Thanks for this thread. To me it would be a more interesting (and instructive!) puzzle to permit shifting the cards around either way so as to maximize the score.
Anyway, here are three scores:
Forcing games I own to be counted as owned: 113
Counting all games as played: 132
Placing games so as to maximize score: 144
A fine point! Here goes...it can't get lower!
Forcing games I own to be counted as owned: 28
Counting all games as played: 38
Placing games so as to maximize score: 48
Marginally better I guess . Thanks again!!!