of them. Each person has to vote for two objects, therefore, they are expected to initiate a primary and a secondary vote. A primary vote corresponds to $x$ points, and the secondary one is $y$ points for each person, but $x$ > $y$ and $y \not = 0$. If an object collects the majority of points, it will be selected. What are the minimum values of $x$ and $y$ so that the polling does not tie for first place? All unknowns here are positive integers.

In general there are no values of $x$ and $y$ that would guarantee no ties. In particular, suppose that $m \geq n \geq 2$, and number the items as $0, \dots, m-1$ and the people as $0, \dots, n-1$. Now each person $i$ can cast the primary vote to item $i$ and the secondary vote to item $i+1\ \mathrm{mod}\ n$. This way, each of the items $0,\dots, n-1$ gets voting score $x+y$, and the remaining items $n,\dots,m-1$ get score $0$. No matter what the values of $x$ and $y$ are, the first $n$ items are tied for the first place.
The same can be done also if $n$ is divisible by some $2\leq k\leq m$. Then we can split people in $k$ groups, and let each group vote as above. Each item $0,\dots,k-1$ ends up with score $\frac n k(x+y)$.
