You're interpreting "I'm morally in the wrong" to mean something like, "Other people will react badly to my actions," in which case I fully agree with you that it would affect my winning.

Actually I am not. I am interpreting "I'm morally wrong" to mean something like, "I made an error of arithmetic in an area where other people depend on me."

An error of arithmetic is an error of arithmetic regardless of whether any other people catch it, and regardless of whether any other people react badly to it. It is not, however, causally disconnected from their reaction, because, even though an error of arithmetic is what it is regardless of people's reaction to it, nevertheless people will probably react badly to it if you've made it in an area where other people depend on you. For example, if you made an error of arithmetic in taking a test, it is probably the case that the test-grader did not make the same error of arithmetic and so it is probably the case that he will react badly to your error. Nevertheless, your error of arithmetic is an error and is not merely getting-a-different-answer-from-the-grader. Even in the improbable case where you luck out and the test grader makes exactly the same error as you and so you get full marks, nevertheless, you did still make that error.

Even if everyone except you wakes up tomorrow and believes that 3+4=6, whereas you still remember that 3+4=7, nevertheless in many contexts you had better not switch to what the majority believe. For example, if you are designing something that will stand up, like a building or a bridge, you had better get your math right, you had better correctly add 3+4=7 in the course of designing the edifice if that sum is ever called on calculating whether the structure will stand up.

If humanity divides into two factions, one faction of which believes that 3+4=6 and the other of which believes that 3+4=7, then the latter faction, the one that adds correctly, will in all likelihood over time prevail on account of being right. This is true even if the latter group starts out in the minority. Just imagine what sort of tricks you could pull on people who believe that 3+4=6. Because of the truth of 3+4=7, eventually people who are aware of this truth will succeed and those who believe that 3+4=6 will fail, and over time the vast majority of society will once again come to accept that 3+4=7.

And similarly with morality.