Swiss system tournaments, a type of group tournament common in chess and other board games, use various criteria to break ties between players who have the same total number of points after the last round. This is needed when prizes are indivisible, such as titles, trophies, or qualification for another tournament. Otherwise players often share the tied spots, with cash prizes being divided equally among the tied players. Some tiebreakers used in other group tournaments are also used in Swiss-system tournaments, while others exploit the particular features of the Swiss system. If the players are still tied after one tie-break system is used, another system is used, and so on, until the tie is broken. Most of the methods are numerical methods based on the games that have already been played or other objective factors, while some methods require additional games to be played. In chess, where results are simply win/loss or draw, strength of schedule is the idea behind the methods based on the games already played: that the player that played the harder competition to achieve the same number of points should be ranked higher. In other games, results may supply more data used for breaking ties. Median / Buchholz / Solkoff The Median system is also known as the Harkness System, after its inventor Kenneth Harkness, or the Median-Buchholz System. For each player, this system sums the number of points earned by the player's opponents, excluding the highest and lowest. If there are nine or more rounds, the top two and bottom two scores are discarded. Unplayed games by the opponents count ½ point. Unplayed games by the player count zero points. Modified Median The Modified Median system is similar to the Median system, except: Players with exactly 50% score are handled as in the regular Median system; Players with more than 50% score have only their lowest-scoring opponent's score discarded; Players with less than 50% score have only their highest-scoring opponent's score discarded . Solkoff This system is the same as the Median system, except that no scores are discarded. Ephraim Solkoff did not invent this system. He introduced it to the United States in 1950, but it was used in England prior to that . Cumulative To calculate this, sum the running score for each round. For example, if a player has (in order) a win, loss, win, draw, and a loss; his round-by-round score