Theoretical and empirical research on performance incentives in tournament settings traditionally focused on effort provoked from competitors and their expected productivity. The parallel between the structure of prize money in professional sports tournaments and relative-performance labor contracts has enabled a wider impact of economics research nominally about sports [see Lazear and Rosen (1981), Rosen (1981), and Ehrenberg and Bognanno (1990)]. Szymanski (2011) provides a review of this literature. However, the incentives for risk management with highly convex tournament payoffs have received more recent attention [see Shmanske (2007) and Nieken and Sliwka (2009)], and the broader implications are important here as well. For example, portfolio managers are universally evaluated by relative performance measures such as return comparisons with other funds in the same class. The structure of payoffs in finance – bonuses, salary, and continued control of other people’s money – is often quite convex: compensation for one very strong year may exceed that for many years of median performance. Concern that this system causes portfolio managers to take excessive risks is at the center of many critiques of the financial industry. Similarly, one win on the PGA Tour captures 18% of the tournament purse, prize money equivalent to five 6th place, ten 15th place, or twenty 25th place finishes. Gratefully, the stability of the world financial system is not at stake in professional golf. Yet risk management on Tour offers several parallels aside from the convexity of payoffs that make for a persuasive field experiment. As Pope and Schweitzer (2011) argue, in PGA tournaments: i) payoffs are clearly defined; ii) performance measures are standardized (e.g., there are always winners); iii) unlike laboratory experiments, the financial stakes are large; iv) competition should limit inefficient strategy or preparation; v) participants are highly experienced.
This paper examines evidence for the rational response of professional golfers to structural and situational risk incentives from a decade of high-stakes PGA events. In particular, I explore the hypothesis that scoring volatility is a strategy, i.e., risk deliberately assumed in response to circumstantial incentives, as distinct from a type. A basic theoretical framework first outlines the objective and constraints a player faces in choosing the level of risk with which to approach each hole of a tournament. Where performance payoffs are convex, the expected earnings maximizing level of risk will exceed that which minimizes a player’s mean score. Observed player distributions of scoring average, volatility, and prize money on the PGA Tour from 2003 to 2012 are presented. Some players earn sufficient money to remain on Tour through steady play with a low scoring average, others with more uneven play that includes a few big paydays. I estimate the long-term rate at which the convex prize money formula in golf tolerates higher mean scores in favor of scoring volatility.
The remainder of the paper identifies situations where players are encouraged to add or subtract risk, and undertakes to measure the response. I focus on two stages of a tournament when risk incentives crystallize. First, on cut day (Friday in most tournaments), players within a few strokes on either side of the eventual cut have dramatically different risk incentives in spite of their close proximity on the leaderboard. No prize money is paid to players who miss the cut, whether by one stroke or ten, while average earnings are historically slightly less than >$25,000 for players that make the cut “on the number.”1 The second stage is the closing stretch of the final round on Sunday, when players near the top of the leaderboard answer to the most convex portion of the PGA Tour’s payoff formula. For a $5 million total purse, the difference in prize money between first and second place is $360,000; between second and third place is $200,000; between third and fourth place is $100,000; between fourth and fifth place is $40,000; between fifth and sixth place is $20,000. This convexity greatly diminishes further down the leaderboard, with linear differences of $10,000 between eighth and fourteenth places, and $5000 between fifteenth and twenty-first. With the exception of an outright leader, who faces concave payoffs, players near the lead have stronger positive risk incentives on the closing holes than those further down the leaderboard. The pervasiveness of ties on a Sunday leaderboard, sometimes involving large groups of players at the same contending score, and scoring gaps between leaderboard positions can complicate the payoff convexity story, even creating concavities in the payoff function away from the lead. This factor is addressed in Section 4 using a direct measure of convexity containing more information than ordinal position alone.
Do players respond to risk incentives in a measurable way, and if so, how does the magnitude of the response compare to risk variation across player types? I look for evidence of the risk-response hypothesis in scoring outcome frequencies and standard deviation, and in direct measures – going for the green on select holes, driving distance, and putting. I also fit a multinomial logistic regression to scoring outcomes controlling for hole difficulty, “risk-reward” opportunity, and other conditioning factors, and do so separately for a sample of 83 elite players. The evidence presented here indicates that players respond measurably on average to risk incentives around the cutline, but much less so to leaderboard position on the closing holes of a tournament. The discussion considers reasons for this apparent lack of responsiveness, including the possibility of loss aversion or other bias from expected earnings maximization.
2 Conceptual Framework
A golfer’s score relative to par on a single hole is a discrete random variable, S, with a probability mass function f(S). In principle S may take integer values s≥–3, but for highly skilled professionals more than 99% of probability weight lies between –1 and +2. The outcomes and probabilities, πs, define the expected value μ=Σsπs and standard deviation
Define the relevant constraint according to the mean and standard deviation of outcomes:
Nearly all professional golf tournaments are contested over 72 holes, so minimizing expected score for each of them in order to minimize expected tournament score is a risk strategy with some appeal. Yet tournament payoffs are convex, and this fact should influence optimal risk strategy for an expected earnings maximizing player. Only one hole is played at a time, and the immediate convexity of payoffs to different outcomes will vary over the course of a tournament, season, and career. One can readily outline a scenario that encourages greater risk even at the cost of a higher expected score. For example, at the conclusion of a tournament, a player near the lead approaches his final hole. The relevant competitor scores are final, so payoffs to different outcomes on the 18th hole are certain. A bogey lowers earnings by $x relative to current position, while a birdie improves earnings by $αx. In considering risks that raise the probability of both a birdie and a bogey (with probability weight shifting from par, other outcomes unaffected for the sake of argument), the expected earnings maximizing player would accept greater bogey relative to birdie probability in proportion to α. Payoffs are convex if α>1, and optimal risk strategy requires an expected score on the 18th hole that is greater than the minimum achievable.
Does the logic of this example extend to earlier holes in the final round, or even to earlier rounds? It is unlikely the incentives facing the same player were fundamentally different when playing the 17th hole a few minutes earlier. His ultimate cumulative score, and therefore earnings, were at that stage a random variable involving compound probabilities of outcomes over two holes, and perhaps a close competitor had not yet finished his round, adding uncertainty to the payoffs. The convex structure of earnings tolerating higher expected scores in favor of volatility would not vanish, however, nor would it back on the 16th hole. Let sh represent a player’s score on the hth hole of a tournament, and
Except perhaps for the final stages of a tournament, the expression in (2) can be expected to summon a very complex decision tree. Not only do expected earnings depend on outcome probabilities implied by risk strategy on each of the remaining holes, but
The ratio of partial derivatives (3), referred from here forward with θ, is the rate at which expected tournament payoffs tolerate a higher mean score with greater volatility on a specific hole. In Figure 1 it is the slope of the second curve shown, representing combinations of μ and σ consistent with a constant level of expected earnings. A player maximizing their expected earnings matches their rate of technical substitution to the slope of the highest achievable earnings frontier.
Convex payoffs imply θ>0, but clearly the returns to risk vary by tournament stage, position, and situation. The final hole Sunday scenario with resolved payoffs of convexity α presents a steeper earnings frontier than a routine Thursday hole with so much of the tournament unresolved. And there are specific circumstances where θ<0 and the frontier downward sloping. On such holes the optimal level of risk will be less than that which minimizes expected score. A rational late-stage leader wants to play conservatively to protect the lead, as the expected cost of falling back exceeds the benefits of extending it. And a player struggling to make a cut2 is incentivized to either reduce risk in the short term to protect his place in the field for the weekend, or compelled to take risks he otherwise would not, depending on immediate proximity to a single aggregate score. The response to such micro-incentives is addressed in Section 4, following an analysis of the long-term macro-incentives of convex tournament payoffs.
3 Risk types
“Consistency is highly overrated. We all want to be consistent as professional golfers, but generally people that are consistent are mediocre… So you want the exciting peaks. And if that means that there are going to be some frustrating days afterwards, so be it.”
Table 1 presents player-level summary statistics for two samples of professional golfers with substantial PGA tournament experience from 2003 to 2012. Data were provided by the PGA TOUR’s ShotLink program, which since 2003 has recorded detailed information on every shot struck in Tour events.4 In addition to the essential outcomes on each hole played, the system details distances (to within an inch, measured by lasers on the course) covered by each shot and remaining to the hole, money earned, and one key coding of players’ strategic intent. The larger sample consists of the 414 players completing a minimum of 100 TOUR rounds over the sample period, a group that accounts for 90% of all holes played over the 10 seasons. The smaller sample includes only the 164 players that earned a minimum of $5 million during the period, all of whom are in the larger sample. The elite group includes veterans who competed in a majority of the 445 total events, as well as young players like Keegan Bradley and Rory McIlroy, both of whom reached $5 million in earnings in fewer than 50 starts. Table 1 outlines the full range of financial payoffs among Tour regulars, from a minimum of <$150,000 total earnings to the nearly $450,000 per event earned by Tiger Woods. Unsurprisingly, players in the elite sample played more events, won and finished tournaments among the top 10 places more frequently, and made more cuts.
|Variable||Players with Min 100 rounds||Players with Min $5 million earnings|
|Earnings per Event||$39,427.89||$38,640.17||$3,053.37||$449,840.60||$66,326.97||$47,290.37||$23,346.20||$449,840.60|
|Hole Scoring Average||0.008||0.031||–0.098||0.196||–0.012||0.020||–0.098||0.051|
|Hole Scoring SD||0.683||0.025||0.624||0.796||0.677||0.022||0.624||0.778|
|Eagle (-) Frequency||0.004||0.001||0.000||0.010||0.005||0.001||0.002||0.009|
|Double Bogey (+) Frequency||0.021||0.005||0.011||0.043||0.020||0.004||0.011||0.037|
|Putts per hole||1.584||0.042||1.407||1.728||1.596||0.028||1.520||1.659|
|Avg Driving Distance (yards)||280.7||6.8||259.0||303.5||281.9||6.7||259.0||301.7|
|Went for it on GFG Frequency||0.488||0.100||0.178||0.747||0.514||0.093||0.281||0.731|
|Final Round Position||37.8||6.6||12.3||58.4||32.5||4.8||12.3||43.3|
Scoring averages presented in the Table reveal that par is far from arbitrary, and the margin between journeyman and elite player is narrow: the mean scoring average is <1% over par in the large sample and slightly more than 1% under par in the smaller sample. The range of scoring means is distorted by a large outlier: the difference between Woods’s –0.098 scoring average and that of the second lowest (Jim Furyk, –0.058) is equal to the difference between Furyk and the 62nd lowest average (Vaughn Taylor, –0.019). Table 1 also includes descriptive statistics for putts per hole, driving distance, and that useful measure of risk intent recorded in the ShotLink data. A portion of holes are classified as presenting an option of “Going for the Green” (GFG) in fewer than the regulation number of strokes, meaning with the second stroke on a par-5 and first stroke on a par-4. All par 5’s are deemed GFG holes, as well as par 4’s sufficiently short that 10% of the field hit tee shots on or with 30 yards of the green perimeter. Variation in the frequency with which players attempt to “go for it” on these holes is quite large, with the most aggressive players at >70% and the least aggressive under 30%. Risk indicators associated with going for the green, driving distance, and putting feature prominently in Section 4.
From Table 1 one can also begin to see the range of scoring variation among players. The lowest scoring standard deviation in the sample belongs to Furyk (0.624), who also has the highest par frequency among the elite sample (0.657). Furyk compiled the second-lowest scoring average as referenced earlier, and earned more than $37 million over the decade. Yet the discussion in Section 2 suggests that Furyk’s steady, low variance profile should not be the only path to financial success in a series of tournaments with highly convex payoffs. If scoring volatility is favored in this environment as expected, earnings will be positively associated with standard deviation conditional on scoring average. Further, a cross-section of player risk profiles and earnings will enable estimation of long-term θ, the average earnings tradeoff between low and volatile scores. Figure 2 depicts the scatterplot of scoring means and standard deviations for the $5 million total earnings sample of 164 players in Panel (a), and an illustrative selection of a dozen players with varying risk profiles in Panel (b). A marker in this space represents a spare characterization of an individual player’s long-run risk “type,” which depends on his fixed characteristics. In Panel (b), the area of each hollow circle marker is proportional to earnings per start. The visible positive correlation between scoring average and standard deviation (0.45) indicates a “price of risk” as measured across player profiles. This is distinct from the rate of technical substitution defined in Section 2, a relationship across “types” rather than the tradeoff confronting an individual.
Earnings payoffs to low and volatile scoring are evoked by comparing marker sizes in Panel (b). For example, note that Adam Scott is rewarded with nearly identical earnings per event as Furyk with a substantially higher scoring average (–0.012 compared to –0.058) accompanied by higher standard deviation (0.692 to 0.623). Consider next two younger players – Spencer Levin and Dustin Johnson. Levin’s first full year on TOUR was 2009, Johnson’s was 2008. Levin had 121 career starts through the sample period to Johnson’s 112. Levin has the lower scoring average, –0.031 to –0.022, but Johnson’s career and per tournament earnings are approximately three times greater. Levin’s is among the steadiest profiles in the sample, with a standard deviation of 0.634. Johnson is one of the longest hitting, most aggressive players on Tour, with a scoring standard deviation (0.717) above the 90th percentile. There is of course more to the difference between the two players than summary statistics reveal, but the comparison seems to illustrate Padraig Harrington’s thinking in the quote at the top of the section. Henrik Stenson, Paul Casey, and Andres Romero are all experienced Tour players who have contended in major championships and qualify on earnings for the elite sample. They constitute much of the northeast frontier of the distribution in Panel (a), succeeding in spite of relatively high scoring averages. No one in the elite sample shares a risk type category with John Daly’s, unique in his own way as Woods. High risk-type golfers succeed because when they play well – and this is clearly not all the time – they have the capacity to “go low” and contend to win tournaments. Evidence from Panel (b) is obviously anecdotal, but a more formal approach for estimating the long-term payoffs to risk is in order.
Table 2 contains parameter estimates from four simple regressions of earnings per tournament on scoring average and standard deviation. The regression equation is
|Earnings per Event||Log Earn. per Event||Earnings per Event||Log Earn. per Event|
|Mean Score (×100)||–6845.4***||–0.166***||–14544.3***||–0.144***|
|SD of Score (×100)||2312.7**||0.0537***||5026.8**||0.0540**|
4 Risk strategies
The empirical evidence to this point establishes scoring volatility’s association with higher expected earnings over the long-term. To the extent that an expected earnings maximizing player could modify his “type” through preparation or mental approach, he should seek to add standard deviation to his scoring profile if this can be accomplished with no more harm to scoring average than a ratio of around 0.35. But this ratio is an average of payoffs to risk over thousands of golf holes, obscuring the diverging incentives within tournament situations. The question remains whether players are able and willing to calibrate risk – shift probability weight away from or toward par – according to circumstances. Risk is a type, but is it also a strategic choice that can be managed hole-by-hole?
4.1 Summary statistics
In order to address this question, it is necessary to shift to a much narrower unit of observation. Table 3 contains summary statistics for the nearly 3 million holes played by all competitors with a minimum of 100 rounds from the 2003 to 2012 PGA TOUR seasons. We see the effect of the traditional second round cut, but also of mid-tournament withdrawals and disqualifications in the declining proportion of holes by round.5 Final round holes represent 18% of the sample. Over the decade of tournaments, 22.5% of holes were par-3, 17.9% were par-5, and the remaining majority par-4, implying a mean par of 71.2 per round. Among all holes played, 2.2% (66,123 holes) were played at zero, one, or two strokes under the cutline over the final 5 holes of a round concluding with a cut, and 2.0% (60,976) were played at three or fewer strokes outside the cutline. The closing 5 holes of a tournament account for 5% of all observations, with just 0.1% (2953) of these played in the lead. Play from positions 2 through 6 over the closing holes represent 0.4% of the sample, with a similar proportion for positions 7 through 11 and positions 12 through 16. Mean position is just under 35th for the 537,606 holes played during a final round.
|Year||2007.3||2.786||2003||2012||Driving Dist. (yards)||280.8||26.1||12||501|
|Round||Went for Green|
|1||0.314||0.464||0||1||Not GFG hole||0.816||0.387||0||1|
|5||0.005||0.067||0||1||Score relative to par|
|Final Round||0.180||0.384||0||1||Eagle -||0.004||0.066||0||1|
|4||0.596||0.491||0||1||Double Bogey +||0.020||0.141||0||1|
|Cut Round Closing 5||Par 3||0.077||0.618||–2||9|
|≤2 strokes inside/on cut||0.022||0.147||0||1||Par 5||–0.314||0.722||–3||8|
|≤3 strokes outside cut||0.020||0.142||0||1||Observations||2,981,448|
|Final Round Closing 5|
|2nd – 6th||0.004||0.066||0||1||Yardage|
|7th – 11th||0.004||0.063||0||1||All||396.84||127.62||99||692|
|12th – 16th||0.004||0.061||0||1||Par 3||189.89||27.02||99||288|
|Final Round Positionψ||34.94||21.66||1||88||Par 4||425.90||43.22||265||558|
|F.R. Position Tiesψ||5.94||3.30||1||29||Par 5||556.28||35.73||433||692|
|Convexity Indexψ||3144||11858||–136951||96094||Field Mean Score||–0.0051||0.241||–1.03||1.10|
|Relative Round Score||–0.008||1.94||–11.59||16.92||Field Score S.d.||0.635||0.104||0.24||1.56|
There are many ties for position on a final round leaderboard, an observation consistent with large fields of closely matched competitors. In final rounds, the average hole is played in a six-way tie for tournament position. This fact, along with the related observation that several strokes can separate leaderboard positions, complicates the narrative of simple convex tournament payoffs in potentially important ways. For example, a $5 million purse pays 3rd position $200,000 less than 2nd position and $100,000 more than fourth, but the convexity is mitigated if a gap of three strokes separates 2nd from 3rd late in a tournament while just one stroke separates 3rd from 4th. It is mitigated further if a group of six competitors is tied for 4th, in which case dropping a shot means splitting 4th place through 10th place prize money. The next variable summarized in Table 3, the Convexity Index, attempts to account for ties and gaps in measuring the convexity of payoffs surrounding a given position. The Index is constructed from a calculation of earnings associated with each observation’s current position (accounting for ties), and with the hypothetical positions the competitor would inhabit with the gain or loss of one stroke against the field. Convexity of payoffs mean that the gain in earnings from a one stroke position upgrade exceeds the loss from a one stroke drop, but there are cases where the difference is negative. Outright leaders gain no advantage from a lower score in this framework, while a higher score can move them into a tie. The same calculation was undertaken to consider two-stroke gains and losses, providing a wider measure of convexity around current position. The Convexity Index is the simple average of the differences between gains and losses associated with one-stroke and two-stroke changes in current position.
One concern in relating tournament position to scoring outcomes is the likely presence of serial correlation within player scores. Position near the top of the leaderboard means by definition a player has scored well to that point. If scores are serially correlated – if players are streaky – there is a threat of attributing the incentive a “hot hand” effect to tournament incentives. The Relative Round Score variable is constructed as a player’s aggregate round score up to the current hole relative to the field, and will be employed to isolate hot hand effects. For example, if the field played final round holes 1 through 15 at an average of one stroke over par, a player approaching the 16th at two under par for the day is assigned a Relative Round Score of –3. Table 3 also reports hole-level statistics for putting, driving distance, and intent on “Going for the Green” holes, all of which are employed as indicators of risk in this section. The most common outcome on the greens is a two-putt (55.8% of holes), followed by a one-putt (37.5%) and a three-putt (3.2%). Both one- and three-putts are often the result of aggressive strategy, and two-putts associated with conservative approach shots and putting. Driving distance is specified as a risk measure primarily because of club selection – hitting driver of the tee is the more aggressive play, while a conservative option frequently calls for a shorter club to play for position or avoid hazards. The categories of “Going for the Green” holes were introduced in the discussion of Table 1. Of the 18.4% of holes played under this designation, just under half (9.1% of all holes) involved the competitor attempting to reach the green in one stroke fewer than regulation. Of those attempts, 22% (0.02/0.091) successfully hit the green. Going for the green is a clear strategic choice that increases the likelihood of both low and high scores.
Overall frequencies of discrete scoring outcomes and summary scoring relative to par are similar to the mean-of-means presented in Table 1. Par is the outcome on 62.9% of all holes. Birdies (19.2%) are more common than bogeys (15.4%), but the overall mean score is brought very close to par by the greater incidence of double-bogeys relative to eagles. Also, double eagles (three under par) are the rarest score in golf, there were just 39 in the decade, while triple-bogeys and worse represent a non-negligible 0.25% of all scores. Average scoring is well over par on par-3 and par-4 holes, but 31.4% under par on par-5 holes. The remaining summary statistics in Table 3 are calculated for the entire field for each hole in each round of a tournament. (For example, all professional competitors play the same tees and hole locations on a course for each round, so yardages do not vary by player.) The total of 36,414 observations for this set includes 72 – one for each hole in each round – in most tournaments, but more in events where multiple courses are used. A Field Mean Score is calculated for each hole in each round. The mean of these means is unsurprisingly close to the overall scoring mean, but the variation is substantial. The easiest holes on TOUR during this period averaged a full stroke under par, and the most difficult a full stroke over par. This measure will be used to control for hole difficulty in the multivariate analysis. The standard deviation of field scoring is calculated from the same set of observations. There are holes in the sample with very little variation in field scores, indicating little opportunity for risk taking. In contrast, holes with high standard deviation in field scores presented opportunity for low relative scores, but also a hazard of high scores. In the golf idiom, these are “risk-reward” holes. Given the questions under investigation, risk-reward holes command special attention.
4.2 Bivariate analysis
Consider first risk indicators around the cutline over the closing holes before the tournament field is cut.6 The cutline provides a natural test of the risk-response hypothesis because of the strong discontinuity of payoffs around a single cumulative score, and because the information signal is typically clear. Players and their caddies (with help from media coverage) are adept at forecasting the eventual cut, which is a function of aggregate scoring and moves at most by one stroke in either direction late on Friday. The transparent case involves a player one stroke off the cutline on their last hole of the round before the cut. A bogey or worse carries no cost in terms of expected earnings, which are zero when a player misses the cut by any number of strokes, while a birdie revives the probability of a substantial payoff depending on weekend play. A player just one stroke better, a negligible difference in overall quality of play to that point, is right on the cutline. There are genuine position improvement benefits from a birdie in this situation, but a very large cost of missing the weekend with a bogey. More generally, the framework developed here indicates players on or just inside the expected cutline should seek to lock in their position for the weekend with conservative play. In contrast, a player just outside the cutline ought to aggressively (even desperately) pursue opportunities to improve the likelihood of making up the necessary strokes, even when the risks carry likelihood of falling further back. In few other situations are tournament micro-incentives so discontinuous and asymmetric.
Figure 3 presents scoring outcome frequencies, mean score, standard deviation, and alternative risk measures for all play around the cutline over the final 5 holes before a field is cut. The measures presented in the Table are weighted for greater influence of later holes, and for “risk-reward” holes.7 The pattern found in Figure 3 is very robust to changes in the weighting formula (including none) and specification of the first “closing” hole. The scoring frequencies in Figure 3A show those players on or just inside the cutline make sharply more pars and fewer under-par scores than players just outside the cutline. In each case the difference in frequencies between play on the cut number and one stroke over – 2.9% points for par and 2.4% points for under-par scores – are statistically significant at the 0.1% level. The clear pattern around the cutline does not hold for over-par scores, however, which show a steady increase in frequency from on to outside the cutline. Over-par frequencies immediately around the cutline are not statistically different. A particular anomaly is the relatively low frequency of over-par scores at +1. The spike in low scores from the same sample indicates a strategic shift toward aggressive play, but the expected cost of that strategy in greater frequency high scores fails to materialize. The frequency of double-bogey or higher (not shown) in this sample is significantly higher than the corresponding frequency on the cutline, but the anomoly is reflected in the comparatively low mean score at +1 as well. However, the risk-response hypothesis makes no clear prediction about mean score around the cutline. Competitors in close proximity to the cut are incentivized to add or subtract risk, either of which should carry the cost of a higher mean (see Figure 1). Whether adding or subtracting risk is more costly depends on the rate of technical substitution on either side of the frontier minimum, an empirical question to be taken up shortly. A more expected pattern holds greater than one stroke above the cutline, where both high and low scores are more frequent. Under-par scores diminish further above the cut and over-par scores climb, indicating an increasingly desperate and costly strategy, and perhaps resignation.
Figure 3B presents direct indicators of risk: scoring standard deviation, frequency of “Going for the Green” on the designated holes, two-putt frequency, and mean driving distance. The four measures provide further evidence that players recognize their position relative to the cutline and respond by making the hypothesized changes to risk strategy. Each of the four measures diverges around the cutline, and differences immediately around it (0 and +1) are all statistically significant with p-values <0.003.8 Of the measures in Figure 3B, only “Went for the Green” captures strategic intent directly. Recall that the overall rate at which players “go for it” is very close to 50%. The sample frequencies on or one stroke under the cutline are <45%, while 51% for the +1 sample and 55% for the +2 sample. The same pattern holds for two-putt frequency, the indicator of conservative play in approaching and playing greens. Two-putt frequency is between 56% and 57% inside the cutline, falling below 55% outside the cutline and below 54% at two and three strokes over. Finally, we observe a pattern of driving distance consistent with risk-adjusted tee shots around the cutline, but the practical difference of a just one or two yards is very small.
The preceding results are strongly consistent with the hypothesis that players on either side of the cutline adjust risk strategy, with those just inside protecting their position and those just outside playing aggressively. The risk environment around the cutline should afford an opportunity to explore the rate of technical substitution: how much does aggressive or conservative play, as measured by changing standard deviation, affect the mean score? The sample means and standard deviations presented in Figure 3 cannot be used alone to estimate this rate. Refer again to Figure 1, which depicts the frontier of feasible combinations of mean and standard deviation for fixed player and hole characteristics. In this framework, players outside the cutline take risks they would not under milder incentives, therefore moving along the upward-sloping portion of the curve. Players on or just inside the cutline avoid risks judged worthwhile under normal conditions, effecting movement backward (right to left) along the downward-sloping portion of the curve. The slope of a representative function cannot be estimated by comparing outcomes across the minimum expected score. Needed is a control sample of holes associated with mild risk incentives, preferably with a matching composition of players and scorecard holes played. The natural control sample is provided by the standard format of a professional tournament. With the exception of a small number of tournaments that use multiple courses in early rounds, players responding to cutline risk incentives on Friday played the very same holes on Thursday. For example, 23,916 out of 27,042 closing holes played on the cutline number at the conclusion of the second round can be paired with a sample of 23,916 observations of the same competitors playing the same holes in the first round. First round play is a strong candidate for tournament situation in which the risk micro-incentives are weakest, since the majority of the tournament lies ahead and the leaderboard has yet to take shape. In a first round, the optimal risk strategy should be closest to that which minimizes expected score.
The mean and standard deviation of scores from the control sample are combined with the cut day “treatment” sample produce an estimate of the average technical rate of substitution for each score relative to the cutline:
where the subscript 1 represents the statistic from the treatment sample, and the subscript 0 represents the control sample. The statistics are calculated using the same weighting formula for influence of closing holes, but are unweighted by field standard deviation. A value of σ1>σ0 indicates a sample of players increasing the average risk with which they played the closing holes of the “treatment” round relative to the same holes the day before. If such an increase in risk carries the expected cost in higher mean scores, the numerator is also positive and the statistic in (5) estimates an average upward slope of the technical frontier. If players just inside the cutline play ultra-conservatively, this too should increase mean score as standard deviation is reduced from the minimizing level. A higher treatment mean accompanying a lower treatment standard deviation produces a negative estimate of the downward slope of a representative frontier. The results of this natural experiment are produced in Table 4.
|Tournament score relative to cutline|
The sample of holes played at two strokes under the cutline over the closing holes before a cut yields a higher mean score (0.019 compared to 0.003) and lower standard deviation (0.669 compared to 0.699) than play on the same holes the previous day. The ratio of –0.55 represents the estimated rate of technical substitution as players in this sample pay a cost in higher mean scoring for reduced risk. The estimate associated with the sample of holes played one stroke under the cutline is steeper at –0.98, both because the increase in mean score is greater and the risk reduction achieved more modest. The sample on the cutline shows very little risk reduction at all, and therefore a larger measured cost of the 0.007 increase in mean score. The +1 sample produced the anomalous scoring pattern in Figure 3A of a significantly greater under-par frequency than the other samples without a significant increase in bogeys. The measured rate of technical substitution for the sample is a good summary of this unexpected result. Players at +1 produced a standard deviation 0.03 points higher on Friday than Thursday, but the mean score actually fell. This group appears to have paid a modestly negative cost for their aggressive play. Results for the +2 and +3 samples are more in line with expectations, as mean score increases of 0.015 and 0.012 are found in combination with rather large increases in standard deviation. The mystery of the +1 cutline score is whether it is caused by sampling variation or if there is a tuned efficiency produced by this particular state: urgent but not desperate, and focused. If there is any truth to the latter, players would do well to extend this mentality to other tournament circumstances.
While the evidence for risk adjustment around the cutline is compelling, a similar effect over the closing holes of a tournament is more difficult to identify. Figure 4 presents the same eight risk indicators from Figure 3 by tournament position (leader though 20th position) over the closing 5 holes of a final round. The same weighting scheme was also applied. To summarize the risk-adjustment hypothesis in this context, the convexity of tournament payoffs is greatest for competitors just behind the leader, between 2nd position and 6th, after which the payoff function flattens as the absolute payoffs diminish. Players competing over the final holes of a tournament in direct pursuit of the lead stand to earn considerably more by moving up one position than lose by falling back one. The convexity should encourage greater risk. However, a tournament leader at the same stage faces very different incentives, with priority to protect the lead with conservative play. There is only modest evidence of such a response in Figure 4. The samples are smaller than in the cutline analysis, ranging from 2969 observations of closing hole leaders to 2108 in 20th position. In the scoring outcomes presented in Figure 4A, leaders and those players in 2nd generate a significantly higher frequency of pars than positions 3 through 11. Leaders do make significantly fewer below-par scores than the sample of players in positions 2 through 6 (p-value 0.02), but this is not matched by fewer over-par scores. Overall mean score by leaders is quite high, which may be consistent with the cost of conservative play, but a clear pattern in positions 2 through 6 is not at all evident.
Figure 4B contains hints of riskier play among close pursuers to the lead, but the results are noisy and hardly conclusive. Standard deviation in scoring peaks in 5th and 6th position, and the standard deviation is significantly lower than that of a combined sample of positions 2 through 6 (p-value 0.013). The frequency of “Going for the Green” is high from positions 2 through 5 relative to the lead before falling again outside the top 5, though leaders do not differ in this measure significantly from the same combined sample (p-value 0.23). The average frequency of two-putts in positions 2 through 6 is more than a percentage point lower than the lead (p-value 0.07). Mean driving distance peaks in positions 2 through 4, but the difference with leaders is not statistically or practically significant.
The absence of clear evidence for risk adjustment over the closing holes of a tournament presents a puzzle. Analysis around the cutline seems to rule out the idea that players are incapable of making such adjustments. Figure 4 provokes the possibility of a systematic bias, in the manner Pope and Schweitzer’s (2011) finding of loss aversion cast doubt on disciplined expected earnings maximization. For instance, the pressure of Sunday afternoon in contention is notoriously challenging, and managing pressure could take priority over risk strategy. When Ben Crane won the 2010 Farmers Insurance Open at Torrey Pines, he refused to look at the leaderboard all day. He learned of his victory when his playing competitor congratulated him on the 18th green.9 Crane is an eccentric case – there is plenty of scoreboard watching – but the story indicates a frame of others have expressed as well. Under pressure, competitors may focus on playing “my game,” reluctant to deviate from type even when a calculating look at the incentives suggests it is worth doing so.
However, there are explanations aside from bias that would either confound empirical measurement of the true risk response or alter the hypothesis about optimal behavior. First, the analysis thus far has not controlled for hole difficulty, the “hot hand” effect, or other factors known or believed to influence scoring outcomes. Second, an ordinal leaderboard may not adequately capture effective position in a tournament. On the course and on television, the leaderboard represents a snapshot in time. Yet players must consider the practical effect of being several holes ahead or behind their closest competitors. For example, if a rival for position still has a likely-birdie par-5 or an especially difficult hole still to play, this influences a players’ understanding of where he stands. Effective position is more fluid and uncertain than the cutline, and the signal may be of insufficient quality to act upon. Third, ties and gaps on the leaderboard further obscure the picture of simple convex payoffs. The remaining pages of this section will attempt to address some of these issues.
4.3 Multivariate analysis
Table 5 presents the results of three variations of a multinomial logistic regression of scoring outcomes on cutline position, final round tournament position, and hole characteristics. The results are presented in marginal effects of each independent variable on discrete scoring outcomes: the change in predicted probability of each outcome associated with a specified change in the independent variable. This requires setting the values at which all other independent variable are held constant, and these selections are detailed in the discussion below. The three versions of the model incorporate alternative specifications of cutline scoring and late final round position, but are otherwise identical. One specification included indicator variables for relative score around the cutline – from two strokes under through three strokes over – on the final 5 holes of a cut round. A second specification collapsed the six cutline indicators to two: (1) two or fewer strokes inside the cutline, including on the number, and (2) three or fewer strokes outside the cutline. Both of these specifications included a characterization of final round position relying on indicators for the lead, 2nd through 6th, 7th through 11th and positions 12th through 16th over the final 5 holes of a tournament. A final specification employed the more parsimonious cutline formulation, and replaced the ordinal final round position with the Convexity Index. For consistency, the Index is set equal to zero except for the closing 5 holes of a tournament. The three alternative specifications yielded identical results for all other variables in the model. To avoid redundancy, Table 5 collapses the results and presents estimates from the separate specifications together.
|Marginal Effects (dProb/dx)||Marginal Effects (dProb/dx)|
|Score < Par||Score = Par||Score > Par||Score < Par||Score = Par||Score > Par|
|Score Relative to Cutline||Closing Holes Position|
|+3||0.003||–0.016***||0.013***||Field Mean Score||–0.496***||0.123***||0.372***|
|≤2 strokes inside/on cutline||–0.005***||0.008***||–0.003||Field Score SD||0.343***||–0.500***||0.157***|
|≤3 strokes outside cutline||0.007***||–0.012***||0.005***||Relative Round||–0.001***||–0.003***||0.004***|
Formally, the model is
where i and h index player and hole, respectively, and the outcomes j=1, 2, 3 represent the discrete scoring outcomes under-par, par, and over-par. The vector of coefficients and explanatory variables is
where Cutline and SunPos denote the alternate specifications of score relative to the cutline and final round position over the appropriate holes, and εih takes the logistic distribution. The compression of scoring outcomes is a concession to the large scale task of computing marginal effects in a sample of 2.9 million holes played. The independent variables in Table 5 have all been previously introduced. Marginal effects sum to zero across scoring outcomes for each independent variable. To use the first variable in the Table as an example, a late cut round hole played at two strokes under the cutline is 0.6% less likely to yield a birdie or better and 0.5% less likely to yield a bogey or worse; the corresponding 1.1% probability weight is shifted to par. The estimates hold all other independent variables in the model constant at specified values. For cutline marginal effects, tournament closing position variables and the final round indicator are all set to zero, par is set to 4, and all other variables to their sample means. In calculating closing tournament position marginal effects, the cutline indicators are zero, a final round par-4 is specified, with the other variables again evaluated at their means.
Results for the disaggregated cutline specification are summarized clearly by the marginal effects of par scores. The three scores on or just inside the cutline are associated with between 0.5% and 1.1% greater probability of par. The three positions just outside the cutline are associated with between 0.8% and 1.6% lower probability of par. In nearly each case, probability weight shifts toward or away from non-par scores in the hypothesized direction. An exception is the +1 anomaly discussed earlier: holes played at one stroke over the cutline generate significantly more under-par scores, but the expected cost of aggressive play – a rise in bogey and worse outcomes – does not appear in the data. When cutline proximity is collapsed to two categories, the result holds with five of six marginal effects significant at the highest level. The results indicate that par is 2% more likely from just inside the cutline than just outside all else equal, with a bit more than half of the probability weight shifting on lower than higher scores.
Just as the multivariate estimates of effects around the cutline sustain findings from Figure 3, estimates for the effect of leaderboard position again fail to provide strong evidence of the risk-adjustment hypothesis over the closing holes of a tournament. Marginal effects are statistically insignificant in all but two estimates, but more importantly there is no general pattern consistent with an adjustment to convex payoffs. Leaders are less likely to break par, but this does not correspond to fewer scores above par. Close pursuers in 2nd through 6th are more likely to break par, and this effect is significant, but there is no measured cost of riskier strategy in greater probability of high scores. This group appears to simply score better. The group in 7th through 11th, the flattening portion of the payoff function, shows a small and statistically insignificant shift toward par.
The Convexity Index was constructed to incorporate a cardinal measure of payoff asymmetry that also accounts for the role of ties and gaps on the leaderboard. If players truly do respond to convex payoffs over the closing holes of a tournament with risk adjustment, and the weak results for position were due to a lack of resolution related to these factors, the index should capture effects ordinal position cannot. There is no evidence in Table 4 for this proposition. Marginal effects are calculated for a $10,000 change in the index, quite a large difference in payoff structure. Yet the results at most indicate a very modest and statistically insignificant prediction of lower scores, not risk adjustment. Taken together, the alternative estimates of risk response to convex tournament payoffs concur: players do not appear adjust risk over the closing holes of a tournament in the way they do around the cutline.
4.4 Individual player estimates
In a final effort to identify risk calibration by individual players, the same multinomial logistic regressions were estimated separately for the 83 players with a minimum of $5 million earnings and at least 20 holes played with the back nine Sunday lead. Two summary measures were calculated for each player: (1) the difference in estimated marginal probabilities of making par associated with “just inside” and “just outside” the cutline; and (2) the difference in the estimated marginal probabilities of making par associated with holding the lead and holding positions 2–6 over the closing 5 holes of a tournament. Both measure the tendency to shift risk according to circumstances, with a greater difference associated with more (fewer) pars when incentivized to reduce (add) risk. Figure 5 presents histograms for these two measures for this elite group. Both measures have small positive means and medians. It is interesting to note some of the players with measures most consistent with active risk calibration around the cutline and over the closing holes. The greatest difference in par marginal probability “inside” and “outside” the cutline is Phil Mickelson, who is 9.4% points more likely to make par when just inside the cutline than he is just outside. Rory McIlroy is second with a difference of 8.1% points, followed closely by Kenny Perry, Luke Donald, and Adam Scott. Each among this handful of players also has a positive difference in par likelihood when leading compared to chasing a top finish. The largest value for this second measure belongs to John Daly, who is a massive 33% points more likely to make par when holding a lead than he is in positions 2–6. This section has established that the average tendency to calibrate risk according to tournament situations is quite small in the full sample of observations, but these individual measures suggest at least some players know how and when to play aggressively.
A PGA Tour professional is highly skilled, experienced, and motivated by large awards to maximize expected earnings. His distribution of scoring outcomes depends largely on talent and other fixed characteristics, but strategic choices are made on each hole – some holes afford more choice than others – that shift probability weight toward or away from par. The long-term scoring profiles of successful PGA TOUR professionals indeed varies substantially when it comes to basic measures of risk. Specifically, somewhat higher mean scores are often consistent with high earnings when combined with sufficient scoring standard deviation. This is almost certainly a result of the convex payoff structure in professional golf, which rewards high finishes more generously than consistent play. The variation in player “types” makes golf more interesting to follow, but is also reflective of other winner-take-all environments. Convex payoffs to relative performance reward risk taking beyond the level that optimizes expected outcomes.
This paper proceeded to address the subtler question of whether and to what degree incentives for risk-taking also generate strategic responses within tournaments. Risk is a type in professional golf, but is it also a strategy that can be calibrated according to tournament circumstances? The evidence presented suggests players measurably alter strategy around the cutline over the closing holes before the field is cut, thereby demonstrating that hole-by-hole risk adjustment is achievable. I provided measures of the rate of technical substitution between scoring mean and standard deviation, with the result that reducing risk – playing more conservatively to protect position – is apparently more costly than adding risk. The puzzle introduced in these pages is why tournament position over the closing holes of a tournament, when the absolute difference in payoffs is largest, generates such an inconsistent and negligible response.
Is this a missed opportunity, a bias from expected earnings maximization? The possibility is intriguing, because if competitors tend to fall back on familiar or heuristic practices under stress, tournament incentives are weakened precisely when the stakes are highest. Or perhaps only stark convexity, as around the cutline, is sufficiently simple to assess and engender a strategic reaction. Further measurement of risk management in tournament settings will help establish the robustness and magnitude of the missing response, and the wider behavioral implications are a worthy area for future research.
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