Competitors or teams in events such as video games, chess, sports, etc. are often ranked using the ELO system. This ranking system is set up so that the difference in ranking between player 1 and player 2 can be used to figure out the probability that player 1 defeats player 2. In particular, if x is the difference in ranking between player 1 and player 2, then a function of the form f(x) = 1/1 + ae-kx , where a and k are positive will tell you the probability that player 1 wins. Suppose that
- When player 1 and player 2 have the same rank, the probability that player 1 wins is 0.5.
- When player 1 is ranked 200 points higher than player 2, the probability that player 1 wins is 0.8.
(a) Determine the values of a and k in the function f(x) using the information above.
(b) What happens to the value of f(x) as x → ∞ (i.e. as x gets bigger and bigger)? Explain your answer mathematically by doing some computations or using algebraic reasoning. Does this make sense in the context of the problem?
(c) What does the difference in ranking need to be so that player 1 has a 99% chance of beating player 2?