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*All-time and Tournament Rankings.*

Each player's prowess in The Banner Saga: Factions is recorded in the Hall of Valor in Strand. There, you can review your overall performance and that of the top players of the game. There is a different listing for all-time and current-tournament performance, accessible by clicking on the appropriate banners.

The most important performance statistic is the player's rating/ranking [1]; this is measured by means of an Elo-ranking system [2], similarly to chess. The Elo-ranking of each player is used by the game's Matchmaker in order to arrange matches between equivalently ranked players. This helps make every battle equally challenging regardless of your ranking!

# Ranking vs. Match type

Your Elo-ranking is affected differently, depending on the match type selected in the Great Hall.

- Quick matches do not affect your Elo-ranking. Play freely, experiment with tactics and earn Renown!
- Ranked matches normally affect your all-time Elo-ranking. For the change in your ranking, refer to the formula section. Ranked match may be against opponents from the Quick match queue; this case results in only a small fixed change in Elo-ranking.
- Tournament matches affect both your all-time and your current-tournament Elo-ranking. These are the matches to give your best show!
- Friendly matches do not affect your Elo-ranking.

Finally, note that Ranked & Tournament matches are open only to specific team-powers, e.g. 6 and/or 12.

# Elo ranking

Upon first arrival in Strand — or upon registering in a new Tournament — each new player is assigned an Elo-ranking of 1000. Then, for each battle won or lost, his Elo-ranking is increased or decreased, respectively. The amount of change in the Elo-ranking depends on the ranking-difference of the two combatants. As a rule of thumb, a win/loss against a higher (or lower) ranked player results in a bigger (or lower) +/- change in your ranking. The maximum allowable change in Elo-ranking is governed by a constant called $K$-factor.

## Formulas

The algorithm of calculating the change in Elo-ranking after a match, is the following. Firstly, the K-factor is calculated independently for each player, $K_i$ where $i={1,2}$, according to their Elo-ranking, $R_i$.

(1)The above formula means that for entry- and mid-level players a $K = 32$ is typically used, while $K = 16$ is reserved for high-level players. Subsequently, given the outcome (or score) of the match for each player, $S_i$, the change in the Elo-ranking $\Delta {R_i}$ is calculated for each player separately:

(2)where $S_i^{e}$ is the expected/estimated score for the $i$-player, given by

(3)A score of $S=0$ denotes a defeat and $S=1$ a victory. A tie (currently unavailable in Factions) would correspond to $S=0.5$, for both players. Evidently, the player scores must satisfy $S_1 + S_2 = 1$.

*Graph: Change in Elo-ranking after a battle.*

## Remarks

- This ranking concept can be explained by the wagering analogy of the
*pot*[3]. Before the match, each player places some if his $R_i$ in the*pot*, with the favorite to win (i.e. the one with the higher $R$) having to put in more. At the end of the match, the winner takes all the pot.

- The Elo-ranking is designed so that when a player reaches his/her
*true*ranking, he/she cannot expect big deviations from it, regardless of the opponents' rankings [4]. In this respect, the estimated score $S_i^{e}$ represents the probability of winning and, statistically speaking, the expected Elo-ranking change is $E[~\Delta R_i~|~R_1,R_2~]=0$.

- The formula for $\Delta {R_i}$ means that when the players are equally matched, $R_1 = R_2$, the Elo-ranking change is $\pm~K/2$ for the winner/loser, respectively. Viewed in terms of the
*pot*analogy (with $K=32$), firstly both players remove $16$ from their $R_i$ contributing to a total of $K=32$ in the*pot*. The winner takes all, so $\Delta R_{\text{winer}} = +16$, while the loser takes nothing, i.e. $\Delta R_{\text{loser}} = -16$.

- When you are playing a Ranked match but your opponent is from the Quick match queue, you can only get a fixed ranking-change of $\Delta R = \pm2$.