Not really sure what you are attempting to do. AFAIK only the cell that calls the UDF can be updated by the UDF. Other cells cannot be updated from the UDF
However, if passing cell references as the arguments to the UDF then the cell references must be ranges, not strings. You can then assign the values from the ranges to string variables as per the following. (Or you could use rng1.value, rng2.value etc directly without first assigning to string variables.
The following example does not update another cell because that cannot be done.
Note: In my regions we use commas (not semicolons) as delimiters between the the arguments so you will need to edit these.
Function GameSet(rng1 As Range, rng2 As Range, rng3 As Range) As String
Dim strPlayer1 As String
Dim strPlayer2 As String
Dim strLoser As String
Dim Temp As String
strPlayer1 = rng1.Value
strPlayer2 = rng2.Value
strLoser = rng3.Value
If strPlayer2 = "Bye" Then
Temp = strPlayer1
strLoser = strPlayer2
End If
GameSet = Temp
End Function