## Thursday, December 19, 2013

### A Sandbagger's Guide to Winning

It takes a great deal of effort to manipulate a handicap.  A player has to generate artificially high scores in rounds that are not in competition. This post examines a sandbagger’s winning percentage as a function of how “many strokes he has in the bag” – i.e., the difference between his manipulated handicap and his true handicap.
Let’s assume Tom and Sam play golf together.  The winner of a match is the player with the lower net score.  Sam has worked his handicap so that his average net score is 1-stroke less than Tom’s.  This does not mean Sam wins every time.  The random nature of scoring lets Tom win when he has an exceptionally good round, or Sam has a bad round.  The probability that Sam wins will depend on the standard deviation of scoring for each player.  For simplicity, it is assumed that each player has a standard deviation of three strokes (i.e., 68 percent of all scores will be between ±3 strokes of the player’s average score).  The probability that Sam wins a match is presented in Table below for various levels of sandbagging.  (Note: The methodology for calculating the probabilities is shown in the Appendix).
Table
Probability That Sam Wins the Match

 Sam’s’ Average Net Score Advantage Probability Sam Wins the Match -1 .59 -2 .68 -3 .76 -4 .82

The best long-term strategy for the sandbagger would be to have only a 1-stroke advantage.  He wins, on average, six out of ten matches.  Tom may not realize he is at a disadvantage since having four wins and six loses is not unreasonable record.  For just a little handicap manipulation (1-stroke), Sam gets an annuity as long as Tom does not figure out the game is rigged.
Tom, however, is not a complete idiot and realizes if his net score was 1-stroke lower he would do better.  He can do this by a) decreasing his average gross score in competition by 1-stroke while maintaining his current handicap, or b) artificially increasing his handicap by 1-stroke while maintaining his average gross score in competition.  Option b) is the easier of the two and the one most often chosen.   In essence, sandbaggers beget sandbaggers.
Sadly, many players do not look at their USGA Handicap as a measure of performance—How good am I?—but as weapon in their arsenal to beat their opponents.    The best evidence of this is watching players post their score.  Are players more jubilant when their trend handicap goes down or up?

Appendix

Suppose there are two scores, X1 and X2 drawn randomly from the normal distributions N(µ1, σ12) and N(µ2, σ22).  We want to find the probability that X1 < X2.
Now X1- X2 is normally distributed with mean,
µ = u1 – u2
and variance,
σ2 = σ12 + σ22
Hence, the variable (X1- X2 -µ)/σ is distributed normally with a mean of zero and a variance of 1 (i.e., the standard normal distribution N(0,1).
And so, the probability that X1 is less than X2, is the cumulative probability up to -µ /σ.   In our example, the standard deviation of scoring for each player is 3.  Therefore,
σ = (32 +32)½   = 4.24
µ = -1
The probability that X1 is less than X2 is
= f(1/4.24)  = f(.24) = .59