spreadsheet demo: https://docs.google.com/spreadsheets/d/

where column M consists games which can be selected in the drop-down menu of column D
then cell E3 contain this formula which detects which checkbox was marked:
=ARRAYFORMULA(IF(A3:A=TRUE; $A$2&":";
IF(B3:B=TRUE; $B$2&":";
IF(C3:C=TRUE; $C$2&":"; ))))
in column F you can enter score as requested and that score is then distributed for each player as requested
then SUMMARY
is calculated based on 4 logic formulas in J3, J4, J5 and K3 which covers for every possible state:
cell J3:
=IF(AND(SUM(G3:G)>=SUM(H3:H); SUM(G3:G)>=SUM(I3:I)); G2&":";
IF(AND(SUM(H3:H)> SUM(G3:G); SUM(H3:H)> SUM(I3:I)); H2&":";
IF(AND(SUM(I3:I)> SUM(H3:H); SUM(I3:I)> SUM(G3:G)); I2&":";
IF(SUM(H3:H)=SUM(I3:I); H2&":"; ))))
cell J4:
=IF(AND(OR(J3=H2&":"; J3=G2&":"); OR(J5=G2&":"; J5=H2&":")); I2&":";
IF(AND(OR(J3=I2&":"; J3=G2&":"); OR(J5=G2&":"; J5=I2&":")); H2&":";
IF(AND(OR(J3=I2&":"; J3=H2&":"); OR(J5=H2&":"; J5=I2&":")); G2&":"; )))
cell J5:
=IF(AND(SUM(G3:G)<=SUM(H3:H); SUM(G3:G)<=SUM(I3:I); G2&":"<>G2&":"); G2&":";
IF(AND(SUM(H3:H)< SUM(G3:G); SUM(H3:H)< SUM(I3:I)); H2&":";
IF(AND(SUM(I3:I)< SUM(H3:H); SUM(I3:I)< SUM(G3:G)); I2&":"; I2&":")))
cell K3:
=SORT({SUM(FILTER($F$3:$F; REGEXMATCH($E$3:$E; G2)))&" points";
SUM(FILTER($F$3:$F; REGEXMATCH($E$3:$E; H2)))&" points";
SUM(FILTER($F$3:$F; REGEXMATCH($E$3:$E; I2)))&" points"}; 1; 0)
of course cells A2, B2 and C2 can be rewritten to match the actual names of players