What is the Greek equivalent of “Skin in the Game”?

Tough one, I agree. And it turns out OP was after Ancient Greek.

For Modern Greek, I agree with Yiannis Papadopoulos’ answer to What is the Greek equivalent of “Skin in the Game”? that “skin in the game” is about emotional investment, not “putting your ass on the line”. It’s easier for me to actually think of negative rather than positive statements.

• Positive: τον κόφτει “it cuts him”—using an archaic form of the verb “cut”, rather than the normal modern form κόβει.
• Negative: αμέτοχος “unengaged, unparticipating” (learnèd); δεν τον κόφτει “it doesn’t cut him”, πέρα βρέχει “it rains far away” (i.e. he’s indifferent), ζαμανφού “I don’t care” [archaic slang: this is of course just French je m’en fous, “I don’t give a fuck”]

Plato?

• τῆς δὲ ζημίας μεγίστη τὸ ὑπὸ πονηροτέρου ἄρχεσθαι, ἐὰν μὴ αὐτὸς ἐθέλῃ ἄρχειν (Republic 347c): But the chief penalty is to be governed by someone worse if a man will not himself hold office and rule.