"Finite Player" is a term coined in 
Finite and Infinite Games: A Vision of Life as Play and Possibility.
Quotations
"Finite players are serious; infinite games are playful.
"Finite players try to control the game, predict everything that will happen, and set the outcome in advance. They are serious and determined about getting that outcome. They try to fix the future based on the past.
"Infinite players enjoy being surprised. Continuously running into something one didn't know will ensure that the game will go on. The meaning of the past changes depending on what happens in the future."
Someone should put a better definition in here.