Esports Math
APPLIED PROBABILITY INSTITUTE // QUANTITATIVE GLOSSARY

Esports Math Glossary

Authoritative mathematical definitions, analytical formulations, and empirical worked examples for 25 foundational concepts in esports probability, game theory, and in-play market trading.

Rating Systems & Volatility

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Elo Rating System

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P(A) = \frac{1}{1 + 10^{(R_B - R_A)/400}}, \quad R_A' = R_A + K \cdot (S_A - P(A))

A zero-sum skill rating algorithm calculating comparative win probabilities based on point differentials. Used as the historical foundation for CS2 and Dota 2 team power rankings.

Worked Case Example: A 200-point Elo advantage (e.g. 1800 vs 1600) yields a theoretical map win probability of 1 / (1 + 10^(-200/400)) = 75.97%.

Glicko-2 Rating System

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g(\text{RD}) = \frac{1}{\sqrt{1 + 3 q^2 \text{RD}^2 / \pi^2}}, \quad q = \frac{\ln(10)}{400}

An advanced Bayesian rating framework created by Mark Glickman that expands classical Elo by modeling uncertainty through Rating Deviation (RD) and skill volatility (sigma).

Worked Case Example: A team with rating 1700 and high RD=180 after a roster overhaul has a wider probability confidence interval (48%-76%) than a stable roster with RD=45.

Rating Deviation (RD)

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\text{RD}' = \sqrt{\text{RD}^2 + c^2 \cdot \Delta t}

A measure of confidence in a player or team's estimated rating, equivalent to one standard deviation in a Gaussian distribution. RD inflates over inactivity intervals and collapses after matches.

Worked Case Example: Following a 60-day tournament break, a team's RD increases from 42 to 98, reducing the model's confidence in their favorite status.

K-Factor Calibration

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\Delta R = K \cdot (S - E)

The sensitivity multiplier determining how many rating points are transferred per match outcome. High K-factors (32+) provide rapid reactivity but excessive noise; low K-factors (16) provide long-term stability.

Worked Case Example: At K=32, an upset win over a favorite awards +24.8 rating points, whereas at K=16 the exact same win awards only +12.4 points.

Map Pool & Veto Game Theory

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Map Pool Depth Metric

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\text{MPD} = \sum_{m \in \text{Pool}} \mathbf{1}_{\{WR_m \ge 0.50\}} \cdot \ln(N_m + 1)

Quantitative index measuring the breadth and statistical viability of a team across the 7 Active Duty maps, weighted by sample volume and winrate thresholds.

Worked Case Example: A Tier-1 team with 5 playable maps above 58% win rate holds an MPD of 8.92, making them heavy favorites in Bo3 playoff brackets.

Permaban Optimization

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\text{BanRate}_m = \frac{N_{\text{bans}, m}}{N_{\text{drafts}}} \ge 0.85

The map a team systematically vetos in the first ban phase across all official fixtures, eliminating exposure to low practice hours and forcing opponents into contested ground.

Worked Case Example: A team with a 92% permaban on Anubis allows them to concentrate 100% of scrim preparation into the remaining 6 maps.

Veto Minimax Game Theory

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\max_{p \in \text{Picks}} \min_{b \in \text{Bans}} \mathbb{E}[\text{WinProbability}(p, b)]

The algorithmic game-theoretic sequence where each team makes ban and pick decisions that maximize their worst-case series payoff against a rational opponent.

Worked Case Example: Team A bans Mirage to neutralize Team B's 80% strength, even if Team A has a 55% win rate on it, because the relative differential (-25%) is lethal.

Best-of-Series Compounding Probability

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P(\text{Bo3}) = p_1 p_2 + p_1 (1 - p_2) p_3 + (1 - p_1) p_2 p_3

The binomial path summation calculating the overall probability of winning a Bo3 or Bo5 series from individual map win probabilities.

Worked Case Example: Winning Map 1 (60%) and Map 2 (55%) with a Decider at (52%) results in an overall Bo3 win probability of 0.60*0.55 + 0.60*0.45*0.52 + 0.40*0.55*0.52 = 58.48%.

In-Play Momentum & Economy

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Pistol Round Cascade Factor

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P(\text{MapWin} \mid \text{WinBothPistols}) \approx 0.742

The structural multiplier where winning a pistol round triggers high-probability conversion of the subsequent anti-eco round, producing an immediate 2-0 score momentum.

Worked Case Example: Winning both pistol rounds in CS2 MR12 yields an empirical 74.2% map win conversion rate across Tier-1 tournaments.

Economy Reset

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\text{LossBonus} = \min(3400, 1400 + 500 \cdot L)

A situation where a team loses a round immediately after winning one, capping their loss bonus at $1,400 and forcing an eco round, creating sustained multi-round EV swings.

Worked Case Example: Winning round 7 but losing round 8 resets Team A's cash to $1,400 per player, granting Team B an 82% win probability on round 9.

Force Buy Equity

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P(\text{Win} \mid \text{Force vs Full}) \approx 0.288

A high-risk tactical investment where a team spends all remaining funds on sub-optimal rifles and pistols instead of saving, yielding ~28-32% upset conversion.

Worked Case Example: A force buy with Galils and Scouts wins ~28.8% against AK/M4 full buys, but losing guarantees an inescapable double-eco.

Eco Round (Save Round)

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P(\text{Win} \mid \text{Eco vs Full}) \approx 0.176

A designated concession round where a team spends minimal funds (<$500) to build a maximum loss bonus bankroll for the next round.

Worked Case Example: Eco rounds convert only 17.6% into round wins, but secure $4,300+ average equipment per player on the subsequent buy.

Map Side Bias Asymmetry

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\text{Bias}_{\text{CT}} = \frac{N_{\text{rounds, CT won}}}{N_{\text{total rounds}}} - 0.50

The structural map design skew that causes round win probabilities to deviate from 50% between Counter-Terrorists and Terrorists (e.g. 54.8% CT on Ancient vs 52.6% T on Anubis).

Worked Case Example: On Ancient, trailing 4-8 on T-side represents a normal expectation; after the half-switch to CT, the team's expected score is 8-4.

Half-Switch Momentum Reset

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\text{Round}_{\text{switch}} = 13 \quad (\text{CS2 MR12})

The mandatory role reversal at round 13 in CS2 MR12 regulation where economies reset to $800 pistol baseline and map asymmetry inverts.

Worked Case Example: A 9-3 lead built on CT-side Nuke often melts away when the defending team struggles on their own T-side pistol round.

Comeback Probability Factor

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P(\text{Win} \mid \text{Score } 4\text{-}8) = f(\text{SideBias}, \text{PistolProb}, \Delta \text{Elo})

The conditional probability of overcoming a substantial half-time deficit (e.g. 4-8 or 3-9) based on map side asymmetry, pistol conversion, and rating skill delta.

Worked Case Example: On CT-heavy Nuke, a favorite trailing 4-8 holds a 34.6% comeback probability if they win the second-half pistol round.

Player Impact & Fantasy Props

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Average Damage Per Round (ADR)

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\text{ADR} = \frac{\sum_{r=1}^N \text{Damage}_r}{N}

The mean health points subtracted from opponents per round played. Unlike KPR, ADR captures assist contribution and chip damage, correlating with true win share at r=0.74.

Worked Case Example: A player averaging 88.5 ADR over 50 rounds generates 18.2% more offensive output than the Tier-1 professional median (74.8 ADR).

KAST Percentage

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\text{KAST} = \frac{N_{\text{Kill } \cup \text{ Assist } \cup \text{ Survived } \cup \text{ Traded}}}{N_{\text{total rounds}}} \times 100\%

The percentage of rounds in which a player achieved at least one Kill, Assist, Survived, or was Traded within 4 seconds of dying. Measures baseline consistency.

Worked Case Example: Elite anchors and supports maintain KAST rates above 75.0%, ensuring they rarely yield 'wasted' rounds without team equity.

HLTV Rating 2.1 Composite

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\text{Rating 2.1} = f(\text{KAST}, \text{KPR}, \text{DPR}, \text{ADR}, \text{Impact})

The industry-standard composite index evaluating competitive CS2 performance by integrating KAST%, KPR, DPR, ADR, and Impact Rating into a centered distribution around 1.00.

Worked Case Example: A Rating of 1.25+ places a player in the top 5% of Tier-1 competitors, while 0.90 indicates sub-average output.

Role-Adjusted Rating (RAR)

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\text{RAR} = \frac{\text{Rating} - \mu_{\text{role}}}{\sigma_{\text{role}}}

A Gaussian z-score standardizing individual performance against specific tactical roles (Anchor, Entry, AWP, IGL, Lurker), removing systematic role bias.

Worked Case Example: An anchor with a raw rating of 1.08 achieves a +0.91 sigma RAR, whereas an AWPer with the same 1.08 rating registers a sub-par -0.33 sigma RAR.

Carry Potential Index (CPI)

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\text{CPI} = \left(\frac{\text{Damage}_{\text{player}}}{\text{Damage}_{\text{team}}}\right) \cdot \left(\frac{\text{Rating}_{\text{player}}}{\overline{\text{Rating}}_{\text{rest}}}\right)

A metric quantifying the degree of statistical reliance a team has on a single star player. High CPI (>1.35) highlights teams vulnerable to solo target-banning and bad individual form.

Worked Case Example: A superstar accounting for 32% of team damage with a 1.35 rating while teammates average 0.95 generates a high CPI of 1.42.

Quantitative Betting & Risk

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Expected Value (EV)

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\mathbb{E}[X] = \sum_{i=1}^n x_i \cdot P(X = x_i) = (P \cdot b) - (1 - P)

The probability-weighted average payout of a wager over an infinite horizon. In esports betting, positive EV (+EV) represents a mathematical edge where model probability exceeds bookmaker implied probability.

Worked Case Example: If Glicko-2 gives Team A a 60% win chance and decimal odds are 1.85, EV = (0.60 * 1.85) - 1 = +0.11 (+11% net yield per unit wagered).

Bookmaker Overround (Vig)

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\text{Overround} = \left(\sum_{i=1}^n \frac{1}{\text{Odds}_i}\right) - 1

The cumulative margin extracted by a sportsbook by pricing reciprocal outcomes below true fair odds. Low overround (sub-4%) preserves player capital, while casual books extract 8-12%.

Worked Case Example: A 2-way match with odds 1.88 on both sides has an overround of (1/1.88 + 1/1.88) - 1 = 1.0638 - 1 = 6.38% vigorish.

Return to Player (RTP)

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\text{RTP} = \frac{1}{1 + \text{Overround}} \times 100\%

The percentage of total wagered turnover mathematically returned to bettors across balanced market distributions. 1win Esports markets benchmark at ~96.8% RTP versus ~92.5% across industry averages.

Worked Case Example: An overround of 3.2% corresponds to an RTP of 1 / 1.032 = 96.89%, leaving only 3.11% house drag.

Variance & Maximum Drawdown

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\sigma^2 = n \cdot p \cdot (1 - p), \quad \text{MDD} = \max_{t \in [0,T]} (H_t - B_t)

Statistical dispersion of betting returns around mathematical expectation and the maximum observed peak-to-trough bankroll contraction. Esports live markets exhibit elevated variance due to sudden economy eco resets.

Worked Case Example: A bettor with a +6% true edge betting 2% unit stakes over 500 bets faces an expected 95% confidence maximum drawdown of 14 to 18 units.

Kelly Criterion (Fractional Kelly)

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f^* = \frac{p \cdot b - q}{b}, \quad f^*_{\text{half}} = \frac{f^*}{2}

The mathematically optimal fraction of bankroll to wager on a positive-expectation proposition to maximize logarithmic capital growth. Half-Kelly (f*/2) and Quarter-Kelly (f*/4) are recommended in esports to hedge against hidden map pool variance.

Worked Case Example: With p=0.62 and decimal odds 1.80 (b=0.80): f* = (0.62*0.80 - 0.38) / 0.80 = 14.5% of bankroll. Half-Kelly mandates a safe 7.25% wager.