“Penalty Game!” only requires your opponent to have 4 cards in hand when it is activated, not when it resolves. So your opponent can activate “Jar of Greed” to have 5 cards, or chain a Quick-Play Spell Card from his hand and have 3 cards, and “Penalty Game!” will still resolve.
If you choose the first effect, neither player can activate a card that would require the opponent to draw during their next Draw Phase, such as “Jar of Greed” or “Disturbance Strategy”.
If the second effect of “Penalty Game!” is chosen, the opponent can still activate the effects of cards that are already face-up on the field, such as “Skull Lair” or “The Dragon’s Bead”.