The Optimal BO5 Prediction Strategy, Applied in the League of Legends Worlds Championships
As the biggest League of Legends (LoL) tournament, Worlds, is right around the corner, analysts and fans alike vividly discuss everything revolving around the game, the teams and the players. A fun topic of discussion which inevitably arises every tournament is one of predictions.
To get players more invested into the game’s biggest event of the year, the LoL team created an in-game platform which allows every player to make their predictions on which team will end up winning each game. Moreover, you actually get hefty rewards for getting the predictions correct. More correct predictions, more prizes to be won! Of course, being the overly-analytical person that I am, I wondered about what could be the optimal prediction strategy which wins me the most rewards.
Worlds 2021 is split into two stages: the group stage and knockout stage. During the group stage, four teams in a group play one another round-robin style, with the two highest-performing teams in each group advancing on to the quarter finals. From the quarter finals onwards, which is the knockout stage, the two teams in that bracket go head-to-head in a best-of-five (BO5) series, and the team which first obtains three wins advancing on to the next bracket, until only one team remains victorious.
Formulating an effective prediction strategy for the groups stage is challenging, because of the variety of teams in one group. Usually, you just predict that the favourites of the match to win that match and you will usually be correct. However, when it comes to the BO5 series, only two distinct teams face off for up to five matches, which allows one to formulate an effective prediction strategy to maximise one’s reward from that series’ predictions.
But before we get into the strategy, let us first understand the specifics of predicting a BO5 series in Worlds 2021.
The Details
In a BO5 series, the team who gets three wins first wins the entire series. As a result, an outcome of the series could perhaps be AABAO, which denotes team A winning the first two games, followed by team B winning one, and finally team A ends the series by winning their third and final game. The last game is not played because the series has already ended, which is denoted by an O.
On the other hand, an impossible outcome of the series is AAAAA, because the series ends once team A wins the first three games. Instead, the outcome would be AAAOO. However, players are allowed to predict the outcome AAAAA, even though that outcome will not be realised. The diagram below shows what you can predict for each game in the series. In total, there are 72 permutations of possible predictions (2³x3²).
Importantly, let’s also look at the possible rewards for predicting a different number of games in the series correctly.
The above table shows the prizes for every number of correct predictions in a BO5 series respectively. Those who are unfamiliar with LoL will not be able to make sense of how much value are these prizes worth, hence, it is important to convert these prizes into a common currency, which makes understanding value much easier.
In this case, a currency which can be used is the Worlds Token, which is conveniently two of the prizes in the pool. In LoL, Worlds Tokens can be used to purchase items such as the Hextech Key and Worlds Orb, which cost 60 and 200 tokens respectively, giving us a reasonable estimate of the value of getting 2 or 4 predictions correct, relative to 1 and 5. However, one cannot purchase the Masterwork Chest using Worlds Tokens. Hence, another currency, in this case Riot Points, a premium in-game currency which can only be purchased using cash, can be used to estimate the value of a Masterwork chest, relative to the Hextech Key, as both can be purchased using Riot Points. Working backwards, the Masterwork Chest, which is the reward for getting 3 predictions correct, can be estimated to be valued at 79.2 Worlds Tokens. Summarizing it all, here is the estimated relative values of the different rewards for one’s predictions:
With all the above information in mind, let’s finally approach the question at hand: What is the optimal prediction strategy to maximise your prizes?
The Approach
To find the answer to that question is relatively simple but tedious. First, we assume that every game in the BO5 series is independent, and that the probability of team A winning every game is equal. Let that probability be p. Next, every possible permutation of predictions of the five games must be considered. These 72 permutations are then evaluated for their expected number of token winnings. In this case, ‘expected’ is a precise mathematical term, which has roughly the same meaning as ‘average’.
To calculate any expected value, the value of an outcome is weighted with the probability of that outcome happening. After considering all the possible outcomes, you can easily calculate the expected value. In our case, we look at a particular prediction which can be made, let’s use AAAAA for simplicity, and superimpose it with all possible outcomes (there are 20 distinct outcomes). The first outcome which could happen is AAAOO. If you were to predict AAAAA, you would get 3 predictions correct, which correlates with a prize worth 79.2 Worlds Tokens. We can also calculate the probability of the outcome AAAOO happening, which is simply p³. By multiplying these two values, we can calculate the weighted value of the outcome, and by summing all the weighted values of all possible outcomes, we get the expected value for that prediction. To find out which prediction is the most ideal, we have to compare the expected values of all 72 possible permutations of predictions for a particular p value, and find the prediction with the greatest expected value.
An Excel spreadsheet can be used to perform the tedious computations mentioned above.
The above diagram shows the expected value calculation for one of the predictions, AAAOO, using an Excel spreadsheet. After setting a value for p, you can easily calculate the expected values for all possible predictions, thereby finding the highest value in the list and matching it to optimal prediction.
The Result
To find the optimal prediction, the expected value for every prediction can be calculated for p=0.05 to p=0.95 with increments of 0.05. The graph of these expected values against the corresponding p value is plotted:
The four important lines to observe for are BBBOO, BBBBO, AAAAO and AAAOO. As can be inferred from the graph, if p>0.85, the optimal prediction is either AAAOO, and if 0.5<p≤0.85, the optimal prediction is AAAAO. Symmetrically, if p<0.15, the optimal prediction is BBBOO, while if 0.15≤p<0.5, the optimal prediction is BBBBO. If the unlikely scenario of p=0.5 occurs, then the optimal prediction is either AAAAO or BBBBO.
An Extension
After relooking at the table for Worlds Token winnings for predicting games correctly, you should be able to observe that the Token winnings are not proportionate to the number of games correctly predicted in that series. In particular, rewards for getting 2 and 3 predictions correct are undervalued, as they should be 100 and 150 respectively, instead of 60 and 79.2 respectively. This influences the overall prediction strategy as it makes one more inclined to aim to correctly predict 4 or 5 games in the series, making a 3–0 prediction greatly beneficial, as a 3–0 outcome has the highest likelihood of occurring, as compared to all the other outcomes.
So how would the results differ if the value of each correct prediction is equal? To investigate this, the reward system for correct predictions will be removed, and the number of correct predictions are counted instead. This means that if you get 1 correct prediction in a BO5 series, you get 1 point, get 2 correct predictions, you get 2 points and so on.
The same excel spreadsheet is used with the above adjustment implemented. The graph of these expected values against the corresponding p value is plotted once again:
Without the skewed reward system, he four same predictions still end up triumphant, even though the graph has flattened out. Now, the point of intersection between the BBBOO and BBBBO lines and the AAAOO and AAAAO lines are about p=0.25 and p=0.75 respectively instead, implying that there are now more situations where predicting a 3–0 victory is more favourable as compared to predicting a 4–0 victory.
Limitations and Future Directions
As stated at the beginning of this article, one of the important assumptions of this approach to identifying the most optimal prediction strategy in a BO5 series is that all the games are independent of each other. This ensures that the probability of the teams winning each game remains constant throughout the series, which greatly simplifies the calculation.
However, this might not be the case in the actual BO5 series. During the series, there will be short breaks given in between every game to allow the team to review the previous game, look at some of the mistakes made, and likely formulate a new strategy for the next game based on their opponent’s previous strategy. This is a crucial component of any BO5 series in most games, and it might influence the probability of winning for subsequent games.
To further refine the calculations, the dynamics of a BO5 series could be better studied, which includes the effect of these ‘review and discussion phases’. Perhaps, Worlds tournaments from the previous years could be used as a reference to find how the probability of a certain team winning changes after winning or losing their last match. That way, an algorithm can be created which takes into account this additional factor, and by integrating it with the Excel spreadsheet used above, it will allow anyone to more accurately determine the optimal prediction strategy for a BO5 series.
