Ukraine might win

Putin's invasion isn't going to plan

Here's a map of the Russo-Ukrainian war situation, as of 26 Feb, from Wikipedia:

Putin's invasion isn't happening as fast as he anticipated. He hoped to win a mostly bloodless victory like the conquest of Crimea in 2014. Even if they didn't get that, the Russians wanted a quick victory, decapitating the Ukrainian leadership in Kyiv, after which they hoped Ukrainian morale would fall and the Ukrainian army would surrender. That doesn't appear to be happening; from what I can gather from the internet, the Russian advance seems to be slower than Putin expected. Ukrainians have rallied round their nation and are preparing to fight bitterly to defend it.

The longer Ukraine can hold on, the more international opinion will go their way (meaning increased outside support) and the higher their morale will be. Conversely, the more Russian loses rise, the slower their advance, the lower their morale will be.

What help might Ukraine get?

Ukraine is already receiving anti-tank missiles and short range anti-aircraft missiles, plus things like fuel and ammunition that are necessary to fight a war. This will no doubt continue.

The West might also arm Ukraine with longer-range anti-aircraft missiles. The short range ones have a vertical range of about 6 km, and modern aircraft can fly up to 20 km high, so short range missiles cannot prevent Russian air superiority (they can, however force them to fly higher, which reduces their accuracy at ground attack, and makes them more vulnerable to longer-range missiles). Since it takes time to train people to use missile systems, these missiles will possibly be crewed by Western volunteers.

And even greater help NATO could give to Ukraine would be air support, for example enforcing a no-fly zone over Ukrainian territory. NATO countries don't want to do this at present, but the longer the war lasts, the more likely they will be to give further support to Ukraine. Even if the no-fly zone is imperfectly enforced, it will still help Ukraine, as they will find it much easier to move their units across the country.

Russia will of course make threats (including of using nuclear weapons), to prevent NATO from implementing a no-fly zone, but at some point the West will realise that if every time Putin threatens nukes the West backs down, then Putin will always win, and will conclude that they have to ignore Putin's threats.

In game theory terms, if every time Putin says "nukes" the West backs down, then Putin will always say "nukes", the West will always back down, and Putin will always win; because the West doesn't want Putin to win, it follows that they need to not back down.

NATO countries don't want to give Ukraine air support at the moment, but the longer the war drags on, the more likely they are to do so.

Another thing the West might do, if they are clever (OK I admit I'm being far-fetched here to suggest the West's leaders might do something clever, but bear with me), is to offer EU and/or US citizenship to Russian soldiers who surrender. Coupled with already low morale this might lead to mass desertions within the Russian army. Even if it doesn't, it will result in Russian soldiers being more closely monitored, which will limit their tactical employment and reduce their operational efficiency. It's really hard to win a war with an army that doesn't want to fight facing a determined opponent.

Sanctions

As well as military help, there will be sanctions. These won't affect the military situation in the short term, but they might affect it in the long term by lowering Russian morale and harming their economy. This might persuade the Russian military or the oligarch class that Putin's a loser, and they may decide to get rid of him.

If the Russian offensive gets bogged down...

If the Russian offensive becomes bogged down, with mostly static front lines, Russian morale plumeting, and mass desertions of Russian soldiers, Putin is in big trouble.

The last time Russian soldiers decided they didn't want to fight an unpopular war, the then Russian leader, Tsar Nikolai II, was deposed and later executed. That might not happen to Putin, but at the very least he will suffer massive lost of face, with his country a pariah state universally reviled and cut off from the world economy. A larger version of North Korea.

Conclusion

I am not certain that Putin will win his war in Ukraine. It's a long shot, but I think there is a significant probability that large amounts of Ukrainian territory will still be in Ukrainian control, and supplied by the West, in a month's time.

Quantifying that:

I think there is a 10% probability that on 26-Mar-2022, the Ukrainian armed forces control >60,000 km^2 (about 10%) of Ukrainian territory, and are being supplied by NATO countries.

If you want to bet on this outcome (using play money) I’ve put the question on Manifold Markets.

Edit 1-Mar-2022: Ukrainian defence minister Oleksii Reznikov is now offering Russian soldiers who defect “full amnesty and 5 million rubles”.