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, *π*_{s}, define the expected value *μ*=Σ_{s}*π*_{s} and standard deviation

of the random variable. Assume for this discussion that

*f*(

*S*) is restricted such that all of the probabilities are specified by given

*μ* and

*σ*. The probability mass function will depend on fixed player characteristics,

*P*, and hole characteristics,

*H*. A player’s fixed characteristics include talent, experience, and the relative strengths and weaknesses of his game. Hole characteristics include length, hazards, rough thickness, green speed and hole position. In addition, a player faces a variety of decisions when approaching each hole that influence the probabilities of carding different scores. These include (1) club selection, e.g., hitting driver or a shorter/straighter club off the tee; (2) the “line” (direction) of longer shots, e.g., shading away from a direct path to the hole in favor of avoiding hazards; (3) attempting to reach the green with a second shot on a par 5, or tee shot on a short par 4, rather than lay-up and approach in the green in the regulation number of strokes; (4) putting aggressiveness, e.g., striking a birdie putt firmly increases the likelihood of holing it, but also of leaving a more difficult par putt. The fact of these choices implies that

*S* is not entirely a function of fixed and random factors; the distribution of probabilities is also under constrained control of the player.

Define the relevant constraint according to the mean and standard deviation of outcomes:

where

Equation (1) defines a player’s capacity to add or subtract standard deviation (risk) to scoring outcomes through the effect on mean score. Under general efficiency of strategy, aggressive measures that increase the probability of a low score will carry a cost in greater probability of a high score, with the standard deviation necessarily increasing. Probability of the score around the mean, typically par, is reduced. Where such a shift of probability weight favors low (high) scores, mean score falls (rises). Mean score is minimized when

*g′*=0, the point where the marginal effects are balanced. Roughly speaking, expected score is minimized when all risks that raise the probability of a birdie more than the probability of a bogey are undertaken, but none of the risks that do vice versa.

Figure 1 depicts the derived shape of

*g*(·), an efficient frontier of (

*μ*,

*σ*) combinations given fixed player and hole characteristics. The minimum mean score and level of risk associated with it are labeled

*μ*^{0} and

*σ*^{0}. I will refer to the slope of this frontier,

*g′*, as the

*rate of technical substitution* (RTS) between risk and mean score, and offer an estimate in Section 4.

Figure 1Theroretical optimal risk bearing. Where the payoff function is convex, expected earnings are constatnt along an upward sloping frontier, and optimal risk, *σ**, exceeds the level that minimizes mean score, *σ*^{0}.

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 $α*x*. 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 *s*_{h} represent a player’s score on the *h*th hole of a tournament, and

his cumulative score up to that point:

Tournament position approaching the hole is

where

represents the vector of cumulative scores to that point for all other competitors in the field. Ultimately, tournament rank maps directly to earnings, so a player’s expected earnings,

*EE*, depend on his current position and the remaining sequence of probabilistic events:

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

represents final competitor scores, and is therefore a vector of random variables. It is tempting to propose that when confronted with this much uncertainty, players fall back on a heuristic rule such as minimizing their expected score, and this may be accurate. However, only one hole is played at a time, and players need only assess the changing earnings prospects of their position with good or bad outcomes on the current hole in order to receive the signal of convex payoffs. Expected earnings are maximized with respect to risk strategy on a specific hole, subject to the constraint in (1) where:

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 cut^{2} 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.

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