We can keep track of the state for each round in the following way, with the yellow box indicating the head of the table. Note that in our diagram, going right means going clockwise around the table, so a player’s right is actually on the left in our diagram.

We will also label the player voting conditions and refer to them by letter in this solution. Smaller numbered conditions take precedence over larger numbered conditions (e.g. A1 takes precedence over A2 which takes precedence over A3).

Mordred sits at the head, so we can put him there. We’ll have to look ahead to fill in the rest of the positions based on who makes each subsequent proposal. Round 1 begins with the player with the lowest Scrabble score. This is Merlin (with a score of 8). Since Mordred goes second, Merlin sits to Mordred’s right (left in our diagram). Percival goes after Mordred in Round 3, so he sits to Mordred’s left. Morgana is supposed to go next, so she sits to Percival’s left, and Arthur is after Morgana, so he sits to Morgana’s left. We now have the initial order.

Round 1

In Round 1, Merlin proposes himself and Arthur. Arthur will vote no because of (A3). Merlin will vote yes because of (E1). Mordred will vote no because of (D2) and (D*) is not triggered. Morgana will vote no because of (G4). Percival will vote no because of (P2). Thus we have in Round 1:

Round 2

In Round 2, Mordred replaces a player with their half-sibling, so this must be replacing Arthur with Morgana and proposing Merlin and Morgana. Arthur will vote yes because of (A2). Merlin will vote no because of (E3). Mordred will vote yes because of (D1) and (D*) is not triggered. Morgana will vote no because of (G4). Percival will vote no because of (P2).

Round 3

In Round 3, Percival proposes two people that have the same number of letters in each of their names as the number of the round of Percival’s next proposal. Looking ahead, we see that Percival will next propose in Round 7 (as Morgana skips her turn), so Percival proposes Mordred and Morgana. Arthur will vote no because of (A3). Merlin will vote no because of (E3). Mordred will vote yes because of (D1) and (D*) is not triggered. Morgana will vote yes because of (G2). Percival will vote no because of (P1).

Round 4

In Round 4, Morgana abstains from proposing so Arthur becomes the proposer. He proposes two of Arthur, Merlin, and Percival. We don’t know which ones yet, so let’s come back to this. Note, however, we can still deduce something about the votes. Arthur’s vote is unknown. Merlin will vote no (E3). Neither Arthur nor Percival voted yes in the previous round so Morgana will vote no on this one (G4). Mordred will vote no (D2) because it’s not possible for (D1) to trigger. Because Arthur is proposing, Percival will vote yes on this one (P3).

Round 5

In Round 5, Merlin brings himself and the other member of Good that wasn’t in the last proposal. Since this depends on the Round 4 proposal, we have to come back to this as well.

Round 6

In Round 6, Mordred proposes the same group as his previous proposal. This is Merlin and Morgana. However, because this is the first proposal that requires three members, as Merlin rightly observes, Mordred has to make a second proposal and he adds himself to the previous proposal. His proposal is therefore Merlin, Mordred, and Morgana. Arthur will vote yes because of (A2). Merlin will vote no because of (E3). Mordred will vote yes because of (D1) and (D*) is not triggered. Morgana’s will vote no regardless of how previous rounds’ votes went because of (G1). Percival will vote no because of (P2).

Round 7

In Round 7, before a proposal is made, Mordred and Morgana switch spots. The player now to Morgana’s left is now Percival, so Percival makes a proposal. At this point, Arthur has been proposed between 1 and 2 times (note that Arthur could not have been proposed in both of rounds 4 and 5), Merlin has been proposed between 4 and 5 times, Mordred has been proposed between 2 and 3 times, Morgana has been proposed between 3 and 4 times, and Percival has been proposed between 0 and 1 time. This means that the two players that have been proposed the most must be Merlin and one of either Mordred or Morgana. Since the last player Percival includes in his proposal is the player to his left (Mordred), and it is implied that this player is not one of the two players that have been proposed the most, the other player that must have been proposed the most must be Morgana. This is confirmed by the fact that had the other player been Mordred, both Mordred and Morgana would have been proposed 3 times, and there would be a tie for second, which would not make sense for the logic puzzle. Thus, Percival’s proposal is Merlin, Mordred, Morgana.

Arthur will vote no because of (A3). Merlin will vote yes because of (E2). Mordred will vote no because of (D1) and he is startled by the duck in this round (D*). Morgana will vote yes because of (G3). Percival will vote no because of (P1).

Round 8

In Round 8, Arthur’s suggestion means we keep the original proposer order, and so Morgana is next. Before the proposal, Merlin and Percival switch places. Morgana manages to get four votes in favor. Morgana is guaranteed to vote yes because of (G3). Percival is also guaranteed to vote yes because of (P3). Merlin would only vote yes if the proposal is the same as the previous round because of (E2), but Mordred is not on the proposed team, so Merlin must vote no. Thus, Arthur and Mordred must vote yes and Morgana must have proposed Percival (as Mordred is not on the proposed team). Morgana must also have included herself and Merlin in the proposal in order for Mordred to vote yes because of (D1) and (D*) is not triggered.

Back to Rounds 4 and 5

Because this vote passes, the quest proceeds and based on what Patsy says at the end (“you succeeded in all three of your quests”), we know that Merlin, Morgana, and Percival are the members of Good. We can use that information to resolve Round 5. “I’ll bring myself and the other member of Good that wasn’t in the last proposal” implies one of Morgana or Percival was proposed in Round 4 and the other was proposed by Merlin in Round 5. This means the Round 4 proposal must have been Percival and one of Arthur or Merlin and the Round 5 proposal must have been Merlin and Morgana.

Note that in Round 4, if the proposal is Arthur and Percival, then Arthur will vote no because of (A1), and if the proposal is Merlin and Percival, then Arthur will vote yes because of (A2).

In Round 5, Arthur will vote no because of (A3). Merlin will vote yes because of (E1). Mordred will vote yes because of (D1). Morgana will vote for because of (G2) as Percival votes yes in Round 4 and no in Round 3. Percival will vote no because of (P2).

Round 9

In Round 9, Arthur proposes the two players that have had the fewest votes in favor of them being on a team (sum the votes for a proposal that included that player). Taking a look at the votes so far:

Arthur has 1 or 2 votes, Merlin has 14 votes. Mordred has 6 votes. Morgana has 15 votes. Percival has 5 or 6 votes. We can actually use this information to resolve Round 4. If the Round 4 proposal was Merlin and Percival, then Percival would have 6 votes and it would be impossible for Arthur to bring the two players with the fewest votes, as Mordred and Percival are tied for second fewest. Therefore, in Round 4, the proposal is Arthur and Percival. Then Arthur will vote no.

Going back to Round 9, Arthur’s proposal is then Arthur and Percival (again). Arthur votes against because of (A3). Merlin votes against because of (E3). Mordred votes against because of (D2) and (D*) is not triggered. There are two proposals this turn (a quest proposal and a marriage proposal!), so Morgana votes against because of (G1). Percival votes for because of (P3).

Round 10

In the final round, Merlin makes the proposal and proposes himself and either Arthur or Percival. Because of Patsy’s statement, and the fact that this is the final proposal, Merlin must propose a team of all members of Good, so his proposal is Merlin and Percival. Arthur votes for because of (A2). Merlin votes for because of (E1). Mordred votes against because of (D2) and (D*) is not triggered. Morgana votes against of (G4). Percival votes against because of (P3).

The final proposals and votes look like this (with 1=YES and 0=NO, and the proposed set of players highlighted in orange)

Reading the votes in clockwise order starting from the head of the table as binary and converting to letters gives the answer ARTHURQWAK.