The first game was played by players who were married to each other. Therefore, the first game could have been between Mr Das and Mrs Das or Mr Nair and Mrs Nair.
Let's consider the first case: The game is played between Mr and Mrs Das. Either Mr Das wins or Mrs Das wins. If Mrs Das wins, the second game will have to be played either between Mrs Das and Mr Nair or Mrs Das and Mrs Nair. Now we have the condition that women have won only one game. Therefore, as per this condition Mrs Das cannot have won the second game. The second game could have been won either by Mr Nair or Mrs Nair. The third game, then, would have to be played between Mr and Mrs Nair. This defies the condition that players married to each other do not play the second and third games. Therefore, Mrs Das could not have won the first game.
This implies that the first game must have been won by Mr Das. Mr Das now has to play with Mr Nair and Mrs Nair. Mrs Das cannot play any more games. Only the first game is played between players married to each other which has already been played between Mr and Mrs Das. Therefore, there cannot be a game between Mr and Mrs Nair. Thus, in his game against Mr Nair or Mrs Nair, Mr Das will be the winner because if either of the Nairs wins then the third game will have to be between Mr and Mrs Nair, which cannot happen. Suppose the second game is between Mr Das and Mrs Nair, then Mr Das wins. But this again defies the condition that women have won one game (Mrs Das has already lost so Mrs Nair has to win). Thus, the second game could not have been played between Mr Das and Mrs Nair. This implies the second game was played between Mr Das and Mr Nair and Mr Das won (if Mr Das loses, Mr Nair will have to play against Mrs Nair which defies the condition). The third game is now played between Mr Das and Mrs Nair and Mrs Nair wins (women have won one game). This possibility works out fine. So, in this case Mr Das has won the maximum of two games (against Mrs Das and against Mr Nair). Mrs Das has lost and so has Mr Nair while Mrs Nair has won one game. The Das' have, thus, won two games while the Nairs have won one game. So, the Das' have won more games than the Nairs.
Now suppose the first game was played between Mr Nair and Mrs Nair, then a similar situation will arise. Mrs Nair would not have won the first game. Mr Nair would have won against Mrs Nair, he would have won against Mr Das and lost to Mrs Das. Thus, Mrs Das would have emerged the winner. In this case the Nairs would have won more games than the Das'.
But the third condition states that the Das' won more games than the Nairs. Therefore, the first game must have been played between Mr Das and Mrs Das and the latter must have lost; the second game was played between Mr Das and Mr Nair and the latter lost; and the third game was played between Mr Das and Mrs Nair, and the latter won. Therefore, Mrs Nair emerged as the winner.