Creating a $\epsilon-\delta$ game is really interesting. Thanks Charles Matthews! BTW, a similar strategy has been stated by Prof.Terry Tao in:Thinking and Explaining, http://mathoverflow.net/questions/38882 (version: 2011-10-12)
One other issue that usually undergrads feel elusive about in $\epsilon-\delta$ method is why is it "for every $\epsilon>0$ there exists a corresponding $\delta>0$" and not the other way round. In this context, the following simple analogy may illustrate the point:
Assuming the discrete maths course is offered to CS students, I will consider software development analogy. In software development, there are essentially two parties. One Developer($\delta$ producer) and the other Client/User($\epsilon$ giver).
We can ask the students which of the below models is preferred:
Model 1: Client gives a specification and developer abids abides by it. That is, client demands for certain feature in her product and developer accordingly makes the product. Analogously, fix $\epsilon>0$ in the range adjust $\delta$ in the domain.
Model 2: Developer gives certain product and client should accept it however pathetic it may be. Analogously, fix $\delta>0$ and expect $\epsilon>0$ to be satisfied.
Model 1 is naturally preferred. And that is our $\epsilon-\delta$ method.
Of course, we can change the setting depending on the target students(engineers/physicists/biologists etc. ).