An interesting game theory problem in Cosmic Encounter

Written as post five of #100DaysToOffload.

Cosmic Encounter is my favorite board game, full stop. The other week I nerd-sniped myself playing it. Here is a simplified game-theoretical description of the problem I ran into.

You and your opponent each secretly select and simultaneously reveal a card with a number on it ranging between 0 and 40. The higher number wins. Before cards are revealed, you may do one or both of the following:

• Declare another positive number X. If the revealed cards differ by X or more, you may add X to the number on your card.
• If your opponent’s card shows at least 15, you may divide it by three and round up.

1. For which numbers can you force a win?
2. What if you have a fixed advantage or disadvantage of up to eight? Advantages are added to your number after all other actions are performed.
3. For which numbers can you force a loss?