Game Theory and Olympic Gold Optimizing Michael Phelps's Chances

Game Theory and Olympic Gold: Optimizing Michael Phelps's Chances to Surpass Mark Spitz Game Theory Final Project Team Yo’ Mama (alphabetizing by last name this time) Kayela Beatty Newton Cheng Lindsay Grimm Kristen Scully Quintin Smith Final project: team Yo' Mama Background At the most elite levels of swimming, athletes usually specialize in one to three events and tailor their training and preparation accordingly. Michael Phelps is an exception. He is competitive on the world stage in 11 events. Phelps's talent and drive are on a level that is rarely seen in swimming. Since age 11, when his coach first recognized him as a special athlete, Phelps has followed a training regimen that has given him versatility across all four disciplines (or "strokes"), not to mention a lethal combination of endurance and speed. Today, at 23, Phelps is still working with the same coach he started with 12 years ago, and the duo has built a long-standing relationship that is also unheard of in the sport. Since he emerged onto the scene in 2003 as a legitimate world-class contender, Phelps has become the poster child for swimming, both in the U.S. and abroad. As the 2004 Athens Olympics approached, the media began comparing Phelps's chances in Athens to the feat of another swimming superstar: Mark Spitz. In 1972, Spitz won seven gold medals at the Munich Olympics, an achievement that was never in danger of being approached before the arrival of Phelps. The offer from Speedo Capitalizing on the media blitz surrounding Phelps before the Athens Games, Speedo – his primary sponsor and outfitter – offered him a bonus of $1 million if he tied or exceeded Spitz's record of seven gold medals during one Olympics. There was one catch: he had to do it in Athens or at the Beijing Olympics in 2008. The deal was a boon to both parties. As Craig Brommers, Speedo's marketing vice president, said in 2004: "The Speedo brand has been around for 76 years. He's the Page 2 of 22 Final project: team Yo' Mama single most important endorser of our brand of all time…if you are a marketer, Michael Phelps is a dream come true."1 In high-profile, professional sports, a $1 million bonus might seem like an insult. But to an athlete whose sport only tends to attract media attention every four years, it would boost his income (the vast majority of which comes from his endorsement contracts and appearance fees).2 While Phelps would certainly welcome a big check from Speedo for a record-setting performance, money is not his only motivation. When asked about what gets him through the daily grind of taxing workouts and challenging competition schedules, Phelps answers: "I hate to lose. I absolutely hate to lose. I can't stand it."3 In Athens, Phelps fell short of the record despite handling the media attention and speculation with relative ease. He won six gold medals and two bronze medals in the following events: Phelps's medal count: Athens 2004 400 individual medley Gold 4x200 freestyle relay 4x100 freestyle relay Bronze 200 individual medley 200 freestyle Bronze 4x100 medley relay 200 butterfly Gold 100 butterfly Gold Gold Gold Gold Four years later, Phelps is once again on the brink of attempting to eclipse Spitz's record. Much has changed since Athens. Despite finishing with a bronze medal in the 200 freestyle in 2004, he now holds the world record in that event. And one of his primary competitors in the freestyle events, Ian Thorpe of Australia, announced his retirement from the sport in 2006. However, the developments since Athens have not all been positive. Shortly after the Games ended in 2004, Phelps was charged with DUI (at age 19) after being pulled over late at night in his 1 2 Ruane, Michael. The Washington Post. August 25, 2004 These include Visa, Omega and PowerBar. 3 Alice Park. Time. August 1, 2004 Page 3 of 22 Final project: team Yo' Mama home town of Baltimore, Maryland. Contrite in his public apology, Phelps emerged from the incident relatively unscathed in the media despite the permanent mark on his record. Then, in November 2007, Phelps broke his wrist when he slipped on a patch of ice while trying to enter a car, and had a pin inserted to accelerate the healing process. During the recovery period, Phelps could not train at his normal level of intensity, which took its toll on his physical and mental conditioning. But he has a second chance to redefine his sport and surpass Spitz. In three months, he and his teammates will be in Beijing. Between now and then, he has to choose which of his 11 events he will swim. Our analysis assumes that Phelps will not limit himself to merely tying Spitz's record, and will therefore swim between eight and 11 events in an attempt to break the record. In press interviews about his attempt at the record, Phelps often says: "I don't want to be the next Mark Spitz. I want to be the first Michael Phelps."4 Which events should he choose to make history? Factors to consider in choosing events To help Phelps decide which events to swim in Beijing, we had to consider a variety of factors that could influence his performance in each event and assigned a numerical value to each factor for each event (see Exhibit A). Using our knowledge of the sport and the way Phelps has performed in previous international meets (the Olympic Games and the World Championships), these numbers represent our best estimates as to how each factor will influence Phelps's probability of winning a given event: Freshness – We assumed that Phelps has a better chance of winning earlier events because he will be "fresher" and at peak performance levels earlier in the competition. Also, he will be less fresh 4 Los Angeles Daily News, July 6, 2004. Page 4 of 22 Final project: team Yo' Mama on days in which he competes in a lot of events, and that consideration is also included in this factor. Dominance – This incorporates Phelps's historical level of dominance over his opponents. For example, although he holds the world record in four events (400 I.M., 200 I.M., 200 freestyle and 200 butterfly) his competition in the 400 I.M. is much weaker than it is in the 200 freestyle, so we assigned him a higher dominance value for the former than we did for the latter. Prep/warmdown – Given the timing of the various events in which Phelps could compete, some events may be more difficult for him to win because of their placement in the overall schedule and the amount of time for him to prepare and recover between events. Media – As a major star on the U.S. Olympic team and a world-record holder, Phelps can expect to receive a large share of media coverage as he attempts to break Spitz’s record. The popularity of certain events and the main competitors can also factor in to how much media coverage any given event receives. Relay preparedness – In the relay events, Phelps will be swimming with three other teammates against teams from other countries. All of the other factors listed above are targeted to assess Phelps's individual chance at winning a gold medal. This factor incorporates the strengths of his relay teammates and how we think they will perform during the relay (based on their past performances and their experience on the international stage). In addition to the probabilities of winning each event, we also needed to account for the volatility associated with these probabilities. To do this, we identified additional factors to create a variance term for each of Phelps's potential events (Exhibit B). Each variance term is expressed as a number between 0 and 100 (From this information, we got the standard deviation by taking the Page 5 of 22 Final project: team Yo' Mama square root of the average of the factors. Then, we divided the standard deviation by 100 to determine the percentage effect on the probabilities we calculated). Competition at U.S. Trials – In order to be eligible to swim an event at the Olympics, participants must earn their spot by competing at the U.S. Trials. In each event, only the top two finishers at the Trials are named to the U.S. Olympic Team. Therefore, the level of competition for the two spots in each event can affect Phelps's chances of success in Beijing. Competition at the Olympics – Obviously, the level of competition in each of the events at the Olympics will impact the variability of Phelps's probability of winning. The more competition there is in a given event, the more the probability of his winning the event can vary. In addition, Phelps's knowledge about certain competitors in specific events can also affect his chances of success. Length of the event – The longer the event, the more strategy is involved. In shorter events like the 100 butterfly and 100 freestyle, swimmers sprint from the start to the finish. In an event like the 400 I.M., swimmers have to choose when to turn up the intensity because it is impossible to sprint for 400 consecutive meters. Therefore, a longer event leads to more variability around the outcome as swimmers choose how they will swim the event before the race and how they will react to the way their competitors are swimming during the race. Timing of the event – Phelps must work within the established event schedule for the Olympics, and the timing of the events as well as the order in which he must swim them can have an effect on his chances of winning. For example, swimming an event 30 minutes after finishing another can vary his probability of winning the event if he does not have time to warm down, if the media is pushing for an interview, or if another complication surfaces for which he did not prepare. This is less of a problem on days where he only swims one event. Page 6 of 22 Final project: team Yo' Mama Outside factors – In addition to the factors listed above, there are a variety of other factors outside of Phelps's control that could impact the probability of him winning specific events. We used this as a catch-all category for factors that did not merit consideration across all events. Portfolio theory Next, we used the principles of portfolio theory to find a mix of events that optimizes Phelps's chance to break Spitz's record. There is a probability associated with winning each event, and we calculated that probability using the factors described in the previous section. There is volatility around the probability of winning each event as well, and we used the second set of factors to arrive at a variance term for each event. In addition, the result of each event has an impact on the results of future events (similar to covariance terms in portfolio theory), so we used our knowledge about swimming and researched the way Phelps has performed in previous international competitions to build two matrices that demonstrate the ways in which each event interacts with the others. The first matrix includes momentum factors (i.e., what mental effect winning or losing one event has on the outcome of subsequent events). The second includes tiredness factors (i.e., what physical effect swimming one event has on the outcome of subsequent events). Just like in portfolio theory, we had to combine the expected returns, variances, and interaction terms of the different events to assess which portfolio would give Phelps the best chance to win the bonus. But we had to adapt the classic equation for the expected return on the portfolio to fit the constraints and nuances of Phelps's situation. In portfolio theory, the expected return on the portfolio is defined as: E(Rp) = ∑ ω*E(Ri) The weighted average of all of the assets in the portfolio equals the expected return on the portfolio. We had to modify this equation in our analysis. We were interested in whether or not Phelps wins or Page 7 of 22 Final project: team Yo' Mama loses a given event – not the expected value of that event (as we would be with an asset). Also, we wanted to estimate the probability of winning eight events given that he was swimming with a portfolio of 8, 9, 10 or 11 events: we were not interested in knowing the probability of Phelps winning every event on his schedule. First, we assumed that the expected probability of winning a given event was distributed approximately normally (with the standard deviation equal to half the variance that we determined using the variance factors in the previous section). Next, we ran a Monte Carlo simulation for each event in Phelps's arsenal in the order that he would swim them in Beijing. If a random number between 0 and 1 fell below the expected probability of winning a given event, we logged it as a "win" for Phelps. Any number above the expected probability was logged as a "loss." Therefore, based on a set of events that had normallydistributed expected probabilities assigned to them – and given that the outcomes of the events were distributed binomially (as a "win" or a "loss") – we then had to include our interaction terms. Each subsequent event was adjusted for the outcomes of the previous events by incorporating a "momentum" factor – which had a positive or negative value depending on the outcome – and a "tiredness" factor, which accumulated and was always subtracted (Exhibit C). We ran this simulation a million times per condition (8, 9, 10, 11 events) to determine which gave him the best chance of winning the bonus. Because we paid careful attention to the sequence of the events, the probability of winning the events and the volatility around that probability, we also wanted momentum to play a large role in our analysis. In swimming, a bad race can sometimes be attributed to physical factors like fatigue, a poor start, or a bad finish, but the mental effects are often more destructive: it is very difficult to clear one's head after a bad swim. To that end, our model builds in an assumption that momentum plays a larger role in the outcomes of events than does accumulated tiredness. Although very Page 8 of 22 Final project: team Yo' Mama unlikely, this suggests that Phelps could lose his first event in spectacular fashion and, because of negative momentum, lose each subsequent event and finish with no gold medals. Running our simulation, we found that our approach, which we adapted from the guiding principles of portfolio theory, suggests that Phelps should swim nine events in Beijing to optimize his chance to break the record. The results are summarized below (Exhibit D shows a portion of the code we used to generate our analysis): Events in portfolio 8 9 10 11 Probability of breaking record 40.3% 42.1% 34.7% 25.7% Determining which events he should drop is relatively straightforward. With all of Phelps's 11 possible events included, his schedule in Beijing would be: 8/9/2008 a.m. p.m. 8/10/2008 a.m. p.m. 8/11/2008 a.m. p.m. 8/12/2008 a.m. p.m. 8/13/2008 a.m. p.m. 8/14/2008 a.m. p.m. 8/15/2008 a.m. p.m. 8/16/2008 p.m. 400 IM prelims 400 IM finals 200 free prelims 200 free semis 200 fly prelims 200 free finals 100 free prelims 100 free semis 200 back prelims 200 back semis 100 fly prelims 200 back finals nothing 100 fly final 4x100 medley relay 100 back prelims 100 back semis 100 back final 200 fly finals 200 IM prelims 100 free finals 200 IM finals 4x100 freestyle relay 200 fly semis 4x200 freestyle relay 200 IM semis 100 fly semis When looking at the expected probability of winning each event and cross-checking it with its location in the schedule (to account for momentum and tiredness), it is clear that his weakest events are the 100 backstroke, the 100 freestyle and the 200 backstroke (in that order). This is also consistent with his historical performance in these events. Given that he will choose to swim nine events, he will drop the 100 backstroke and the 100 freestyle from his Olympic program. Page 9 of 22 Final project: team Yo' Mama A note on sensitivity Changing the value of the factors within the probability analysis and variance analysis do not have a measurable effect on the results. The larger drivers are the interaction terms for momentum and tiredness. Because many of the assumptions we made around momentum and tiredness are based on human behavior and are therefore unpredictable, we did not find it useful to run a sensitivity analysis on these levers. We knew that we wanted "momentum" to be weighted more heavily than "tiredness" (because physical and mental considerations are included in the former, whereas the latter is something that Phelps tries to anticipate and prepare for in his training). But we did not want to spend too much time trying to differentiate between, say, a ratio of 2:1 (momentum : tiredness) and 4:1, because that would have required an understanding of human behavior that was beyond the scope of this project. Therefore, based on the personal experience of one of our team members – who was a Division I swimmer and later a Division I coach – we assumed the ratio to be 3:1 in this part of our analysis (and we adjusted that assumption in the next section). Game theory From the portfolio analysis in the previous section, we have determined a schedule that optimizes Phelps's chances of breaking Mark Spitz’s record. Assuming that Phelps, his longtime coach, and all of Phelps's competitors have done a similar analysis of their own, this schedule should be clear to them and everyone should know what he will swim. Phelps, however, does not announce his event schedule until the day before U.S. Trials (when he is required to). If the analysis is so clear and accurate, what would be the reason behind Phelps's secrecy? The answer is game theory. By announcing his schedule, he would effectively be releasing information to his U.S. rivals that helps them prepare for the Trials. It also releases information about his relative confidence in winning a Page 10 of 22 Final project: team Yo' Mama given event (relative to winning another event in his arsenal). Why give that information away for free if you can extract some additional information from your rivals by not releasing it? In setting up the game theory for this analysis, two elements require explanation and consideration: 1) a preparation technique called tapering and 2) the timing of the U.S. Olympic Trials. All swimmers use a preparation technique called tapering. Weeks before a major competition, swimmers gradually reduce the volume and intensity of their training to give their muscles time to repair and recover. In combination with special nutrition and mental preparation, this technique allows them to perform at their very best during the competition. Since tapering is a process that happens over an extended period of time – and since it corresponds to reduced aerobic capacity because of less volume and intensity – most swimmers only choose to taper once per season. It is possible to taper for one competition, briefly build the intensity back up, and taper again, but the second taper is usually less effective than the first. With reference to the timeline below, the U.S. Olympic trials is just six weeks before the actual Olympic Games, forcing most swimmers to choose whether to taper for the Trials or the Olympics. (Aside: The timing of the Trials is a constant source of contention among coaches and swimmers in the U.S. Many swimmers would prefer the Trials to be much earlier in the season so that they can spread out the time between the Trials and the Olympics and attempt two tapers. But scientific studies have been done by the Olympic Training Center in Colorado Springs and show that, in holding the Trials close to the Olympics, it ensures that the top swimmers will focus their taper on the Olympics and not on trying to taper twice. This discord demonstrates a mis-alignment of incentives: U.S. swimmers want to make sure they qualify for the Olympics. The U.S. Olympic Swimming Committee wants them to be at their best for the Olympics.) Page 11 of 22 Final project: team Yo' Mama Since only two U.S. swimmers per event are named to the Olympic team, swimmers may choose to taper for the Trials to improve their chances to get on the team, but potentially compromise their performance at the Olympics. Alternatively, swimmers can choose not to taper for the Trials, risk not making it onto the team, but perform at their best at the Olympics (assuming they made it onto the team, of course!). Phelps's game Enter Michael Phelps. Since he is such a dominant player, he will almost surely qualify for the Olympic team in all the events in which he chooses to participate (without tapering), leaving only one other slot available to make the team for his events. By keeping his schedule secret until a few days before the trials, he forces the other swimmers to decide – months before the Trials even occur – whether to taper for the Trials or not. By using the principles of game theory, we can illustrate why this is a dominant strategy for Phelps. In the analysis below, we considered the game of announcing versus not announcing for three different scenarios. The pay-off tables for each scenario were sorted so a higher number always represents a better outcome for the respective players. Page 12 of 22 Final project: team Yo' Mama Scenario 1: Strong US competition in a relatively weak event for Phelps (200 backstroke, 100 backstroke) Phelps Phelps doesn't announce, rival tapers Phelps announces, rival tapers Phelps doesn't announce, rival doesn't taper Phelps announces, rival doesn't taper Rival Phelps announces, rival doesn't taper Phelps doesn't announce, rival doesn't taper Phelps announces, rival tapers Phelps doesn't announce, rival tapers 4: 3: 2: 1: 4: 3: 2: 1: Taper ( 3, 2 ) Announce Rival Don’t Taper Taper ( 1, 4 ) Phelps ( 4, 1 ) Don’t Announce Rival Don’t Taper ( 2, 3 ) When rivals know they are strong in comparison to Phelps, it is in their best interest to hold off on tapering so they will be able to perform at peak levels in Beijing. Scenario 2: Weak US competition in a strong event for Phelps (400 I.M., 200 butterfly) Phelps Phelps doesn't announce, rival tapers Phelps announces, rival tapers Phelps doesn't announce, rival doesn't taper Phelps announces, rival doesn't taper Rival Phelps announces, rival tapers Phelps doesn't announce, rival tapers Phelps announces, rival doesn't taper Phelps doesn't announce, rival doesn't taper 4: 3: 2: 1: 4: 3: 2: 1: Page 13 of 22 Final project: team Yo' Mama ( 3, 4 ) Taper Announce Rival Don’t Taper Taper ( 1, 2) Phelps ( 4, 3 ) Don’t Announce Rival Don’t Taper ( 2, 1) When rivals know they are weakly matched against Phelps, they will choose to taper. Scenario 3 - Strong US competition in a strong event for Phelps (200 I.M., 200 freestyle, 100 butterfly, 100 freestyle). Phelps Phelps doesn't announce, rival tapers Phelps announces, rival tapers Phelps doesn't announce, rival doesn't taper Phelps announces, rival doesn't taper Rival Phelps announces, rival doesn't taper Phelps announces, rival tapers Phelps doesn't announce, rival doesn't taper Phelps doesn't announce, rival tapers Taper ( 3, 3) 4: 3: 2: 1: 4: 3: 2: 1: Announce Rival Don’t Taper Taper ( 1, 4) Phelps ( 4, 1 ) Don’t Announce Rival Don’t Taper ( 2, 2) Page 14 of 22 Final project: team Yo' Mama In a strong event for both Phelps and his rival, the rival should choose not to taper. By playing his dominant strategy of not announcing his event schedule, Phelps unlocks valuable information using this game theoretic approach. Phelps is now in a position to predict how his primary competitors will adjust their taper schedules in response to his secrecy under the three possible scenarios. We can now refine our portfolio analysis and re-evaluate which (and how many) events he should swim. Incorporating tapering into the portfolio analysis Using the same approach that we did for the initial portfolio analysis, we ran our simulation again, but this time added a "taper" factor into the model. We defined the taper factor to be an effect on the probability of winning a given event. This effect was only relevant for the events in which there is significant competition in the U.S. (see chart below). For all of the other events in which Phelps does not face tough U.S. competition, we assumed that the initial case would hold (i.e. that they would taper for the Trials) and did not add a "taper" factor. The toughest U.S. rivals are of primary concern to Phelps right now for two reasons: first, they stand in the way of his qualifying for the Olympics in the events he chooses to swim (since only two U.S. swimmers can qualify for the team per event). Second, they are the only rivals whose behavior he can affect between now and Beijing. Most countries have already selected their Olympic swim teams, and Phelps will not be competing against any of his international rivals between now and Beijing. The "taper" factor changes the expected probability of winning certain events in Beijing, but it does so for different reasons depending on the event: Page 15 of 22 Final project: team Yo' Mama Expected probability of winning if: Rival tapers for Rival tapers for U.S. Trials Beijing 85.0% 80.0% 62.5% 32.5% 57.5% 50.0% 67.5% 62.5% Event 400 individual medley 200 backstroke 200 individual medley 100 butterfly In the 400 I.M., Phelps is so dominant that even if he does not convince his primary U.S. competitor, Ryan Lochte, to taper for U.S. Trials, his chance to win the gold medal does not reduce by much. It accounts for the fact that Lochte will be sharper in Beijing if he tapers for Beijing, but it does not have enough of an impact to drastically reduce his chance to win gold in Beijing. The same logic holds for the 200 I.M. (where Lochte is again his biggest U.S. competitor) and the 100 butterfly (where Ian Crocker, the world-record holder in the event and a fellow American, is his toughest competitor). The 200 backstroke is a different story. Under the initial portfolio analysis – where we determined that he should swim nine events – this is the last event he will decide to add to his roster. While it is a strong event for him, he has two tough competitors, both of whom are from the U.S.: Ryan Lochte and Aaron Piersol. Forcing the hands of one or both of them would give him a lot of information about his chances in Beijing: if one or both of them tapers for U.S. trials, his chances in Beijing improve. If one or both of them choose to wait and taper for Beijing, his chances of winning a gold medal collapse. Therefore, once he knows that his decision not to release his events ahead of time will not affect their decision to taper, he should choose to remove the 200 backstroke from his event schedule and instead focus on the eight remaining events. Adding the "taper" factor to the portfolio analysis, we now see that swimming eight events yields the highest probability of Phelps breaking the record: Page 16 of 22 Final project: team Yo' Mama Probability of breaking record 25.9% 21.6% 16.4% 12.6% Events in portfolio 8 9 10 11 With the "taper" factor included, the probability of breaking the record drops significantly as compared to the initial analysis. This is because of the momentum factor. With some of his U.S. rivals choosing to taper for Beijing (because of Phelps's decision not to announce his event schedule), several of his races will be closer in Beijing than they would have been if his U.S. rivals had decided to taper for the U.S. Trials. Although he will get positive momentum from winning a given event, if the race is closer, we assumed that the momentum effect would be reduced. A close race will make it look like he is fighting to win races that he would have dominated had some of his rivals tapered for U.S. Trials. Although we still weighted our "momentum" interaction term more heavily than our "tiredness" interaction term (as we did in the initial analysis), we reduced the value of the "momentum" weight per event (from 3 to 2) while keeping the "tiredness" weight per event the same. What if Phelps attempts to tie the record rather than break it? Knowing how competitive Phelps is and how motivated he is to be "the first Michael Phelps, not the second Mark Spitz," we are confident that his mental model is based upon a goal of breaking Spitz's record and not merely tying it. Increasingly, male and female swimmers are continuing to improve well into their 20s (and even into their early 30s). Phelps will be 27 years old at the next Olympic Games and will therefore have another legitimate chance to tie or break the record if he does not do it in Beijing. For these reasons, we believe that, in 2008, Phelps will swing for the fences and try to break the record. Page 17 of 22 Final project: team Yo' Mama However, we did run our analysis under the assumption that he will merely try to tie the record in Beijing. In that case, and keeping all of our factors and other assumptions the same, we see that, under the first scenario (before incorporating the game theory analysis with tapering), Phelps has the highest probability of tying the record if he swims eight events: Events in portfolio 8 9 10 11 Probability of tying record 60.1% 48.3% 37.7% 28.2% When we incorporate our "taper" factor into the model, we find that – if he is trying to tie the record – he should choose to swim eight events under this second scenario as well: Events in portfolio 8 9 10 11 Probability of tying record 41.9% 31.4% 23.7% 18.4% The implications of this are interesting. Phelps, using game theory, knows that swimming eight events will maximize his chances to break Spitz's record. But if he finishes second (or worse) in a given event, he will also have the peace of mind to know that there is still a high probability that he will be able to tie the record. This could have a profound effect on his mental state and is effectively a strategic hedge: if he loses one event, the $1 million is not lost. Rather, he has set himself with the highest chance to to tie the record, which also earns him the bonus. Had he gone with the non-game theoretic plan of swimming nine events, his chances to tie the record (after losing one event) are lower than under the eight-event condition. Page 18 of 22 Final project: team Yo' Mama Conclusion These are nerve-wracking times for Phelps and his coach. With the bulk of his training under his belt already, the next few months will be about fine-tuning his mental and physical preparation and reducing distractions as much as possible. Through it all, though, Phelps must send a message to his competition that he is gearing up for a monumental eight days in Beijing. In the weeks leading up to U.S. Trials, "an official 'media blackout' will begin, and aside from a few interviews poolside after competitions, Phelps will be silent. It's a plan…to get him prepared for Beijing."5 While Phelps will certainly use that time to focus on the task ahead of him, it will also give him the added benefits of game theory. By not announcing his U.S. Trials schedule, Phelps can predict how some of his toughest competitors will react, and he can use that information to refine his mental model. Swimming eight events will give him the highest probability of winning eight gold medals or seven gold medals. He wants to break the record, but in the event that he falls short of that goal, he will still be in a position to share the spotlight with Spitz. While this might not be an ideal outcome for the ultra-competitive Phelps, he will be $1 million richer and will have another shot at eight gold medals in 2012. 5 Van Valkenburg, Kevin. "No Excuses as Phelps Prepares for Beijing." The Baltimore Sun, April 30, 2008. Page 19 of 22 Final project: team Yo' Mama Exhibit A: factors included in expected probability of Phelps winning an event Order of events 400 individual medley 4x100 freestyle relay 200 freestyle 100 backstroke 200 butterfly 4x200 freestyle relay 100 freestyle 200 backstroke 200 individual medley 100 butterfly 4x100 medley relay Freshness factor 10 9 8 6 7 5 6 6 3 6 5 Dominance factor 9 4 5 3 10 8 4 3 8 5 7 Prep/warmdown factor 9 8 8 3 9 4 7 8 3 9 9 Media factor 6 6 7 9 7 8 8 8 9 7 7 Relay prep factor x 6 x x x 8 x x x x 7 (Average*10)/100 proxy for E( R) 85.0% 66.0% 70.0% 52.5% 82.5% 66.0% 62.5% 62.5% 57.5% 67.5% 70.0% Scale: 1-10; low numbers are bad, high numbers are good. Exhibit B: factors included in the variance of the probability of Phelps winning an event Order of events 400 individual medley 4x100 freestyle relay 200 freestyle 100 backstroke 200 butterfly 4x200 freestyle relay 100 freestyle 200 backstroke 200 individual medley 100 butterfly 4x100 medley relay Competition at Trials 5 50 40 80 20 10 30 80 40 30 10 Competition at Olympics 60 90 80 85 50 30 80 90 40 40 60 Length of event 60 30 30 70 20 40 60 30 30 50 60 Timing of event 10 10 20 70 50 50 70 70 80 30 20 Outside factors 40 40 20 5 5 30 10 10 10 50 50 Avg (proxy for var) (see note below) 35 44 38 62 29 32 50 56 40 40 40 ** To arrive at standard deviation, we took the square roots of these numbers and divided by 100 to arrive at a percentage effect on probability. Scale: 1-100; low numbers correspond with low volatility Page 20 of 22 Final project: team Yo' Mama Exhibit C: interaction matrices ("momentum" and "tiredness") MOMENTUM 400 individual medley4x100 freestyle relay 400 individual medley 1 4x100 freestyle relay 0.6 1 200 freestyle 0.5 0.6 100 backstroke 0.5 0.6 200 butterfly 0.4 0.5 4x200 freestyle relay 0.4 0.5 100 freestyle 0.3 0.5 200 backstroke 0.2 0.3 200 individual medley 0.2 0.3 100 butterfly 0.1 0.2 4x100 medley relay 0 0.1 200 freestyle 100 backstroke 200 butterfly4x200 freestyle relay 100 freestyle 200 backstroke 200 individual medley 100 butterfly 4x100 medley relay 1 0.8 0.6 0.6 0.6 0.4 0.4 0.3 0.2 1 0.6 0.6 0.5 0.4 0.4 0.3 0.2 1 0.8 0.6 0.5 0.5 0.4 0.3 1 0.6 0.5 0.5 0.4 0.3 1 0.6 0.8 0.5 0.4 1 0.8 0.6 0.5 1 0.6 0.5 1 0.6 1 TIREDNESS 400 individual medley4x100 freestyle relay 400 individual medley 1 4x100 freestyle relay 0.5 1 200 freestyle 0.4 0.5 100 backstroke 0.3 0.5 200 butterfly 0.3 0.3 4x200 freestyle relay 0.3 0.3 100 freestyle 0.3 0.3 200 backstroke 0.2 0.2 200 individual medley 0.2 0.2 100 butterfly 0.1 0.1 4x100 medley relay 0.1 0 200 freestyle 100 backstroke 200 butterfly4x200 freestyle relay 100 freestyle 200 backstroke 200 individual medley 100 butterfly 4x100 medley relay 1 0.9 0.5 0.4 0.6 0.3 0.3 0.2 0.1 1 0.4 0.4 0.3 0.3 0.2 0.1 0 1 0.9 0.5 0.3 0.3 0.2 0.2 1 0.6 0.4 0.4 0.3 0.1 1 0.3 0.3 0.1 0.1 1 0.9 0.5 0.3 1 0.6 0.3 1 0.3 1 Page 21 of 22 Final project: team Yo' Mama Exhibit D: screen capture of computer code Page 22 of 22