Re-examining the bias and market inefficiency in eSports betting markets: On the application of multivariate regression model

2.1 Data sources

The data consist of 2730 odds which come from 7472 games. The odds we used come from the online cash betting website Bet365. The data contains the two most mainstream types of games in eSports at present: MOBA and FPS. Since each type of eSports has tons of different games, we choose the games that have a larger market share in each category for this study. According to the survey report of Eilers & Krejcik [16], League of Legends account for 38% of the sportsbook betting volume, compared to 29% for Counter-Strike: Global Offensive and 18% for Dota 2. In addition, we also included one mobile MOBA game, King of Glory.

There are significant differences between eSports and traditional sports. Specifically reflected in:

2.2 Odds and commission

In this paper, we use the following mathematical notation: α₁ denotes the odds for the favorite, while α₂ for the underdog. These odds can be obtained through online betting website, and 1 < α₁ < α₂. For a $1 bet, the expected return from betting on the favorite is: E ( R F ) = ( α 1 - 1 ) P F - ( 1 - P F )(1) where p _F is the subjective probability of a favorite winning, while p _U  = 1 - p _F is the subjective probability of the underdog winning.

Similarly, the expected return from betting on the underdog is: E ( R U ) = - P F + ( α 2 - 1 ) ( 1 - P F )(2)

Bookmakers want to earn a risk-free commission c, which is a positive number, by balancing the money bets on both sides. That means the expected return from both sides should equal -c: { - c = ( α 1 - 1 ) p F - ( 1 - p F ) - c = - p F + ( α 2 - 1 ) ( 1 - p F )(3)

After solving the above system of quadratic equations in two variables, we can get: { c = 1 - α 1 α 2 α 1 + α 2 = α 1 + α 2 - α 1 α 2 α 1 + α 2 P F = α 2 α 1 + α 2(4)

We could define favorite be the smaller one between α₁ and α₂. For a simple $1 bet model, the actual return from betting on the favorite can be calculated from the money lines for each game: A F = { α 1 - 1 , if the favorite team wins - 1 , if the favorite team loses(5)

Similarly, the actual return from betting on the underdog is: A U = { α 2 - 1 , if the underdog team wins - 1 , if the underdog team loses(6)

Occasionally, bookmakers will give an initial commission for a specific game. As the betting volume grows, bookmakers could offer higher odds by reducing the commission in order to attract more bettors, and therefore earn more money. In order to reduce the model complexity caused by different commissions, we use the odds 24 hours before the match starts, where c = 0.07.

In order to investigate whether the actual return from betting on the underdog equals the expected return from betting on the underdog, we can state the null hypothesis and the alternative hypothesis as below: H 0 : μ A ( R U ) = μ E ( R U ) H 1 : μ A ( R U ) ≠ μ E ( R U )(7)

As mentioned earlier, we have fixed commission c, then we have E (R _F ) = E (R _U ) = - c. In other words, the population mean of E (R _U ), denotes as μ_{E(R U )}, is known. Now we could use a one-sample t-test to determine if there is a significant difference between the means of E (R _U ) and A (R _U ). We could calculate the appropriate statistic: t = x ¯ A ( R U ) − μ E ( R U ) S 2 A ( R U ) n − 1(8) where the x ¯ A ( R U ) denote the mean of actual return from betting on the underdog, denote the variance of actual return from betting on the underdog.

Given a significance level α, rejection of the null hypothesis H₀ supports the idea that the means of A (R _U ) and E (R _U ) are statistically different. On the other hand, bettors may be biased toward favorites or underdogs.

Similarly, we can also investigate whether the presence of market bias influences bets on different types of games, tournaments or match types. Let A (R _U ) and A ( R U ′ ) be an actual return from betting on the underdog of different tournaments we want to compare. We can state the null hypothesis as below: H 0 : μ A ( R U ) = μ A ( R U ′ )(9)

Therefore, the alternative hypothesis is: H 1 : μ A ( R U ) ≠ μ A ( R U ′ )(10)

Since in our data, the sample size is greater than 30, and data points are independent of each other, z-test can be used to complete this task. For the population standard deviation is unknown, we could calculate the appropriate statistic: z = ( x ¯ A ( R U ) − x ¯ A ( R ′ U ) ) − ( μ A ( R U ) − μ A ( R ′ U ) ) S 2 A ( R U ) n A ( R U ) + S 2 A ( R ′ U ) n A ( R ′ U )(11) where the x ¯ A ( R U ) , x ¯ A ( R U ′ ) denote the mean of actual return from betting on the underdog of different tournaments, , denote the variance of actual return from betting on the underdog of different tournaments.

Given a significance level α, rejection of the null hypothesis H₀ supports the idea that the means of A (R _U ) and A ( R U ′ ) are statistically different. Additionally, bettors may be biased when they bet on the underdog in different tournaments.

Table 2 shows the summary statistics of the data. The results show that the average actual return on bets placed on favorites is –0.087, while the average actual return on bets placed on underdogs is –0.061. Both the values are very close to the expected return, which is -c(-0.07). From this point of view, it seems to be possible to conclude that the entire betting market is very efficient.

However, note the bold numbers in Table 2, which present a large bias in some tournaments. Some results in MOBA games show a large reverse favorite-longshot bias, even in some cases the actual return on bets placed on the underdog are positive numbers, which implies a bettor can earn money by simply placing bets on the underdog. Interestingly, on the contrary, in Counter-Strike: Global Offensive, the results show the presence of a large favorite-longshot bias.

Even more interestingly, the distribution of the betting bias is completely different among eSports titles. In League of Legends, the reverse favorite-longshot bias occurs only in the regional tournaments, which are the lowest-level tournaments in our tournament lists. Similarly, such reverse favorite-longshot bias occurs only in the lowest tournament in King of Glory. On the contrary, the favorite-longshot bias occurs only in the MSI, which is an international tournament. In Dota 2, the reverse favorite-longshot bias only occurs in the top tournament The International 10(TI). In Counter-Strike: Global Offensive, the favorite-longshot bias occurs in high-level tournament, like PGL Major and Blast Premier final. We will discuss and explain the reasons in the following section.

In terms of robustness, the Weighted Least Square (WLS) regression is better than Ordinary Least Square (OLS) regression in cases when the data points have different variances due to the different sampling or experimental conditions [2]. In this section, we discuss how to use WLS to further quantitatively analyze the impact of various factors on the game results. Assume there are m samples and n features, the linear regression model is expressed as:

y i = X i T w + e i(11) where y is the dependent variable, X is the feature vector, w is the vector containing the regression coefficients, and e represents the observation error. Note that the first element of the vector x is 1 to represent the interception.

In our dataset, homoscedasticity is not guaranteed and observation errors are in fact not identically distributed. In this case, the covariance matrix of observation errors is represented as: C = [ σ 1 2 0 ⋯ 0 0 σ 2 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ σ m 2 ](12) where the diagonal elements are not identical and each observation has its own variance. Lack of homoscedasticity means the model will not have the lowest mean square error (MSE). In addition, model coefficients and standard errors will be inaccurate and hence their inferences and any hypothesis testing based on them will be invalid. Weighted Linear Regression (WLS), also called WLS (Weighted Least Squares) model, is a generalization of linear regression where the covariance matrix of errors is incorporated into the model. Hence, it can be beneficial when we are dealing with heteroscedastic data. Here, we use the maximum likelihood estimation (MLE) method to derive the weighted linear regression solution. The dependent variable y in the linear regression model is a multivariate normal random variable. Therefore, the MLE can be derived as: w ˆ = arg max w 1 ( 2 π ) m | C | exp ( - 1 2 ( y - Xw ) T C - 1 ( y - Xw ) )(13)

Solving it with respect to w gives us the solution of WLS: w ˆ = ( X T C - 1 X ) - 1 X T C - 1 y(14)

Additional information is available in the citation [24].

3.1 Subdividing the ranges of probabilistic

Table 3 provides full-sample results from t-tests for $1 underdog bets. We broke down the sample by subjective win probabilities at the closing money line in 4% increments. As discussed in Section 2, we have fixed c, then we have E (R _F ) = E (R _U ) = - c = -0.07, and the variances are 0. Therefore, the one-sample t-test, discussed in Equation (8), was used to determine t value that may show rejection of the null hypothesis H₀ : μ_{A(R U )} = μ_{E(R U )}. The results in Table 3 show that the t-test for all lines and each group p _U accept the null hypothesis except in the group 0.32 < p _U  ≤ 0.36. However, we can’t reject the hypothesis in groups 0.30 < p _U  ≤ 0.34, and 0.34 < p _U  ≤ 0.38. Thus, no systemic market bias appears in the full sample.

A bettor can bet at any time, where commission and odds are different at every moment. In our $1 betting model, we assume that a bettor always places their bets at a fixed time (for example, 24 hours before the game starts) with a fixed commission for each bet. Thus, according to Equation (4), when we got the odds of a favorite α₁ and fixed c, the odds of an underdog α₂ can be calculated by the following formula:

α 2 = 1 - c 1 - 1 - c α 1(12)

This can eliminate the uncertainty caused by random changes in the commission. In fact, the t-test results are highly dependent on the commission c. Smaller commissions could give higher returns for bettors, while also increasing risk for bookmakers. Table 4 shows the average actual return results of all samples by giving different c. The second column shows that decreasing the value of c enlarges the gap between the actual return and expected value, which means increasing the possibility that the mean of actual return betting on the underdog does not equal the mean of expected return. Figure 1 shows the relationship between the value of t value and c.

OPEN IN VIEWER

Fig. 1

t values versus different c values.

The commission used in the different betting markets varies a lot. For example, the commission in EGB.com, which is a cash betting market, is approximately 0.09. While the commission in VPgame.com (a skin betting market) is only 0.019. Why VPgame.com can provide such a low commission? And how does it make sure that the mean of actual return equals the mean of expected return? The answer is that VPgame.com does not set the odds until all bets are placed on each side of a wager. As a result, the bookmakers can calculate an accepted odd with a very low commission without any risk. Unfortunately, such a system does not apply to the mainstream cash betting market. Thus, we will not discuss this system.

Some geographic regions have far more tournaments and professional teams than others, such as Counter-Strike: Global Offensive, most of the top teams have come from the European or North American region. Therefore, regionalism may be one potential source of market bias in eSports gambling. Recent studies have demonstrated that, in games such as CS:GO, bettors who placed their wagers on non-European favorites competing against European underdogs achieved average returns exceeding bookmaker commissions, resulting in a positive profit (Sweeney et al., 2021). In this section, we seek to confirm whether similar patterns exist in other types of esports, such as League of Legends (e.g., LoL). Given the dominance of the LCK and LPL regions in the League of Legends arena, our objective is to ascertain whether such conclusions hold true across such esports titles.

Table 5 demonstrates the results of the mean actual return betting on favorite and underdog. When LPL or LCK favorites were playing matches against underdogs non-LPL + LCK, the actual favorite win rate is much higher than expected (as calculated in column 2 using formula 4). In some cases, the actual win rate is 10 percentage points higher than expected (LCK/other with p _U 0.38). Meanwhile, the mean actual return from betting on the underdogs will be much lower than expected. The p-value of the t-test results shows a clear favorite-longshot bias, in other words, bettors overbet the underdogs and underbet the favorites. The bias is very large when LPL + LCK plays games against heavy underdogs that come from non-LPL + LCK. In some extreme cases, for example, p _U 0.38 when the favorite team comes from LCK, or p _U 0.28 when the favorite team comes from LPL + LCK, the actual return betting on the favorite would incur positive returns (see the bold numbers).

No bias was observed when non-LPL + LCK favorites competed against LPL + LCK underdogs. The findings further indicate that during the 2021 season, both LPL and LCK outperformed the expectations of the cash betting market, with LCK emerging as the most dominant region compared to any other.

Firstly, we will discuss the differences between considering a BO-n match as a whole and individually. Let takes a BO5 match in the grand final of TI10 for example. When we record these 5 games individually, the average actual return value of betting on the favorite and the underdog are –0.452 and 0.737 (see Table 6 for more details). When we consider the 5 games as s whole, we will get the mean actual return value –1 and 1.895. Note that the former makes the mean actual return value close to 0, which is unreasonable. For the reason that a favorite loses to a heavy underdog (in this case, p _U  < 0.28), the betting on the underdog should get more benefit.

Table 7 shows the result of different match formats. It seems that the results show an overbetting on the favorite in the match format of BO1, BO2, and BO7. In order to measure whether the bias is large enough, we use z-test discussed in Eq.(11) to compare each pair of match formats. The z-test results of actual return from betting on underdogs are depicted as a matrix in Table 8. The matrix shows that there are no systemic biases between each pair of match formats.

Surprisingly, this result is quite different from what was expected. As it is generally assumed that short match format (BO1 and BO2) are more prone to surprises, especially in MOBA games. An unexpected result or worse strategy in the ban/pick phrase the MOBA game is likely to lead to the loss of the match. On the contrary, the outcome of the long match format should be favorable for the favorite. Note that the match format of BO7 only exists in King of Glory, therefore, this result cannot be considered as the reason for the match format, but the characteristic of the game title itself. Another possible reason is that the sample size of BO7 is too small to be convincing enough.

Similarly, to measure whether the bias is large enough among different levels of tournaments, we use the z-test to compare each pair of tournaments. In each eSports title, we divide the tournaments into 3 levels: the top level, the major level, and the regional level. Take League of Legends for example, the top level is S11, the most important international tournament for this game. The major level, MSI, is another major international event. The regional level includes 4 major regional leagues: LPL, LCK, LEC, and LCS.

The z-test results of the average actual return are depicted as a matrix in Table 9. The upper triangular part of the matrix is the statistical value of the average actual return from betting on underdogs, while the lower part is the favorites. Bold numbers indicate p < 0.05, while the significant level α is set to 5%. Meanwhile, the t-test is used to compare each tournament with the expected return E (R _U ).

There are generally significant differences between the regional level and higher events. This bias is more pronounced in MOBA games, such as League of Legends and DOTA 2. Especially in Dota2, each pair of tournaments show a large bias. In Counter-Strike: Global Offensive, bias only appear between the top-level event and the regional-level event. Not enough evidence has been found in King of Glory. One possible reason may be that almost all the teams of King of Glory come from China, therefore, different teams are familiar with each other, and the differences between different levels of events are relatively smaller than other eSports titles.

In summary, the bias between tournaments supports the conclusion we got in Section 3.1: the distribution of the betting bias is completely different among different eSports titles and different levels of tournaments. This conclusion inspires us that it is necessary and meaningful to analyze the tournaments of each level separately.

So far we already know that match format and tournaments can have a significant effect on betting market bias. The following WLS models can quantify the magnitude of their impact. These factors can be modeled as: y = w 0 + w 1 x bo 1 + w 2 x bo 5 + w 3 x top + w 4 x region + w 5 x n + e(13)

In the formula above we consider 3 independent variables: match format (x_bo1, x_bo3, x_bo5), tournaments (x _top , x _major , x _region ), and number of games (x _n ). y is the dependent variable of mean actual return when betting on the underdog, and e is the identically distributed normal error. Note that the match format and tournaments are represented as a dummy variable that takes only the value 0 or 1 to indicate the absence or presence of some categorical effect. To avoid multicollinearity, we removed the dummy of bo3 and major as the base category against which the other categories are compared.

The data used to train the model is consist of 732 odds which come from 1430 League of Legends games played in the 2020–2021 seasons. Table 10 shows that the P-value for the F test statistic is less than 0.05, providing evidence against the null hypothesis that w₀ = w₁ = w₂ = w₃ = w₄ = w₅ = 0. It means that at least one coefficient in our WLS model is none-zero, therefore our linear model is statistically significant.

Specifically, the coefficient of x_bo5 and x _region are statistically significant. The negative coefficient x_bo5 indicates that the average return of betting on the underdog of games with match format BO5 is lower than games with match format BO3. Such results strongly support the assumption we mentioned in section 3.4: short-match formats are more prone to surprises, especially in MOBA games. Similarly, the positive coefficient x _region indicates that the average return of betting on the underdog of regional tournaments is higher than major tournaments. The reasons for this result are various, such as the level of the participating teams, prize pool, etc.

4.1 Discussion

It’s clear that we found both favorite-longshot bias and reverse favorite-longshot bias in the online cash betting market on eSports. Such biases widely exist in all kinds of eSport titles in our data. In a typical efficient betting market, the average actual return on a bet would be approximately equal to the average expected return on a bet, which should be the value relative to the commission, -c. The results show that the gap between actual return and expected return is so large that in some extreme cases, the value could be a positive value, which means a bettor can get positive returns using a simple betting strategy. We discuss the reasons for the biases from four aspects: commission, region, match format, and tournaments. When we look deeper into the match format and tournament, we found that long match format is the important factor of long-short bias, while regional tournaments is the important factor of reverse long-short bias in League of Legends. Some biases we found are different to the previous literature. The reasons for these differences are as follows: firstly, we took all the maps in a BO-n match format as a whole, for the reason that maps in a BO-n match are closely related. Secondly, we fixed the commission. Our simplified $1 betting model is that a bettor always places their bets at a fixed time (for example, 24 hours before the game starts) with a fixed commission for each bet. This can eliminate the uncertainty caused by random changes in the commission. Finally, most of the tournaments we got are from high-level leagues. The reason is that all the information on high-level tournaments is more open and reliable. The present findings proved that it is necessary to analyze the tournaments of different levels separately.

4.2 Limitation

While this study contributes valuable insights into market biases in eSports betting, it is essential to acknowledge certain limitations that can guide future research endeavors:

Data Availability: Our analysis heavily relies on the availability of historical betting data, which may be limited due to confidentiality or data accessibility issues. This constraint may affect the comprehensiveness of our findings.

Longitudinal Analysis: Our research primarily provides a snapshot of market biases during the study period. A longitudinal analysis tracking changes over time would offer a more dynamic understanding of these biases.

Multiple linear regression: Multiple linear regression assumes a linear relationship between the independent variables and the dependent variable.It provides information about the relationships between variables, but it may not provide insights into the underlying mechanisms or reasons for observed relationships.

Based on our findings and the existing gaps in the field, several potential avenues for future research are delineated below:

Data Sources and Analysis Tools: As the eSports industry evolves, explore new data sources and advanced analytical tools for studying market biases. Leveraging big data analytics, machine learning, and AI may provide new insights such as match outcome prediction [7].

Behavioral Factors: Investigate the psychological and behavioral factors that contribute to market biases in eSports. Understanding the cognitive biases of bettors, their information processing, match-fixing, game balance of eSports, and decision-making strategies can provide a more comprehensive picture of why these biases occur.

Cross-Cultural Analysis: Investigate how market biases in eSports differ across various regions and cultures. Cultural factors, such as attitudes towards risk and gambling, may play a role in the prevalence and nature of biases.

Fan Engagement: Explore how market biases and betting activities impact fan engagement in eSports. Investigate the relationship between betting behavior and viewership, as well as the potential ethical implications of these connections.

4.3 Applications

Our findings hold several noteworthy implications and potential applications.

Firstly, our research sheds light on the persistence of market biases in the emerging domain of eSports betting. Understanding the existence and drivers of these biases can inform financial analysts and investors seeking to predict and navigate eSports-related consumption and investments [18].

Secondly, our study has practical relevance for the eSports industry itself. With eSports rapidly gaining popularity worldwide, the insights gained from our research can help operators, tournament organizers, and teams better comprehend the dynamics of eSports betting. They can utilize this knowledge to make informed decisions regarding sponsorship deals, tournament structures, and fan engagement strategies.

Thirdly, the regulatory landscape surrounding eSports betting is still evolving. Our findings regarding market biases can assist policymakers in crafting effective regulations and safeguards to ensure fair and transparent betting practices.

Lastly, the broader applicability of our research extends to other emerging markets and industries. The study of market biases and behavioral patterns in non-traditional domains can serve as a template for researchers and analysts exploring similar dynamics in areas such as virtual currencies, online gaming, and prediction markets.

In summary, the significance of this research extends beyond the realm of eSports betting. Firstly, it contributes valuable knowledge to the fields of finance, economics, and behavioral finance by shedding light on market biases in non-traditional betting markets. Secondly, our findings not only enhance our understanding of behavior in eSports markets but also provide new paradigms for research in broader market behavior and prediction studies. Lastly, we believe that this study can serve as an inspiration for other fields. The study of market behavior is not confined to finance alone but can be applied across diverse domains to enhance decision-making and behavior analysis. We encourage fellow researchers to draw from our methods, exploring market behavior in other contexts, and advancing the frontiers of knowledge.