Chapter 21 Quiz: Modeling Combat Sports and Tennis
Test your understanding of rating systems, style matchup analysis, surface effects, physical attribute modeling, and live win probability for individual sports.
Question 1. What is the standard Elo expected score formula for Player A against Player B?
Answer
The expected score is $E_A = 1 / (1 + 10^{(R_B - R_A) / 400})$, where $R_A$ and $R_B$ are the current Elo ratings of Players A and B respectively. A rating difference of 400 points corresponds to approximately a 91% expected win probability for the higher-rated player.Question 2. Why are optimal K-factors for MMA Elo systems (100-200) much larger than for tennis (20-32)?
Answer
MMA fighters compete only 2-3 times per year, so each fight must carry substantial weight in updating the rating for the system to remain responsive. Tennis players compete 60-80 times per year, providing enough data for smaller, more conservative updates. The lower frequency in MMA means fewer data points are available, so each observation must contribute more information to the rating estimate.Question 3. What three parameters define a player in the Glicko-2 system, and what does each represent?
Answer
The three parameters are: (1) $\mu$ (mu) -- the player's rating on the Glicko-2 internal scale, representing estimated skill level; (2) $\phi$ (phi) -- the rating deviation, representing the uncertainty or confidence in the rating estimate; and (3) $\sigma$ (sigma) -- the volatility, representing the degree to which the player's true ability tends to fluctuate over time.Question 4. How does inactivity affect a fighter's rating in the Glicko-2 system compared to a standard Elo system?
Answer
In Glicko-2, inactivity causes the rating deviation ($\phi$) to increase, meaning the system becomes less confident in the fighter's rating without changing the rating itself. When the fighter returns, their result has a larger effect on their new rating, and their opponent's rating is less affected by the result. In standard Elo, inactivity has no inherent effect unless a manual decay mechanism is added; the system treats a fighter returning from a two-year layoff identically to one who fought last month.Question 5. What is a style matchup matrix, and how is it used to adjust Elo-based predictions?
Answer
A style matchup matrix is an asymmetric matrix where entry $M_{ij}$ encodes the performance adjustment when a fighter of style archetype $i$ faces a fighter of style archetype $j$. The adjustment is applied on the log-odds scale: $P_{\text{adj}}(A) = \text{logistic}(\text{logit}(P_{\text{Elo}}(A)) + M_{s_A, s_B})$. The matrix is populated empirically by computing the average deviation between actual outcomes and Elo-expected outcomes for each style pairing across historical data.Question 6. Name the six MMA style archetypes used in the chapter's classification system.
Answer
The six archetypes are: (1) striker -- high volume standing offense; (2) wrestler -- strong takedown offense and defense; (3) grappler -- submission-oriented ground game; (4) balanced -- no dominant dimension; (5) counter_striker -- defensive with selective offense; and (6) pressure_fighter -- high-volume, forward-moving style with lower defense.Question 7. In tennis, what are the three primary playing surfaces, and how do they differ in terms of ball behavior?
Answer
The three surfaces are: (1) Hard court -- medium bounce height and speed, most "neutral" surface with variation between venues; (2) Clay -- ball bounces higher and slower, producing longer points and rallies, favoring stamina, topspin, and defensive play; (3) Grass -- ball bounces lower and faster, producing shorter points, favoring serve dominance and aggressive net play. Each surface systematically advantages or disadvantages certain playing styles.Question 8. What is the surface affinity score, and how is it calculated?
Answer
The surface affinity score measures how much a player's performance on a specific surface deviates from their overall performance, normalized by their overall win rate. It is calculated as: $\text{SA}(p, s) = (\text{Win\%}(p, s) - \text{Win\%}(p, \text{all})) / \text{Win\%}(p, \text{all})$. A player with a 60% overall win rate who wins 72% on clay has a clay affinity of +0.20 (+20%), indicating they are substantially better on clay relative to their overall level.Question 9. How does a surface-adjusted Elo system blend overall and surface-specific ratings?
Answer
The system uses a logistic blending weight: $w = n / (n + N)$ where $n$ is the number of matches the player has played on that surface and $N$ is a threshold parameter (e.g., 50). The blended rating is $w \times R_{\text{surface}} + (1-w) \times R_{\text{overall}}$. With few surface matches, the blended rating is close to the overall rating. As the player accumulates surface-specific matches, the blended rating converges toward the surface-specific rating. This prevents overfitting to small surface samples while allowing genuine surface effects to emerge.Question 10. What is the typical magnitude of the reach advantage in MMA, expressed as win probability per inch?
Answer
Each inch of reach advantage beyond a threshold (typically 2-3 inches) is associated with approximately a 1-2 percentage point increase in win probability, after controlling for overall rating. The effect is modeled as: $\Delta_{\text{reach}} = \beta_{\text{reach}} \times \max(0, r_A - r_B - \tau_{\text{reach}})$, where $\beta_{\text{reach}}$ is typically 0.008-0.015 on the probability scale. The effect is nonlinear and most pronounced in striking-heavy matchups.Question 11. Describe the age curve for MMA fighters. At what ages do fighters typically peak, and how does the decline progress?
Answer
MMA fighters typically peak between ages 26-30, where the trade-off between physical attributes (speed, power, reflexes) and accumulated skill/experience is optimal. Before age 26, fighters are still developing technically (+2% to +5% improvement). Between 31-33, gradual decline begins (-3% to -8%). Ages 34-36 see significant decline (-8% to -15%). At 37+, steep decline occurs (-15% to -30%), with high variability -- some fighters maintain ability while others deteriorate rapidly. The decline accelerates due to cumulative brain trauma, reduced training intensity, and declining physical attributes.Question 12. Why are weight cuts relevant to combat sports modeling, and what are the measurable effects of severe weight cuts?
Answer
Weight cuts are relevant because fighters routinely cut 10-20% of their body weight before weigh-ins. Severe cuts (more than 15% of walk-around weight) are associated with reduced chin durability, slower reaction times, and decreased cardiovascular endurance, particularly in later rounds. The model applies a penalty when the cut exceeds a threshold (typically 12% of walk-around weight). An important paradox is that fighters who miss weight and fight heavier tend to perform better in the actual bout, because they enter the cage larger and more hydrated, though financial penalties and sanctions limit this.Question 13. In tennis live modeling, what are the two fundamental parameters needed to compute exact match win probability, and how are they updated during the match?
Answer
The two parameters are: (1) $p_s$ -- the probability that the server wins a point on their serve, and (2) $p_r$ -- the probability that the returner wins a point on the opponent's serve (equivalently $1 - p_s$ for the opponent). They are updated during the match using Bayesian updating: $p_s^{\text{posterior}} \sim \text{Beta}(\alpha_0 + \text{service points won}, \beta_0 + \text{service points lost})$, where $\alpha_0$ and $\beta_0$ encode the prior from career statistics.Question 14. How is the probability of winning a tennis game from deuce calculated?
Answer
If $p$ is the server's point-win probability, the probability of winning a game from deuce is $P(\text{win from deuce}) = p^2 / (p^2 + (1-p)^2)$. This follows from the fact that at deuce, the server must win two consecutive points (probability $p^2$), or the state returns to deuce after one point each (probability $2p(1-p)$), creating a geometric series that sums to the stated formula.Question 15. What are the three components of the MMA live win probability model?
Answer
The three components are: (1) Round survival probability -- the probability that the fight reaches the end of the current round without a finish, based on pace of action, types of strikes landed, ground position, and historical finishing rates; (2) Scorecard probability -- the probability that each fighter wins on the judges' scorecards if the fight goes to decision, updated round-by-round using the 10-point must system; (3) Finish probability -- the probability that a finish (KO/TKO/submission) occurs in the current round and the conditional probability of which fighter finishes. These combine as: $P(A \text{ wins}) = P(\text{finish}) \times P(A \text{ finishes} | \text{finish}) + P(\text{no finish}) \times P(A \text{ wins} | \text{no finish})$.Question 16. Why does the indoor/outdoor distinction matter for tennis modeling, and which player profiles benefit from indoor conditions?
Answer
Indoor hard courts play faster than outdoor hard courts because there is no wind to slow the ball, humidity is controlled, and court speeds are typically faster. This benefits serve-dominant players and big hitters, as the faster conditions reduce the returner's reaction time and make serve breaks less likely. The surface-adjusted Elo system can incorporate an indoor speed bonus for players whose profiles suggest serve dominance. Players who rely on defensive movement, heavy topspin, and extended rallies tend to perform relatively worse indoors.Question 17. What is the knockout vulnerability index, and what variables does it incorporate?
Answer
The knockout vulnerability index quantifies a fighter's susceptibility to being knocked out or stopped. It is calculated as: $\text{KO\_Vulnerability} = \alpha_0 + \alpha_1 \times \text{KO\_losses} + \alpha_2 \times \text{strikes\_absorbed} + \alpha_3 \times \text{age}$. The key inputs are: number of career KO/TKO losses (each loss indicates previous vulnerability), total significant strikes absorbed (cumulative brain trauma proxy), and age (which amplifies vulnerability). Higher values indicate greater vulnerability. The progression of chin deterioration is typically irreversible and accelerating.Question 18. Explain why the matchup matrix is asymmetric. Give a concrete example.
Answer
The matrix is asymmetric because the advantage that Style A has over Style B is not necessarily the same magnitude as the disadvantage Style B has against Style A. For example, the advantage a wrestler has against a striker (+0.06) may not equal the disadvantage a striker has against a wrestler (-0.06 in the chapter example, but this is coincidental). A grappler versus a wrestler might have a different adjustment than a wrestler versus a grappler, because the specific mechanical interactions differ depending on who is initiating the grappling exchanges and who is defending.Question 19. What role does altitude play in tennis performance, and what is the physical mechanism?
Answer
At high altitude, reduced air density decreases drag on the tennis ball (drag force is proportional to air density $\rho$, which drops with altitude). At 2,500 meters, air density is approximately 73% of sea level, reducing drag by about 27%. This makes the ball travel faster, making serves harder to return. Additionally, the reduced Magnus effect makes topspin less effective. The combined result is faster, flatter tennis that favors aggressive, flat-hitting players and big servers, while disadvantaging topspin-heavy clay court specialists.Question 20. How should retirements and walkovers be handled in a tennis Elo system?
Answer
Retirements and walkovers carry partial information and should be handled differently from completed matches. A retirement indicates the retiring player was likely losing, but the match was not completed under normal competitive conditions. Common approaches include: (1) ignoring retirements entirely and not updating ratings; (2) treating retirements as normal results but with a reduced K-factor (e.g., 50% of normal K); (3) treating walkovers (pre-match withdrawals) as non-events that carry no information about relative skill. The choice affects predictive accuracy and should be optimized empirically.Question 21. What promotion-tier-based initialization scheme is suggested for MMA Elo ratings?
Answer
The chapter suggests the following initial Elo ratings by promotion tier: UFC (debuting from top regional) = 1500, UFC (from Contender Series) = 1450, Bellator/PFL = 1350, ONE Championship = 1400, major regional promotions (LFA, Cage Warriors) = 1250, and minor regional = 1100. These starting points reduce the number of fights needed for ratings to converge by providing informative priors about fighter quality based on the competitive level of their promotion.Question 22. In the best-of-three set format versus best-of-five, why does the longer format favor the stronger player?
Answer
The longer format favors the stronger player because the probability that the better player wins increases with the number of sets, just as the probability that the better team wins a playoff series increases with the series length. In a best-of-five, a player with a 55% probability of winning any individual set has a higher probability of winning three sets before their opponent does compared to winning two sets in a best-of-three. This is analogous to the law of large numbers: more trials reduce the influence of randomness, allowing the higher-probability event to occur more reliably.Question 23. How are momentum and fatigue modeled in the tennis live win probability framework?
Answer
Both are modeled as time-varying adjustments to the base point-win probability: $p_s(t) = p_s^{\text{base}} + \delta_{\text{momentum}}(t) + \delta_{\text{fatigue}}(t)$. Momentum reflects the empirical observation that after winning consecutive points or games, a player may temporarily elevate their performance (though evidence is mixed). Fatigue reduces serve speed, court coverage, and error resistance, particularly in long matches (4th and 5th sets), hot conditions, and after deep runs in previous rounds. Fatigue effects are most pronounced in best-of-five Grand Slam matches.Question 24. What is the "finish multiplier" concept in combat sports Elo, and why is it used?
Answer
Finish multipliers scale the K-factor based on how the fight ended. For example, a KO/TKO multiplier of 1.25 means the rating update is 25% larger than for a standard decision win. A submission earns a 1.20 multiplier, unanimous decisions use 1.00, while split decisions use 0.85. The rationale is that dominant finishes convey more information about the skill gap between fighters than close decisions. A fighter who knocks out their opponent has demonstrated a larger performance advantage than one who won a split decision, and the rating update should reflect this.Question 25. Where are the structural advantages for analytical bettors in individual sports markets compared to team sports markets?