Problem 1: Getting the Rules Right Suppose you are a member of an Olympic doubles badminton team. The setting we are considering is the 2012 Summer Olympics in London and your team is one of 16 teams competing. The teams are divided into 4 groups, A, B, C, and D, with four teams assigned to each group. Your team (YT) is in group A. In the preliminary round of competition, the four teams in each group compete in a round robin tournament (each team plays a match against the other 3 teams in their group). The two teams with the best records from each group move to an elimination tournament. In the elimination tournament, the eight teams from the round robin tournament are matched for a quarterfinal. The four winners of the quarter-final matches advance to a semifinal. The two winners of the semifinal matches advance to a final match. The winner of the final match receives the gold medal; the loser receives the silver medal. The losing teams from the semifinal play each other to determine the bronze medal winner. Teams are matched in the elimination round as shown in the chart below, where A1 and A2 are the teams from group A with the best and second best record, respectively, in the group A round robin tournament. The other slots are determined the same way. Quarter Finals Semi-Finals Final A1 C2 B1 D2 A2 C1 B2 D1 Suppose that your team (YT) is about to play the final round robin match of the tournament against another group A team (AA). Both teams are already guaranteed a slot in the quarter finals. The winner of this game will take the A1 slot and the loser will take the A2 slot. All other slots in the elimination round have been determined. However, there was a surprise upset in group D so that a highly ranked team has ended up in slot D2, and a low ranked team has taken the D1 slot. Your coaches estimate: The winner of this round robin match has a 20% chance of advancing to the finals (winning at least a silver medal). - The loser of this round robin match will be matched against lower ranked teams in the elimina- tion tournament and has a 60% of advancing to the finals (winning at least a silver medal). If Your Team (YT) and the other team (AA) both play well (W), you each have a 50% of winning this match. If one team chooses to play badly (B) and the other team plays well (W), then the team that plays badly will surely lose. If both teams play badly, the judges will surely notice and both teams will be disqualified. (a) Create both an extensive form and a normal form game where the players are your team (YT) and the other team (AA). Each has the option to play well (W) or badly (B). The teams choose simultaneously (prior to the beginning of the match). The payoffs are the probabilities of winning at least a silver medal. Problem 1: Getting the Rules Right Suppose you are a member of an Olympic doubles badminton team. The setting we are considering is the 2012 Summer Olympics in London and your team is one of 16 teams competing. The teams are divided into 4 groups, A, B, C, and D, with four teams assigned to each group. Your team (YT) is in group A. In the preliminary round of competition, the four teams in each group compete in a round robin tournament (each team plays a match against the other 3 teams in their group). The two teams with the best records from each group move to an elimination tournament. In the elimination tournament, the eight teams from the round robin tournament are matched for a quarterfinal. The four winners of the quarter-final matches advance to a semifinal. The two winners of the semifinal matches advance to a final match. The winner of the final match receives the gold medal; the loser receives the silver medal. The losing teams from the semifinal play each other to determine the bronze medal winner. Teams are matched in the elimination round as shown in the chart below, where A1 and A2 are the teams from group A with the best and second best record, respectively, in the group A round robin tournament. The other slots are determined the same way. Quarter Finals Semi-Finals Final A1 C2 B1 D2 A2 C1 B2 D1 Suppose that your team (YT) is about to play the final round robin match of the tournament against another group A team (AA). Both teams are already guaranteed a slot in the quarter finals. The winner of this game will take the A1 slot and the loser will take the A2 slot. All other slots in the elimination round have been determined. However, there was a surprise upset in group D so that a highly ranked team has ended up in slot D2, and a low ranked team has taken the D1 slot. Your coaches estimate: The winner of this round robin match has a 20% chance of advancing to the finals (winning at least a silver medal). - The loser of this round robin match will be matched against lower ranked teams in the elimina- tion tournament and has a 60% of advancing to the finals (winning at least a silver medal). If Your Team (YT) and the other team (AA) both play well (W), you each have a 50% of winning this match. If one team chooses to play badly (B) and the other team plays well (W), then the team that plays badly will surely lose. If both teams play badly, the judges will surely notice and both teams will be disqualified. (a) Create both an extensive form and a normal form game where the players are your team (YT) and the other team (AA). Each has the option to play well (W) or badly (B). The teams choose simultaneously (prior to the beginning of the match). The payoffs are the probabilities of winning at least a silver medal
