Tuesday, June 25, 2013

Why You Win (or Lose) at Skins


Introduction – “Skins” is a very popular golf game.  The game is simple in concept.  A group of players play a hole and if one player makes the lowest score, he wins a skin.  At the end of the round, a pool of money is divided among the players who won skin(s).  The popularity of the game derives in part from the ability of a player to win a skin even if he isn’t playing well. Skins, however, is not like a lottery where everyone has an equal chance to win.  A player’s success will depend upon 1) the Skins method (i.e., how skins are determined), 2) the distribution of handicaps among the competitors, and 3) the cap on the maximum handicap allowed.  Each of these factors is discussed in turn.
Skins Method – There are seven commonly used methods for playing skins:
1. Gross Skins – A skin is awarded only if one player has the lowest gross score on a hole.  The value of a skin is equal to the betting pool divided by the number of skins.
2. Net Skins – A skin is awarded only if one player has the lowest net score on a hole.[1]    The value of a skin is equal to the betting pool divided by the number of skins.
3. Net Skins Played at ½ Handicap – This is the same as Method 2 except the player only receives one-half of his handicap.  If this results in a player having a “n +1/2” handicap, the player deducts a one-half of a stroke on the hole with a n+1 stroke allocation.  For example, a player with a handicap of 7 ½ , would get ½ stroke on the 8th handicap allocation hole. The value of a skin is equal to the betting pool divided by the number of skins.
4. Separate Gross and Net Skins – Gross and net skins are determined at full handicap.  The value of a gross skin is equal to half the betting pool divided by the number of gross skins.  The value of a net skin is equal to half the betting pool divided by the number of net skins.
5. Equal Gross and Net Skins – Gross and net Skins are determined at full handicap. The value of any skin is equal to the betting pool divided by the total number of skins.
6. No Double Win – This is the same as Method 5 with one exception.  A player cannot win a gross and net skin on the same hole. 
7. Net Skins, but Gross Skin Takes Precedence – If a player has a gross skin on a hole, he is awarded a “net skin” even if other players had net scores equal to the player’s gross score.  In essence some net scores are better than others.  On closer inspection, this method is identical to Method 6 as long as a player cannot receive more than one handicap stroke on a hole.[2]  Therefore, this method is eliminated from the analysis.
To examine the equity of each method a simulation model was built.  The probabilities of a 5, 10, and 15 handicap player were estimated (see Appendix).  In the first analysis, it was assumed that 5 players of each handicap level were competing. The expected return (the amount returned per dollar bet) for each group is shown in Table 1 below.
Table 1
Expected Return per Dollar Bet

Method
5-Handicap
10-Handicap
15-Handicap
1. Gross Skins
$1.50
$.97
$.53
2. Net Skins
$.77
$1.04
$1.19
3. ½ Handicap
$1.18
$.89
$.93
4. Separate Gross and Net
$1.14
$1.00
$.86
5. Equal Gross and Net
$1.13
$1.00
$.87
6. No Double Win
$1.19
$.92
$.89

The lower-handicap player has a distinct advantage in all methods except Method 2 (Net Skins).  Methods 3, 4, 5 and 6 are essentially equivalent in terms of equity. 

Handicap Distribution – The previous analysis had an equal number of players in each handicap group.  A player’s chances are improved if the field does not include many players with handicaps similar to his.  To demonstrate this effect, estimates of the expected return are made when there is 1 player at a handicap level and 7 players in each of the other levels.  Table 2 presents the expected return when there is only one player at the designated handicap level.

Table 2
Expected Return with Only One Player at the Designated Handicap Level

Method
5-Handicap
10-Handicap
15-Handicap
1. Gross Skins
$1.97
$1.02
$.40
2. Net Skins
$.98
$1.34
$1.57
3. ½ Handicap
$1.47
$.1.06
$1.3
4. Separate Gross and Net
$1.48
$1.18
$.99
5 Equal Gross and Net
$1.49
$1.17
$1.03
6. No Double Win
$2.19
$1.11
$1.24

Under Method 6, for example, if there is only one 5-handicap player (instead of five) the player’s expected return jumps from $1.19 to $2.19.  This is due to the player’s birdies being more likely to win a skin when he doesn’t face competition from other 5-handicap players.

Cap on Maximum Handicap – In some skins games, the maximum handicap allowed is 18 or one handicap stroke per hole.  To examine the equity of this cap on handicaps, five additional players with 25-handicaps were added to the simulation model. The expected returns for each handicap level are shown in Table 3.

Table 3
Expected Return per Dollar Bet

Method
5-Handicap
10-Handicap
15-Handicap
25-Handicap
1. Gross Skins
$2.02
$1.22
$.68
$.09
2. Net Skins
$.64
$.97
$1.29
$1.10
3. ½ Handicap
$1.34
$.99
$1.10
$.57
4. Separate Gross and Net
$1.33
$1.09
$.98
$.59
5 Equal Gross and Net
$1.36
$1.08
$.97
$.59
6. No Double Win
$1.41
$1.00
$.92
$.67

As can be seen, the expected returns for 5-, 10-, and 15-handicap players have been increased with the addition of the five 25-handicap players.  The 25-handicap player does not do well under most scenarios.  It should be noted, however, that not all 25-handicaps are alike.  A 25-handicap who has a large variance in his hole scores (i.e. he has more pars and more triple bogeys than assumed in the model) will do better than Table 3 suggests.  Under reasonable assumptions about the hole-score variance of a 25-handicap player, however, it is unlikely this player will do as well as the 5-handicap player.   
 
Method 5, No Double Win, is usually defended in terms of equity.  “A player should not get two skins for one hole,” it is argued.  Double skins, however, are most likely to go to the medium handicap player.  The elimination of double skins gives an even greater edge to the low-handicap player.  This is demonstrated in Table 4 which shows the total number of skins by handicap for the two methods.

Table 4
Total Skins by Handicap (1,000 trials)

Method
5-Handicap
10-Handicap
15-Handicap
25-Handicap
5. Equal Gross and Net
4372
3500
3112
1855
6. No Double Win
3563
2502
2332
1698
 
 
 
 
 
Percent Reduction in Skins
18.5%
28.5%
25.1%
8.5%

The “No Double Win” provision eliminates a larger percentage of skins for the 10- and 15-handicap groups than it does for the 5- and 25-handicap group.  Method 6 basically shifts money to handicap groups at the extreme ends from the handicap groups in the middle. Whether this is more equitable depends upon the eye of the beholder.

Conclusion – Skins is golf’s mini-version of Powerball.  The odds are stacked against most players, but it is still fun to play.  Played for nominal stakes, a player should not be too concerned about his expected losses.  If a few low-handicappers are turning the weekly skins game into an annuity, however, a player should consider switching his contribution to a tax deductible charity.                                         

Appendix

Score to Par
5-Handicap
10-Handicap
15-Handicap
25-Handicap
-2
.005
.003
.000
.000
-1
.140
.090
.060
.010
0
.450
.350
.250
.150
+1
.310
.380
.430
.380
+2
.070
.140
.200
.300
+3
.020
.030
.040
.100
+4
.005
.007
.020
.060

 



[1] If a player has a handicap less than or equal to 18, his  gross score on a hole is reduced by one stroke to obtain his net score if his handicap is less than or equal to the stroke allocation for that hole. Otherwise, his net score is equal to his gross score.
[2] Under Method 7, if a player has a gross skin, he receives a “net skin” since a gross skin takes precedence. All net skins are awarded a “net skin” unless the player also had a gross skin on that hole.  In essence, Method 7 is Method 6—gross and net skins, but a player cannot win both on a hole.

1 comment:

  1. Very interesting and informative! These are exactly the sorts of questions we have been wondering about at www.AustinSkins.com, but I lack the skills required to run such simulations. However, we have been experimenting and tracking the results for about 3 years. We are currently on the 4th iteration of our game. We originally started out with a model that was a hybrid between types 2 (net skins) and 3 (half handicap). We gave 70% handicaps and paid net skins only. We actually found that this was slightly skewed toward the higher handicappers. This was balanced out by offering a "low net" prize, which was skewed toward the low handicappers because it was only 70% handicapped. Because of the skew toward the high handicappers in skins, we lowered the percentage used for skins and found that it drastically changed the outcomes and favored low handicappers way too much.

    After that, we tried doing skins at 70% and low net at 90% handicaps, but that proved to be an administrative nightmare. Despite that fact, the very small sample size that we have suggests that this could be a very equitable system. We've thought about building an app that could help us calculate the scores and payouts in real time, but until we end up doing something like that, this method is too difficult to administer as a practical matter.

    Now, we are doing 85% for both skins and low net. As I mentioned earlier, the equity drops off for anyone above about 15 or 16 handicap. As you mention above, some players with higher handicaps who have a high standard deviation in their average hole score relative to par, can win skins in our game, but they are the exception. And we do have such players that continue to play with us because they do actually win.

    I had been wondering about the equity of no-double-win and had considered implementing that method to see if it would aid in the goal of equity, but looks like it would not, based on your findings. Thanks for saving me the effort!


    ReplyDelete