March Madness
More March Madness
Just when you thought it was safe to play in the office NCAA basketball pool, along comes a new twist on an old gambit: optimal (?) bidding
By Edward H. Kaplan and Michael J. Magazine
Betting on sports events is enormously popular. Some sports, such as horse racing, would probably not continue to exist in the absence of associated betting. In addition to standard bets on the outcomes of sporting events, friendly "pool" contests have evolved in which participants try to predict the outcome of some event (such as the Super Bowl, the World Series, the NCAA Basketball Tournament [often referred to as "March Madness"], the Wimbledon Tennis Championship or the Master's Golf Tournament).
Pools associated with the NCAA Men's Basketball Tournament are especially popular. Typically, the objective in such pools is to predict the winners of the 63 tournament games a priori to maximize the number of points gained from correct predictions (where different numbers of points are awarded for different winning picks). Joining the structure of the NCAA tournament with the probabilities that different teams will defeat other teams and the point structure for any given pool enables the construction of models that select prediction strategies that maximize the expected number of points gained in the pool [1, 2].
A rather different pool structure applies not only to the NCAA Tournaments but to many other sporting events as well. In the context of the NCAA Tournament, the pool works as follows: pool participants pay a fee to enter the pool and receive a fixed number of points with which to bid for teams. Prior to the start of the tournament, a sealed bid auction is conducted for all of the teams in the competition, with participants constrained by the total number of points with which they can bid. Whoever bids the highest for any given team "owns" that team. At the end of the tournament, the prize money (comprised of the pool fees) is split among the tournament winners in some pre-specified fashion (for example, 60 percent to the owner of the tournament champion, 20 percent to the runner up, and 10 percent to each of the losing semi-finalists in the "Final Four").
Presumably, there are better versus worse ways of bidding for teams in such a pool. If all pool participants have equal information regarding which teams are likely to do well in the tournament, for example, then the bidding problem is essentially dominated by game-theoretic concerns: how should I bid, given that my opponents are symmetrically asking how they should bid (and conducting the same analysis as I am to answer the question)? On the other hand, if one player in the pool has an information advantage, such as better estimates of the likelihoods that each team would "win, place or show" in the tournament, it might be possible to exploit that advantage to determine better bids.
The two authors of this article realized this possibility when one of us (Mike) told the other (Ed) about a pool he had participated in for years. What made this more interesting was that the pool participants were all trained in operations research. Was there a way to model this contest that could lead to better bidding strategies? We decided to conduct an experiment to find out.
Preliminaries
Mike has played in this pool for many years, so we will describe the contest from Mike's point of view. Mike and the other players (henceforth "suckers") pay an entry fee of $100 for the right to allocate 1,000 points across the 64 teams in the NCAA Men's Basketball Tournament. Any sucker (and Mike) can place any non-negative, integer bet of up to 1,000 points on any team, subject to the constraint that the total number of points allocated per sucker (and per Mike) cannot exceed 1,000. Suckers can also place a "field" bet, meaning that they win the rights to all unclaimed teams (teams that no one bid for) if their field bet exceeds all other field bets.
The payoffs in this pool are as follows: the person who owns the tournament champion gets 60 percent of the total entry fees, with 20 percent, 10 percent and 10 percent going to the owners of the tournament runner-up and the two losing Final Four teams, respectively. Presumably ties are possible (since two players could bet the same maximal points for the same team) in which case any winnings would be split. The idea is to enter a set of bids with the goal (prayer!) of making as much money as possible. An unmodeled (but perhaps even more important) objective is to prevent your competitors from winning their preferred teams (leaving them to stew over their wasted bids as a result).
The Basic Ideas
There are two basic ideas in the model (see the Appendix for details). The first is that, compared to the suckers, Mike can gain a better understanding of the expected payoff associated with owning any team. This is because for any team, Ed has figured out the probabilities of that team winning the tournament, losing in the final game of the tournament, or losing in a semifinal game [2].
The second basic idea is that the behavior of the suckers can be modeled statistically. Such an assumption is not totally unreasonable if all suckers have essentially the same information about the basketball teams in the tournament.
Estimating Bidding Behavior
Estimating the fraction of all points that any sucker will bet on any team is a key requirement of the model. As with the stock market, past behavior is no guarantee for future performance, yet we used the betting data from the 1999 and 2000 pools to shape our model. Since the teams that appear in the tournament are different from year to year, it is necessary to model bidding amounts as a function of some team attribute. All tournament teams are seeded from 1 through 16 (with the #1 seeds recognized as the tournament favorites), thus one idea would be to see how the amount bid on teams varied with the seeding of teams in the tournament. However, though the actual strength of tournament teams can vary considerably over time, seeding values cannot. A statistic more reflective of team strength is the Sagarin rating. This rating, regularly updated over the course of the season for each team, is meant to reflect the expected number of points teams will score in a game. A plot of the fraction of all points bid by Mike and the suckers in the 1999 and 2000 tournaments versus the pre-tournament Sagarin ratings appears in Figure 1. As is clear from the figure, teams with higher Sagarin ratings did receive higher bids.
Figure 1
In-Sample Experiments
Using the data in the figure above as a guide, we asked how well Mike would have done in the 1999 and 2000 NCAA pools using the model. Recognizing that it would not be fair to presume the exact betting amounts as a function of the Sagarin ratings, we instead divided the teams into five groups as shown in Table 1, and worked with average bids within those five groups. Such an approach enables the Out-of-Sample Experiment we will report below.
| Sagarin Rating | Fraction of Points Bid |
| Less than or equal to 80 | 0.00004 |
| 80+ — 85 | 0.00097 |
| 85+ — 90 | 0.00747 |
| 90+ — 92 | 0.05070 |
| 92+ | 0.12470 |
Table 1. Point-Bid Probability as a Function of the Sagarin Rating
Using the Sagarin ratings to determine the probabilities that any team would defeat any other team as described in Kaplan and Garstka [2],we computed the probabilities that each team would survive each round of the tournament. Combining these with the bidding probabilities from Figure 1, we constructed a dynamic program inside Excel to see what bids would result if the goal was to maximize the expected payoff from the pool, and compared these bids to those that actually occurred. To save computing time on Ed's laptop, we forced the model to allocate points in even multiples of five.
Table 2 shows the results. In 1999, had the model competed against Mike and the five suckers, it would have bid the amounts and owned the 20 teams shown in Table 2. Two of these teams (Ohio State and Michigan State) reached the Final Four, which would have returned 20 percent of the pool. It is amusing to note that the highest bid for Auburn (a No. 1 seed) actually offered was three points, in contrast to the 155 points offered by the model. On the other hand, the model's bid of 15 points for Ohio State (a Final Four team), which also received a maximum bid of three points, seems more reasonable. And, the model's bid of 155 points for Michigan State is spectacular in retrospect, as this Final Four team received a maximum actual bid of 140 points. In his actual bidding in the 1999 tournament, Mike ended up owning five teams, but none of those reached the Final Four.
| 1999 Teams Owned |
Points Bid | 2000 Teams Owned |
Points Bid | |
| Indiana | 15 | Arizona | 5 | |
| Auburn | 155 | Wisconsin* | 5 | |
| UCLA | 15 | Dayton | 5 | |
| Detroit | 5 | Louisville | 5 | |
| Ohio State* | 15 | Gonzaga | 5 | |
| Missouri | 5 | Utah | 5 | |
| Iowa | 15 | Ball State | 5 | |
| Arkansas | 5 | Maryland | 5 | |
| Gonzaga | 5 | Auburn | 5 | |
| Florida | 15 | Kansas | 5 | |
| Miami (FL) | 15 | DePaul | 5 | |
| Purdue | 5 | Oklahoma | 5 | |
| Wisconsin | 15 | Oregon | 5 | |
| Michigan State* | 155 | North Carolina* | 5 | |
| Mississippi | 5 | |||
| Villanova | 5 | |||
| NC Charlotte | 5 | |||
| Arizona | 15 | |||
| Washington | 5 | |||
| Miami (OH) | 5 |
Table 2. Model Results from In-Sample Experiments (*team reached Final Four)
Similarly, in the 2000 pool, the model would have owned 14 teams. In a repeat performance, two of these teams (Wisconsin and North Carolina) reached the Final Four of the tournament, which again would have returned 20 percent of the pool. In the 2000 tournament, however, both of the Final Four teams selected by the model were actually part of the winning "field bid" in the pool. In other words, neither Mike nor the five suckers allocated any points directly to either of these two teams! Also, none of the relatively large bids placed by the model (e.g. 205 points for Michigan State, 190 for Stanford) resulted in winning team ownership (Michigan State went for 631 points while Stanford went for 622). In Mike's actual 2000 bidding, he won ownership of 12 teams, one of which (Florida) reached the tournament final (returning 20 percent of the pool to Mike).
Out-of-Sample Experiment
To further test the model, we turn to the 2001 Tournament. Using the relationship between bid amounts and Sagarin ratings detected from the 1999 and 2000 data (i.e. Figure 1 and Table 1) with the 2001 Sagarin ratings to model the bidding behavior of the suckers along with the Kaplan and Garstka model to produce team-specific chances of finishing in the money, Table 3 reports the resulting winning bids. The model would have owned 11 teams, but unfortunately would not have won any money as none of these teams reached the Final Four. The model did, however, end up owning a reasonable portfolio of teams, and some of these in the face of substantial bids from the actual pool participants (for example, Kansas received a maximum bid of 62 points while Kentucky received a maximum bid of 52 points). Mike's actual bids in the 2001 tournament rewarded him with the ownership of 20 teams. In a performance competitive with the model, however, none of Mike's teams reached the Final Four.
| 2001 Team Owned | Points Bid |
| Creighton | 5 |
| Kentucky | 70 |
| Wisconsin | 15 |
| Georgetown | 5 |
| Virginia | 15 |
| Temple | 5 |
| North Carolina | 70 |
| NC Charlotte | 5 |
| Kansas | 70 |
| Xavier | 5 |
| Wake Forest | 15 |
Table 3. Model Results from the Out-of-Sample Experiment
Conclusion
The results of our experiments suggest that the model proposed can lead to improved pool performance, though as with most endeavors there are few guarantees. Note, however, that the model had a more difficult task than the actual players: the model competed against six bids (whereas Mike only had five suckers to contend with).
There are some aspects of the model where we could have proceeded more carefully. For example, our characterization of bidding behavior as a function of Sagarin ratings is very simple; perhaps a deeper analysis would have revealed more intricate dependencies. Perhaps we should have attempted to model the maximal bids directly instead of inferring them from order statistics arguments. Perhaps we should have attempted to incorporate field bets directly into the analysis.
One thing you can be sure of is that having made this analysis public, Mike will not use this model in this year's pool! However, there is nothing to stop the rest of you from trying to improve your own performance in basketball or similar office pools via the incorporation of modeling ideas such as those discussed above.
References
- Breiter, D.J. and Carlin, B.P., 1997, "How to play office pools if you must," Chance, Vol. 10, No. 1, pp. 5-11.
- Kaplan, E.H. and Garstka, S.J., 2001, "March madness and the office pool," Management Science, Vol. 47, No. 3, pp. 369-382.
Edward H. Kaplan is the William N. and Marie A. Beach Professor of Management Sciences at the Yale School of Management, and professor of Public Health at the Yale School of Medicine. Usually he writes about problems such as HIV prevention and bioterror response logistics, but he still hopes to get rich via Mike. Michael J. Magazine is the associate dean for Faculty and Research, professor of Quantitative Analysis and Operations Management, and Ohio Eminent Scholar at the University of Cincinnati. Normally he writes about problems in operations and supply chain management, but he still hopes to make a bundle off of Ed.
OR/MS Today copyright © 2003 by the Institute for Operations Research and the Management Sciences. All rights reserved.
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