The strategic jump-the order effect on winning “The Final Three” in long jump competitions

1 Introduction

A common argument for changing rules within a given sport is that the change will make the sport event more exiting or increase fairness. Wright (2014) and Kendall and Lenten (2017) provide surveys of changes in rules from various sports and their impact upon competitors’ behavior. Both papers present examples of when rule changes, aiming to increase excitement, have led to unexpected – and in some cases – unfair consequences. In this paper we add an additional sport to their list of examples: long jumping. To make the long jumping international competitions more exiting, a new tournament format, “The Final Three”, was introduced in 2020. Given the number of prequalified contestants in a competition, the three highest placed athletes, after a sequence of attempts (in general five attempts, henceforth rounds), were qualified for the sixth round (the final henceforth). The medals among these three athletes were then distributed according to their achieved result in the final, irrespective of what distance they jumped in previous rounds. This rule differed from the former applied rule, where the winner was the one who made the single longest jump over all rounds in the competition. A consequence, albeit not unforeseen, of this change in tournament rule, is that the winner may not be the athlete who had the best overall result achieved in the competition. According to the organizer Diamond League, the motive for changing the rule was to award athletes who have the ability to perform under the most intense pressure, bringing more drama to the competition as nothing would be decided until the last jump.^[1] Alongside the criticism that the new rule was unfair because the first five rounds by the finalists become void in the final, objections were also raised because the rule weakened the athlete’s incentive to perform his best in the first five rounds, since ending up in third place was as good as ending up in first place in order to reach the final. Following the consultation with athletes, coaches and meeting organizers, the Wanda Diamond League General Assembly decided in December 2021 to revise The Final Three format.^[2] The properties of this revised Final Three format and its expected impact on fairness and excitement are discussed in the last section in this paper.

Even though the former format of The Final Three, as described above, likely is to be put aside in the future, the format’s properties are nevertheless interesting to analyze. There will probably always be a debate about how to build drama and excitement into the long jumping competition, which in its traditional format may be perceived by some as an endlessly repetitive jumping. After all, the format of The Final Three bears a resemblance of the format applied today in running tournaments in track and field, where runners qualify to a final in a sequence of elimination heats. Regardless of how fast a runner runs in previous races, the runner with the fastest time in the final will be the designated winner.

The design of The Final Three though suggests that the athlete’s performance in previous jumps, i.e., the athlete’s placement after five rounds, could have a direct impact on the probability of winning the competition. There are two reasons for this assertion. First, in case of a tie in the final, the best performance from the previous five rounds separates the athletes. Second, the mutual placement of the finalists after five rounds determines the order in which the finalists jump in the final. The athlete who is placed third after five rounds begins, the second-best performer is next, while the leading athlete after five rounds finishes the final. This order suggests an advantage for the leader after five rounds, since it is possible to adapt the risk of making a foul, an illegal result, depending on the other two finalists’ performance in the final. Likewise, it could be argued that the third placed athlete has a disadvantage of going first in that no results of the rivals yet exist. We believe the jumper adapts the risk of making a foul by the position of the distance marker, from where the jumper starts the approach run. By moving the marker backwards, this risk obviously declines. However, this comes with the cost of lowering the probability of a good result. The reason for that is that the takeoff foot, most likely, will be far from the official take-off line, from which the distance of the jump is measured. Thus, there is a trade-off between having a low risk of making a foul and a high probability of making a good result.

The main purpose with this work is to investigate to what extent game theory may guide us as to the possible advantage of being the first, the second or the last jumper in The Final Three. By deriving the probability each of these jumpers have of winning the competition, we obtain a measure indicating how advantaged or disadvantaged a jumper is ending up on third place rather than on first place in the qualification rounds, given that his achieved result in the qualification does not count in the final. Our approach also makes it possible to highlight and quantify the importance of both strategic thinking and of having a low variation in the length of the approach run.

In the paper we present a stochastic model for the outcome of a long jump, where the derived distribution of the outcome depends on the position of the distance marker. Given our stochastic model, we characterize a sub-game perfect equilibrium and derive the equilibrium win probabilities for the three finalists in The Final Three. The competition is viewed as a non-cooperative rational strategic sequential-move game where the three finalists are players with asymmetric information about the achieved outcomes of other players’ moves.^[3] Their decision making is interdependently related and all of them aim to maximize the probability of winning the final by making a strategic choice of the position of the distance marker. Data from the long jumping final from the Olympics in Tokyo in 2021 is used to estimate parameters in our model.

The rest of the paper is organized as follows: Section 2 provides a literature overview. In Section 3, a stochastic model for the result of a long jump is presented. Section 4 describes the data collection and displays estimates of certain parameters included in our model, based on the collected data. In Section 5 equilibrium win probabilities are derived, while the findings in the paper are discussed in Section 6.

2 Literature

The first part of our work is related to the literature on modeling the jumper’s problem in the long jump competition to decide his optimal aimed take-off line, an issue analyzed in some early papers. In Ladany, Humes, and Sphicas (1975), a general model is developed to determine the aimed optimal take-off line which maximizes the jumper’s expected distance in a competition with three rounds. In their model, the jumper’s choice of take-off line is not conditioned on the jumps of his competitors. Given their collected data from 10 consecutive training sessions of an athlete, they solve the problem by simulation of a specific case. Assuming that both the entire jumping distance, and the distance between the tip of the take-off foot and the take-off line aimed for, are normally distributed, the authors derive an analytical solution, which provides a valid approximation of the simulated result. A similar model is applied in Ladany and Singh (1978) in which the jumper’s objective is to maximize the probability of jumping a given distance. This policy is then compared with the case when the objective is to maximize the expected distance jumped. Using the results from the training sessions obtained in the previous study by Ladany, Humes, and Sphicas (1975), the finding is that the later policy – to jump the longest jump – is inconsistent with the desire to maximize the probability to jump a given distance. Ladany and Mehrez (1987) extend the problem of maximizing the probability of jumping a given distance in a single jump to a 3-trial competition case. They show that the optimal location of the aimed take-off line, maximizing the probability to jump a given distance in a single jump, also maximizes the same probability in the 3-trial case. However, to prevent erroneous values of the optimal location of the aimed take-off line due to the aforementioned truncation of the distribution of the distance between the tip of the take-off foot and the aimed take-off line, they solve the maximization problem by simulation, leaning on the estimates from Ladany, Humes, and Sphicas (1975). Ladany and Mehrez (1987) find that when the targeted distance is relatively short in relation to the jumper’s ability, the jumper can select an aimed take-off line farther behind the official take-off line and still enjoy a probability close to 1 of success. Conversely, when the targeted distance is relatively long, a deviation from optimal aimed take-off line, closer to the official take-off line, has a little effect upon the very low probability of success.

The contribution of our work to this literature is an extension of the model by Ladany and Mehrez (1987). We derive a probability distribution of the jumper’s score which captures the fact that one of the underlying distributions – the score of a legal jump – is truncated.

The second part of our paper relates to the relatively large body of literature on the design of sporting contests and competitors’ performance (for a survey, see Szymanski 2003). More precisely, the structure of The Final Three in the long jumping final, directs our interest towards the literature on the effect of the order of actions on performances in sports with sequential moves. Several factors are put forward to explain the observed differences in competitors’ performances, e.g., psychological pressure, learning effects, cognitive effects, and strategic interaction. The observed performance in penalty shootouts in soccer, where the kicks are taken alternately by the two teams – A and B – according to ABAB, has attracted attention in a row of papers. A survey of empirical studies on penalty shootouts, provided in Csató (2021), indicates that there exists a first-mover advantage of kicking first due to the mental pressure the player taking the second kick is put under. Penalty shootouts are also applied in many ice-hockey leagues determining the winner if a match is still drawn following overtime. The shooting is in general ordered ABAB, where the home team decides who will start the shootout. Kolev, Pina, and Todeschini (2015) analyze data from shootouts from the National Hockey League and find a lower winning frequency for the team shooting first. The serves in tennis tiebreak have an order close to the Prohuet-Thue-Morse sequence. Analyzing data from larger tennis tournaments, Cohen-Zada, Krumer, and Shapir (2018) find no systematic advantage to any player serving first or second in tiebreak.

Brady and Insler (2019) analyze recorded data from the PGA tours on playing partners in golf to examine whether there is an order effect when both players’ golf balls are positioned relatively close to each other when playing from fairway or putting on green. They find evidence of a second-mover advantage, which they motivate is consistent with a learning effect. Krumer, Megidish, and Sela (2017) analytically show that competing in the first round and in the last round in a round-robin tournament with three symmetric competitors is more advantageous than any other ordering. Winning in the first round affects the continuation values of the competitors, providing the winner a higher continuation value of winning than her opponents. Krumer and Lechner (2017) find this prediction to be in line with their analysis of the observed outcome in Olympic wrestling and in UEFA/FIFA soccer cups, in which the group stages are arranged as round-robin tournaments. González-Díaz and Palacios-Huerta (2016) show that most chess matches – where two players play an even number of games against each other, altering the color of pieces – is won by the player playing with the white pieces in the first game. Playing with the white pieces in chess confers a strategic advantage. Therefore, the player playing with the black pieces in the first game is more likely to lag in the match and thus being at a psychological disadvantage, affecting the player’s cognitive performance.

In the sports referred to above, a competitor’s best response to an outcome, caused by her opponent’s move in the sequential game, is always to try to shoot a goal, hit the ball in the hole, win a new match, etc., irrespectively of her opponent’s outcome. However, in the long jumping final, a competitor will condition her best response on the information she has about her opponents’ performances, by optimally positioning her distance marker.

A similar situation can be found in many running tournaments at track and field events, where runners first compete in a sequence of qualification heats prior to the final. A runner advances to the final either by being among the top placed runners in his heat or having one of the fastest times of those who did not advance by place, regardless of the heat (“a lucky looser”). Therefore, in a sequence of qualification heats, runners in later qualification heats are better informed about times required to reach the final than are runners in earlier heats. Hill (2014) analyses data on runners’ performance in International Association of Athletics Federations (IAAF) 5000-m events, held 2001–2011, where runners compete in two separate qualification heats. Given the runners’ asymmetric information, the analysis does not support the hypothesis that there is an advantage or a disadvantage of competing in either of the two qualification heats.

3 A stochastic model for the result of a long jump

In this section a stochastic model for the outcome of a long jump will be presented, taking the effect of the athlete’s attitude towards risk of making a foul jump into account. The athlete controls the risk by changing the position of the distance marker indicating the start of the approach run towards the official take-off line (henceforth the foul line), from which the athlete tries to make as long a jump as possible before landing inside a sandpit.

For a jump to be legal the toe of the athlete’s shoe needs to be behind the foul line while launching. Otherwise, if the toe crosses the foul line, the athlete has made a foul jump that doesn’t count. Long jumps are measured from the foul line to the impression in the sandpit, closest to the take-off board, made by any part of the athlete’s body, while landing.

For a certain athlete, X is defined to be the distance of the approach run, i.e., the distance of the predetermined number of steps the jumper takes from the marker to take-off. We assume X to follow a N μ X , σ X 2 . We define γ + μ _X to be the distance from the marker to the front edge of the foul line. Thus, the distance from the point of take-off to the foul line, denoted by T, can be written as T = γ + μ _X  − X, where a fair jump requires T ≥ 0, while T < 0 defines a foul jump. It follows that T ∼ N γ , σ X 2 . Thus, γ is interpreted as the expected distance from the take-off foot to the foul line. Assuming the athlete has full knowledge of the distribution of X, he is in control of the value of γ depending on where he places the marker.

Furthermore, we define Y to be the entire distance of the jump, from the point of take-off to the first landing point of the athlete. Y is assumed to follow a N μ Y , σ Y 2 . In addition, define V* = Y − T. Now, the score V is defined by

Thus, V can be considered a mixed random variable with a continuum of positive^[4] real values as possible outcomes as well as a qualitative outcome.

Assuming Y and T to be independent, the pdf of V*|T ≥ 0, denoted by f V * | T ≥ 0 ( v * ) , can be shown to be

where erf(⋅) is the error function.

The probability of making a foul is given by 1 − Φ γ σ X , where Φ(⋅) is the cumulative distribution function of a standard normal. Consider the function

The probability of having a score better than or equal to v 0 * is

In Figure 1 the function g v * is illustrated graphically for μ _Y  = 8.00, σ _X  = σ _Y  = 0.1 and for two different values of γ (γ = 0 dashed line; γ = 0.20 solid line), where meter is the unit of measurement. Note that the function g v * is not a probability distribution function and the area under the graph is smaller than 1. For the dashed curve the area is smaller than for the solid curve, resulting from a larger probability of making a foul.

Figure 1:

Graphical illustration of g v * , the continuous part of the distribution for the mixed random variable V for μ _Y  = 8.00, σ _X  = σ _Y  = 0.1 and for two different values of γ (γ = 0 [dashed line]; γ = 0.20 [solid line]).

An athlete choosing a value of γ = 0 takes a large risk of making a foul, here P f o u l = 0.5 , to have a reasonable chance of making a good score, say better than or equal to 8.00. A value of γ = 0.20 means the athlete’s risk of making a foul is small, here 0.023, and from Figure 1 it should be evident that the probability of making a descent result, say 7.70 or better, is quite large compared to the case where γ = 0. However, this comes with the cost of lowering the chance of reaching a very good result, since for γ = 0.20 the probability of getting a result better than or equal to 8.00 is smaller than for γ = 0. Thus, by choosing the value of γ, i.e., by choosing the position of the distance marker, the athlete balances the trade-off between the risk of making a foul and the chance of a long legal jump.

4 Data

The present study uses data from the final in long jumping at the Olympics held in Tokyo in 2021. Data were collected by reviewing video clips of both men’s and women’s long jump finals. Measurements on the two variables T and Y defined in Section 3 are of interest, i.e., the distance from the take-off foot to the foul line and the entire distance of the jump.

Except for two jumps – one jump for each of two female athletes – it was possible to record measurements on T for all legal and foul jumps. For legal jumps, measurements on Y could be perfectly recorded by adding the measurement on T to V*, defined in Section 3 as the score, i.e., the distance from the foul line to the first landing point of the athlete. However, for foul jumps which make up about 30 % of all jumps, the score was not formally registered. For these cases a visual assessment of the score was made.^[5]

The purpose of the data collection is to get reasonable values, for world class long jumpers being representative for competing in The Final Three format, on parameters included in our stochastic model. Three parameters are of interest. These are the expectation and the standard deviation of the entire distance of the jump, μ _Y and σ _Y , respectively, as well as the standard deviation of the distance from the point of take-off to the foul line, σ _X .

In Table 1 estimates of these parameters are displayed for those top athletes in the final who had the possibility of making six jumps instead of three.^[6] The mean and the standard deviation of the entire distance of the jump, y ̄ i and s Y i , are shown in the first and second row, respectively. In the third row the standard deviation of the distance from the point of take-off to the foul line, s T i , is displayed.^[7] Within brackets the number of observations used in the calculation is found.

The last three columns in the table show the minimum, the maximum, and the mean values of the estimates. The mean values are used to characterize a typical world class long jumper, while the minimum and maximum values constitute ranges within which most world class jumpers’ parameter values are believed to lie.

5 Deriving equilibrium win probabilities

This section takes the model presented in Section 3 and the parameter estimates in Section 4 as the point of departure when analyzing the sequential game and calculating equilibrium win probabilities for each of the three competitors.

5.1 Modeling the sequential game

The competitors (henceforth players) are denoted by A, B and C, where the order in which they jump in the final round is descending. We assume they all aim to maximize the probability of winning the competition by making a strategic choice of the value of γ. It is assumed that each player has full knowledge about his own capacity as well as the two opponents’ capacity in terms of individual specific distributions of X and Y along with V.^[8] In addition, we assume all players to be aware of each other’s goal to maximize the probability of winning.

Figure 2 presents the ordering of players’ jumps as a game tree. Player C starts jumping and makes either a foul, f _C , or scores V ̄ C . Then, at each of the nodes B ₂, A ₁, A ₂, and A ₃ there exist three possible outcomes of a player’s jump: (i) a foul (f); (ii) a legal jump scored lower than the leading score V ̲ ; (iii) a legal jump scored higher than the leading score V ̄ . At node B ₁, the jump by player B will either be a foul or a legal jump with a leading score. The game contains seven subgames of which one has a trivial solution. If player C and player B make fouls, player A automatically wins the final irrespectively of her achieved result (A ₀). The last row indicates the winning player.

Figure 2:

The ordering of jumps and possible outcomes in the long jumping final.

To solve for the combination of the three players’ strategies that are a best response to each other, i.e., the equilibrium in such a game, the method of backward induction will be used.

Therefore, a stagewise solution of the equilibrium probabilities is performed. A rough description is as follows. First, the strategy of A is obtained, given the best score after the first two jumps by C and B, where at least one of these jumps is legal corresponding to the nodes A ₁, A ₂and A ₃ in Figure 2. Second, the strategy of B is derived. Two separate cases are examined depending on C has a legal jump or not (nodes B ₁ and B ₂). For both cases information about A’s strategy from the first stage is considered as well. Third, we obtain the strategy and calculate the equilibrium win probability of C considering the response from B and A described in stage 2 and 1, respectively (node C). Fourth, the equilibrium win probability of B is derived, by considering the two separate cases described in stage 2. Finally, in the fifth stage, A’s equilibrium win probability is derived.

In parallel to the theoretical description of how to obtain the equilibrium win probabilities, to illustrate we provide a numerical example. In this example all three finalists are assumed to have identical capacities, with parameters μ _Y  = 8.14, σ _Y  = 0.11 and σ _X  = 0.10 all measured in meters. The parameter values are set to the mean values for males in Table 1, to represent the capacity of a male world class long jumper. The choice of having identical capacities is the primarily interesting case, since we would like to isolate the effect of the order in which the three finalists jump.

5.2 Jumping last – the strategy of player A

Player A observes the achieved result by player C and player B and will condition his strategy on the best score achieved among these two players. As mentioned above, no strategy is needed for player A to win the final, if the other two players make fouls. Denote the best score from the previous jumps by v 0 * . Now, the probability for A to end up as the winner can be written as

p A | γ A = P V A ≥ v 0 * | γ A = ∫ v 0 * ∞ g v A * | γ A d v A * ,

where the probability depends on three fixed known parameters; μ _Y , σ _X , and σ _Y , and a parameter γ _A , the expected difference from take-off to the foul line, whose value can be affected by A by changing the position of the distance marker. Facing v 0 * , A chooses the value of γ _A that maximizes the probability of getting a score better than or equal to v 0 * . That is, the maximization problem

max γ A ∫ v 0 * ∞ g v A * | γ A d v A *

is solved for given values of μ _Y , σ _X , and σ _Y , and for different values of v 0 * . For μ _Y  = 8.14, σ _Y  = 0.11 and σ _X  = 0.10, the solution to the problem in terms of γ _Aopt , the optimal choice of γ _A , for different values of v 0 * is shown in Figure 3a, and the corresponding win probabilities, denoted by p _Aopt , are shown in Figure 3b.^[9]

Figure 3:

(a and b) The relation between γ _Aopt (jumper A’s optimal choice of the expected distance from the point of take-off to the foul line) and v 0 * (3a) with meter as the unit of measurement on both axis, and corresponding win probabilities p _Aopt (3b) for μ _Y  = 8.14, σ _X  = 0.10 and σ _Y  = 0.11.

Figure 3 reveals that if A is to beat 7.80 for example, his best choice of γ _A is 0.15, resulting in a win probability of around 0.83. Instead, if the best score to beat is as good as 8.20, he needs to gamble in the sense that the expected distance from the point of take-off to the foul line, is small. Here, γ _A  = 0.05 does the trick to maximize the win probability, yielding a value for p _Aopt of about 0.07.

5.3 Jumping second – the strategy of player B

Player B needs to be prepared to be able to solve two maximization problems, depending on C makes a foul jump or not.

Define the implicitly given function p Aopt = h v 0 * , presented as the solution to player A’s maximization problem, given in the previous subsection. Starting with the case of a foul jump made by player C, i.e., player B’s optimal choice is conditioned only on his conjecture about player A’s choice of strategy, the probability that player B wins the final given γ _B , denoted by p B | γ B ′ , can be written as

p B | γ B ′ = ∫ 0 ∞ ( 1 − h v B * ) g v B * | γ B d v B * .

For our numerical example the integral above is solved numerically and in Figure 4 the win probability for B for different choices of γ _B is presented.

Figure 4:

The probability of player B winning the final for different choices of γ _B , conditioning on player C makes a foul jump and player A maximizes his chance of winning.

The maximum probability for B to win is about 0.473 attained at γ _B  = 0.12. These values are the solution to the maximization problem

max γ B ∫ 0 ∞ ( 1 − h v B * ) g v B * | γ B d v B *

and are denoted by p B ′ and γ B ′ , respectively. The figure also reveals the importance of being strategic to increase the chance of winning. For example, a poor choice of γ _B , say γ _B  = 0.01, reduces the chance of winning by 0.12, compared to a perfectly strategic choice, since the win probability drops to 0.35.

Now, we turn to the case when player B also must respond to the score v C * achieved by player C, besides conjecturing the strategic behavior by player A. Let V _Aopt be the score made by player A, assuming A is acting in an optimal way. This score is considered a random variable by B, when deciding on the optimal value of γ _B . Then, the probability that player B is the winner, given v C * and γ _B , denoted by p B | γ B , v C * , can be written as

p B | γ B , v C * = P V B ≥ v C * , V B > V Aopt | γ B = P V B ≥ v C * | γ B P V B > V Aopt | γ B , V B ≥ v C * = P V B ≥ v C * | γ B P V B * > V Aopt | γ B , V B * ≥ v C * = ∫ v C * ∞ g v B * | γ B d v B * 1 1 − F V B * v C * | γ B × ∫ v C * ∞ ( 1 − h v B * ) f v B * | γ B d v B * ,

where F V B * ( ⋅ ) is the distribution function of V B * .

Figures 5a and b shows the solution to the maximization problem

max γ b p B | γ B , v C *

for our numerical example.

Figure 5:

(a and b) The relation between γ _Bopt (jumper B’s optimal choice of the expected distance from the point of take-off to the foul line) and v C * , the score by C (5a) with meter as the unit of measurement on both axis, and corresponding win probabilities p _Bopt (5b) for μ _Y  = 8.14, σ _X  = 0.10 and σ _Y  = 0.11.

In Figure 5a, values of γ _Bopt , the optimal choice of γ _B for different values of v C * , are shown. The corresponding win probabilities, denoted by p _Bopt , are shown in Figure 5b.

Not surprisingly, the better score made by C, the larger risk of making a foul jump is player B willing to take to maximize the probability of winning, as γ _Bopt is decreasing in v C * .

5.4 Jumping first – the strategy and the equilibrium win probability of player C

In a similar way as V _Aopt is defined, let V _Bopt be the scores in the final round made by player B. Player C considers both these scores as random variables, when deciding on the optimal value of γ _C . In addition, define the implicit function s v C * , the relation between the probability that player B beats player C and the score by player C, v C * . Then, the probability of player C being the winner given γ _C , denoted by p C | γ C , , can be written as

p C | γ C = P V C > max V Aopt , V Bopt | γ C = ∫ 0 ∞ ( 1 − s v C * ) ( 1 − h v C * ) g v C * | γ C d v C * .

The integrals are solved numerically for our numerical example and in Figure 6 the win probability for player C for different choices of γ _C is shown.

Figure 6:

The probability of C winning the final for different choices of γ _C .

The maximum probability for player C to win, the equilibrium win probability p _C , is about 0.318 attained at γ _C  = 0.09, denoted by γ _Copt . Thus, there is a minor disadvantage of going first, about 1.5 percentage points lower probability of winning compared to the case where the probability of winning is the same for all three finalists. Figure 6 also highlights the importance of player C acting strategically. A deviation from γ _C  = 0.09 might substantially lower his win probability.

6 Discussion

To what extent a long jumping competition has become more exciting after the introduction of a new tournament format is not a matter of question in this study. Yet, the original design of The Final Three has features of a tournament with a play-off: after a series of rounds – in which competitors may perform very mixed results – the competitors’ achieved scores are compared, and the contestants with the highest scores then battle for winning in a one-shot game. Unless the two first jumpers in the final have not fouled, the outcome of the final hinges on the last jump.

The single decisive jump in the final can be said to amplify two features in a long jump competition. First, the one-shot round generates a more transparent strategic interaction between the athletes than was the case in the old tournament format, where winning the competition was conditioned on the competitor’s best achievement in six rounds. Second, athletes may experience more psychological pressure being aware of that they only have one chance to beat a rival’s prior or conjectured performance. In our work, we have focused on the question what effect the sequential order of jumps has upon each athlete’s probability of winning the final. For this purpose, we extend previous probability models on long jumping, e.g., Ladany and Mehrez (1987), by developing a stochastic model for the outcome in long jumping which captures the truncation of one of the underlying probability density functions. Assuming athletes having equal capacities, i.e., symmetric players, we make use of backward induction to analyze the subgame perfect equilibrium in this sequential game. Given the values we assigned to the distributional parameters, based on data from the long jumping final in Olympics in 2021, we find for men, assuming identical capacities, the equilibrium probabilities of winning the final when jumping first, second and last to be 0.318, 0.331 and 0.351, respectively. For women the results are almost identical, 0.319, 0.331 and 0.350. Hence, the athlete who achieves the best score in the initial rounds prior to the final, will have a relatively small advantage over the other athletes of jumping last.

In the light of the competitor’s small difference in favor of being the best jumper in five rounds, it is easy to understand the claim that this new tournament format is not as fair as the older tournament format. Since the advantage of jumping last in the final rests on an ordinal ranking of each competitor’s best achieved result from the earlier rounds, our relatively small, estimated reward of jumping last in the final may appear insufficient to compensate an athlete for having made an outstanding performance in one or in several of the five rounds prior to the final.

However, it should be emphasized, that the sequential one-shot jumping in the final implicates that the component of strategic interaction is lifted out of the last jumper’s decision process. Unlike the other two competitors, whose optimal strategies are based on their prior information about the capacities of the subsequent jumper(s), the last jumper’s optimal choice – where to position the distance marker – does not require such information. The decision problem will only be to maximize the probability to score at least a given distance. If the other competitors’ priors are wrong, our derived result of the last jumper’s probability of winning the final, likely forms a lower estimate of the probability to win. As indicated by our numerical analysis, small deviations from the optimal choice of the parameter γ set by the other two finalists, e.g., due to misjudgments of the prior of the last jumper’s capacity, significantly reduce their probabilities of winning the final.

To facilitate our analysis, it is assumed that each athlete’s objective is to win the final, that is, we have not incorporated athletes’ placing considerations when formulating their strategies. It goes without saying, that the presence of prize money in long jumping competitions likely affects the athletes’ choice of strategies. In such case, an analysis would be based on modeling athletes’ expected utility and alternative attitudes towards risk, which is a challenge for future research.

Our prediction of a last mover advantage in The Final Three rests on two effects. First, in case of a tie in the final, the best performance from the previous five rounds separates the athletes. Second, the athlete jumping last will, unlike her competitors, condition her jump solely on the other two finalists’ actual performance in the final. However, we have not separated the two effects in our analysis. Developing approaches to split up these two effects is a theme for further research.

As mentioned in Section 1, The Final Three format has been revised as this paper is written and the new format has already been implemented. In its revised format, two major changes are to be seen. It remains that only the top three jumpers after five rounds will get a sixth attempt, but a jumper’s achieved results from all her sixth jumps will be counted when distributing the medals among the three finalists. Also, the order in which the three remaining athletes jumps in the sixth round is redrawn, where the best-placed athlete after round five goes first, followed by the athlete placed second and third, respectively. These changes are interesting, and they invite us to reflect on how they may affect fairness and excitement in the long jumping final.

The first change – the inclusion of the best result from all sixth attempts when ranking the athletes – is likely motivated by fairness considerations. Increasing the number of attempts, will increase the probability that the jumper with the highest capacity wins the final. The change also places less emphasis on the importance of strategic behavior in the final, i.e., the positioning of the distance marker. Given the achieved score after the initial five rounds, the athlete’s objective in the last attempt will now be trying to improve the standing score rather than to avoid making a foul and – which was the case in the original Final Three – facing the risk of ending up at third place. In other words, the expected distance from the take-off foot to the foul-line is likely to be smaller for the finalists in this revised form of The Final Three than was the case in its previous form. Hence, in addition to a down-size of the strategic part, this change also comes with an expectation of an increasing number of extraordinary results in the final round, albeit at the prize of more fouls. Moreover, although a low variation in the approach run length is still preferable, its importance has decreased with the change. A jumper with a large variation can afford a higher risk of a foul jump in a single attempt, since not only the last jump counts.

The second change – reversing the order of the jumps of the three finalists – is a natural consequence of the first change, bringing excitement to the final. Letting the third placed jumper from the five initial attempts to jump last in the sixth attempt, will keep the uncertainty of the distribution of the medals to the very last jump in the competition. It is true that there may exist an advantage of jumping last also in the revised form of The Final Three, which thus would harm fairness. However, in the light of the results we have obtained in this work on equilibrium probabilities to win The Final Three, as it originally was designed, the reduction in fairness, due to the reversed order of jumps, is small and therefore a low price to pay to maintain the excitement throughout the competition.