Alternating Sets against Two Better Opponents

 

From my 2008 archive, while writing to avoid grading finals. Many examples of this paradox involve duels and death, so I share this with a Tennis example, because I'm all about love.... (ALL my students would agree).



Trying to avoid actually grading semester exams, I was playing around with a problem from F. Mosteller's classic, "Fifty Challenging Problems in Probability." Problem two concerns a three-set tennis match in which the player, a youth named Elmer, will alternately play against his father and the club pro, given the club pro is a better player than the father. He will win a prize if he can win two consecutive matches of the three. The question is whether he is better of playing the sequence father-pro-father or pro-father-pro.
The counter-intuitive part is that his odds of winning are better if he plays the better player, the pro, more often. (This reminds me of Parrondo's paradox, which I will try to write about some day) Here is a simple explanation (I hope). If we let f represent the probability he wins against his father, and p the probability he beats the pro, then to get two consecutive wins, he must win the first two or the last two, so we need the probability win, win added to the probability of Lose, win, win.

For the sequence father-pro-father the probability of success is fp + (1-f)pf. We can distribute the 1-f to get fp + fp - fpf, and factoring fp out of each term we get fp(2-f).

If we do the same with the sequence pro-father-pro we just interchange the p and f to get pf(2-p). Now since we are given that p is smaller than f (he is LESS likely to beat the better player) we see that 2-p must be larger than 2-f, and so the pf(2-p) is the higher probability.

The advantage when actually calculated is very small. For example, If the probability of beating his father is .4, and the probability of beating the pro is .2, his probability of winning in the father-pro-father sequence is about .128 . By switching the order to play pro-father-pro his probability of success increases to .144(which is twice what it would be if he only played a three set match against the pro).
What happens if we change these probabilities of success, but keeping the order so that the pro is better than the father? 

Letting p remain at .2 and raising his level against his father to .5 improves his chance of success to 18%. In fact, if we substitute the value .2 into the expression pf(2-p) we get .2f(1.8).... the probability of success is a linear equation, .36f. If we let his probability against his father go to one, you can see that the has a 36% chance of winning if he goes Pro-father-pro. But if you look at the other sequence, fp(2-f), and again substitute in the .2 value against the pro, the equation is quadratic, .4 f - .2f².

Notice the pro-father-pro probability (green) intersects the father-pro-father probability (red) at (0,0) and (.07ish) .  This is the region when beating dad is less likely than beating the pro.  At father =.2, the probability of any order is like playing all three games against the pro, (or the dad  at this point since both have the same probability of beating him.

 Visualizing the graph of this negative quadratic, we can see that he vertex will occur when f= 1, so that is the maximum, and when f=1 we get a probability of success of .2, exactly what his probability against the pro in a single game would be. This makes sense if you consider that if he ALWAYS beats his dad, the match really depends on the one game in the middle against the pro.

How would a fourth game alter the mix? does it matter in which order he plays? It would seem that with four games both orders might be equal, but lets look at the possible winning paths for each.
When the order is p-f-p-f two wins can happen with probabilities pf + (1-p)fp + (1-p)(1-f)pf. Now compare the probability when the order is reversed, fp + (1-f)pf + (1-f)(1-p)fp. Note that all the terms except the second are the same. Once more the fact that 1-p must be larger than 1-f (because f is greater than p) leads us to conclude the best order is to play the pro first.