Meaning of = in antisymmetric relation We have a set of students.
$$
\{\text{Bob}, \text{John}, \text{Tom} \}
$$
Scores of a exam were:
$$
\begin{align}
\text{Bob} & : 5 \\
\text{John} & : 10 \\
\text{Tom} & : 10 \\
\end{align}
$$
In the real life, we can sort by scores and say students are ordered.
$$\text{Bob} \le \text{John}$$
$$\text{Bob} \le \text{Tom}$$
$$\text{John} \le \text{Tom}$$
$$\text{Tom} \le \text{John}$$
But order relations are said to be antisymmetric.
$$
xRy \land yRx \Rightarrow x = y
$$
In the example,
$$
\text{John} \le \text{Tom }\land \text{Tom} \le \text{John} \Rightarrow \text{John} = \text{Tom}
$$
So, John and Tom must be equal, if the example is an ordered set. But it is only the scores that matches. If you compare the elements, they are different and John $\ne$ Tom.
Is it allowed to compare scores and say $\text{John} = \text{Tom}$? Can't we say that the example is an ordered set?

Adding some comment to make my question clear:
As nachosemu said, I am doing
$$
x R y \Leftrightarrow score(x) \leq score(y), \,\,\, x,y\in \{\text{Bob}, \text{John}, \text{Tom} \}
$$
For this example, should I read antisymmetric relation as comparing elements:
$$
xRy \land yRx \Rightarrow x = y
$$
or as $=$ in $\lt, =, \gt$:
$$
xRy \land yRx \Rightarrow score(x) = score(y)
$$
 A: In this case,
The (partial order) relation R is defined:
$$
x R y \Leftrightarrow score(x) \leq score(y), \,\,\, x,y\in \{Bob, John, Tom\}
$$
After this, you are comparing this relation with another relation R':
$$
x R' y \Leftrightarrow x = y, \,\,\, x,y\in \{Bob, John, Tom\}
$$
which is not the same one. Here, R' is an equivalence relation.
A: Starting with the bottom line: You can say either $\mathop{score}(\mathrm{John}) = \mathop{score}(\mathrm{Tom})$, or $\mathrm{John} \sim \mathrm{Tom}$ (not equal, cause they are different persons, but equivalent). Here $\sim$ is simply defined as meaning "having the same score".
You can also define an ordering directly on the students by $x\lesssim y\iff score(x)\le score(y)$ (this is basically what you're doing in the question). If you do this, what you get is in fact a total preorder on the set of students. A total preorder is the same as a total order, except it's not required to be anti-symmetric. But we can say $x\lesssim y\wedge y\lesssim x \iff x\sim y$. This is closely related to a total order (which is anti-symmetric): If you group the students into equivalence classes corresponding to points, you get a total order of two sets
$$
\{\mathrm{Bob}\} <  \{\mathrm{John, Tom}\}
$$
As such, total preorders are total orders of equivalence classes. See also the top picture on the linked wiki-page.
A: The terminology "ordered set" does not really have a fixed meaning. This seems to be a perfectly reasonable confusion in the original question. It is especially confusing as the terminology "partially ordered set" does almost always mean the same thing. That is, a set together with a relation which is reflexive, antisymmetric and transitive.
There is no general agreement about what exactly an "order" is, and even more confusingly the word "total" can mean different things in combination with other words. For example, the difference between partial orders and total orders is not the same as the difference between partial functions and total functions even though all four kinds of thing are binary relations.
As other answers have already made clear, a given set can have lots of different binary relations. When modelling the world, more than one relation may capture something useful.
