It was reported early on all over the internet that it was a "murder for hire" plot. This implies that Tillie supplied them with something in exchange for this being done. So it was Tillie's idea, she wanted Lyle dead so she could run away with them. IMO she lied to try to save herself by saying "oh but I didn't want him dead so they said well give him pills". BS.
http://www.syracuse.com/crime/index.ssf/2015/06/da_joyce_mitchell_murder-for-hire_prison_escape.html
"DA confirms Joyce Mitchell discussed murder-for-hire plot with escaped inmates
A prison worker charged with helping two convicted murderers escape from a maximum-security facility had discussed with them a murder-for-hire plot involving her husband, a district attorney confirmed Wednesday.
Clinton County District Attorney Andrew Wylie said at a news conference that Joyce Mitchell talked to inmates Richard Matt and David Sweat about the possibility of them killing her husband, Lyle."
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http://7online.com/news/officials-murder-for-hire-plot-connected-to-prison-escape/792075/
"Officials: Murder-for-hire plot connected to New York prison escape"
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"Authorities have said Mitchell, who got close to the men while working with them in the prison tailor shop, discussed killing her husband.
"Joyce Mitchell tells us that was discussed between her and Matt and that upon their escape, they were going to return back to Joyce Mitchell's home at which time Matt and Sweat were going to kill her husband," said Andrew Wylie, Clinton County district attorney."