It seems clear, from Littlefinger’s discussion with Sansa (III: 767-769), that he arranged for the poison to be secreted in the hairnet he had Dontos bring to her for a third party to pick up and drop into Joffrey’s drink at an opportune time. In general various Tyrells are blamed for the attempt. Lady Olenna was certainly the only one who had an opportunity to take the poison from the hairnet, as she adjusted it when meeting Sansa and Tyrion outside of the feasting hall (III: 672, 673). But it’s argued that Margaery, being right next to Joff and his cup, had a greater opportunity to place the poison in his cup with the added argument that, given that she was expected to share drinks with him from the chalice, it’d be safer if she herself put in the poison. Some suggest Ser Garlan, Margaery’s brother, may have been the true culprit. Although he was seated even further from the newlyweds than Lady Olenna was (III: 674), he was near the chalice when it was briefly left unattended for the pie-cutting; however, so was Lady Olenna (III: 681). It should be noted that an alternate theory has been put forward, suggesting that Lady Olenna or another Tyrell was actually targetting Tyrion, based on the fact that Joffrey succumbed to the poison shortly after eating from the piece of pie given to Tyrion (III: 683). Given how Littlefinger is quite unsurprised by Joffrey being dead, it seems he was entirely unaware of the possibility that the poison might be given to Tyrion instead of Joffrey. This does not necessarily negate the possibility, but it does make it seem rather less likely. Joffrey was clearly a much greater concern for the Tyrells than Tyrion and his marriage to Sansa were. Finally, it is almost certain that Lord Mace is completely unaware of his family’s involvement in Joffrey’s murder.