Eu esperava que eu tivesse ganhado o jogo - ME ou MYSELF?

Blink,

O pronome a ser usado é um 'object pronoun'. Observe a seguir.

I expected you to have won the game.
I expected him to have won the game.
I expected her to have won the game.
I expected us to have won the game.
I expected them to have won the game.


Porém, quando você estiver referindo a você mesmo, não é preciso usar nem "me" nem "myself" com 'expect'. Veja como fica.

I expected to have won the game.

Bons estudos!

Mas não pode usar um pronome mesmo que seja para se referir a mim após o expected?

It sounds much better if you say:

I expected (that) I would win the game.
I expected (that) he/she/you/they would win the game.

This form is also implicitly about the past, and all we need is the infinitive:

I expected to win the game.
I expected you/him/them/etc. to win the game.

For a present expectation, looking forward:

I expect to win the game
I expect I/you/they/etc. will win the game.

Mas não pode usar um pronome mesmo que seja para se referir a mim após o expected?

Usar o pronome da forma sugerida na pergunta faz com que a sentença não fique natural. Não me recordo de "me" ou "myself" serem usados com "expect" dessa forma.

So, which seem right? (I have two personal favorites.)

1. I expected someone to have argued with me about my earlier post.
2. I expected someone to argue with me about my earlier post.
3. I expected someone would have argued with me about my earlier post.
4. I expected someone would argue with me about my earlier post.