How would you give your students a tournament as assessment? I am currently teaching calculus and I thought it would be a good idea to do some kind of limit (and later derivative/integral) calculation tournament. The thing is... The students only seem to give their best if the activity is an actual assessment so I was wondering how can I prepare a tournament as a fair assessment.

The main issue is that as an assessment I want every student to be able to get the maximum score (otherwise it wouldn’t be fair), however since it is a tournament only one student can get the first place... 

So I was wondering how one can organize a tournament in a way that is fair for all students? There are some remarks to take in consideration:


There should be a prize/incentive for attaining a better place in the tournament compared to the other students.
Everybody should be able to get the maximum score.
There should be some kind of rule which determines whether a student is out the tournament or not.
Every student needs to participate.
The whole tournament shouldn’t take longer than two hours; I can’t spend too much class time on that.


So with those remarks in mind how would you guys organize a tournament? I'd like to compile some suggestions before planning it.
Eager et al. describe how they use flipped classrooms to teach a mathematical biology course. As part of their course, they use "competitions" to motivate students. They discuss some of your concerns in a method paper:


  One of the difficulties in teaching a class via the flipped classroom is how to properly assess and examine student learning [3]. Because the class is very modeling focused, and modeling is a time-consuming process, traditional exams are very difficult to administer in a course like MTH 265. The authors’ solution to this problem is to do away with traditional
  exams in favor of both individual quizzes and group mathematical modeling competitions.
  The individual quizzes involve the students choosing two small modeling questions from a
  group of three questions and solving them with help from their class notes, homeworks,
  past computer programs and video lectures. These quizzes are given over a 55-minute time
  period. The questions in these quizzes are structured in such a way so that, after the student
  chooses which two problems to work on, he or she can solve both questions in a combined
  40 minutes—meaning that some (but not all) of the modeling process is given to them. These
  exams provide the students with an incentive to learn the material on their own (which is
  nontrivial in a flipped course with a large group component), as well as the motivation
  to become proficient in using some of the elementary skills a mathematical or quantitative
  biologists would need to possess (e.g. diagram building, script writing/amending, elementary
  algebra and calculus, scientific writing).
  
  The mathematical modeling competitions stemmed from the successful 24-hour Wisconsin
  Mathematical Modeling Challenge at UW-L, as well as the success of similar activities in
  other mathematical biology courses [8]. These competitions complement the individual, inclass
  quizzes by providing students with more open-ended modeling problems and a longer
  timespan to produce their solutions. These modeling problems can ofen be introduced with
  only a few paragraphs of text—with few or no equations. The students, in groups, then
  have 24 hours to come up with an appropriate model for this system, as well as answer
  the biological question posed. The group of students with the best solution is rewarded
  with perfect homework scores until the next modeling competition. The class has either
  an individual quiz or a modeling competition at the end of every other week for a total of three individual quizzes and three modeling competitions during the semester worth roughly
  30 percent of each student’s final grade.


Full citation (in case link breaks):  Eric Alan Eager, James Peirce &amp; Patrick Barlow (2014) Math Bio or Biomath? Flipping the mathematical biology classroom, Letters in Biomathematics, 1:2, 139-155, DOI:
10.1080/23737867.2014.11414476

As a disclosure, I have collaborated with Eager and gave guest lectures in his Bio Math course about my own research.