How was this player going to survive the third game?

I don't think the old man lied about his motivation to play, other than needing the money. He genuinely wanted to play kids' games and feel alive and young again. He even says so, that "it's more fun to play than to watch".

He must invariably have accepted that he was not guaranteed to win the third game. But this follows suit with him being on death's door anyway and having little interest in living a slightly longer, boring and terminally ill life.

Just how rich people sometimes shoplift for the sheer thrill of it, not because they can't buy the items, the old man plays the game for the fun and thrill of it, not because he needs to play them to survive.

Note also that he would've been patently incapable of playing the last two games, so it makes sense that he intentionally bowed out during the 4th game, effectively granting a win to someone who had been very nice to him for the most part.

I don't think his plans were to survive. I think he cared less at that point about death. He just wanted to play and enjoy the games.

This is clear from the scene at the end when old man says after getting so much rich there is nothing that brings pleasure. So I am assuming it was his way of having fun.

And as far as the marble game is concerned I think we know he was clearly winning but kept faking that he is losing.

Spoiler

If you watch the episode he was never in danger of losing. All of the other contestant's handcuffs were connected by a padlock. His was not.

If you see carefully when this game finished (when they won) he wan not handcuffed at all, looks like he knew they combination of the lock. And if they loose and found out that the old-man is not hand-cuffed they could have easily changed the rule saying if someone was able to free himself or herself from the rope, they will not be terminated, like they changed the rule when someone was not member of the team they are not terminated.

The other answers may be correct about Player 001's overall mindset, but they miss the point that in this particular game, he knew he was almost guaranteed to win - and he says this to his team at the time.

At first glance, this game looks like it would be extremely dangerous even for someone who was able to prepare and knew what to expect, even for someone who had organised the games himself. The deaths of the players on each losing team are clear, visible, and unavoidable. In the first two games, if Player 001 had lost, his death could have been faked by a guard firing a blank, as actually happened in the fourth game. In the third game, the very act of losing causes the losers to die, without the guards doing anything.

But you're forgetting how our heroes' team actually wins, even with three women and an old man against a team of stronger men. It's all a matter of strategy, and that strategy comes from Player 001. As he says in the episode itself:

Tug-of-War needs more than just raw strength. All you need is a good strategy in Tug-of-War and combined with good teamwork, you’ll be able to win when you are against stronger teams.

Okay, when I was young, we always like[d] Tug-of-War. It’s a game I know well. And back then, I would always win, even if there was a wrestler on the other team, and it seemed the odds were stuck against us. Listen closely, and I’ll tell you how my team and I were able to win even when it seemed impossible.

First off, having a good leader is very important. The person is at the front and keeps an eye on how the other team is performing. And the rest of the team focuses on the back of their leader’s head and follows their lead. If the leader seems weak or looks like their spirit is beginning to falter, then the game is already over.

And then at the end of the rope, you’ll need to have someone strong and dependable like the anchor of a ship. After that, it’s all about how you arrange the rest of your team. If one player is on the right side of the rope, then the next one should be on the left, all the way down the rope. Both of your feet should be facing straight forward, and then, hold the rope in your armpits. That way, everyone can put in all of their strength.

Finally, and this is absolutely the most important thing. Once the game begins, for the first ten seconds … you have to hold your ground. You should lean back, practically lie down. Push your lower abdomen up to the sky, as hard as you can. And throw your head back to the point where you can almost see the groin of the person behind you. If you do that, the other team won’t be able to pull us to their side. Just hold like this for ten seconds. Then the opposing team will start to get frustrated because they are thinking, “Why won’t they budge?” Because they believed that they were much stronger. If you can hold out for that long, you will be able to catch a moment in your opponents’ hold, and their rhythm breaks.

_{transcript source}

Even without revealing that he's the mastermind behind all the games, he's stating openly that he feels sure of winning this game, that his team always won Tug-of-War even against physically much stronger opponents. It's his strategy and his confidence that enables the whole team to survive. Even without knowing who he really is, it makes sense that his age, although a disadvantage in terms of brute strength, comes with experience which can help him win games. That's also how he could so easily beat Gi-Hun at marbles, even in a game that seemed to be pure chance: it's all about strategy and understanding people.

So he would've known from the start that the first four games weren't a big risk for him. The fifth one would have been - or at least, even if he was told the right way to step, everyone behind him would have survived too, which would've ruined half the point of the game - and that's probably why he eliminated himself in the fourth game rather than earlier or later.

I posted part of this in a comment, but I'm going to expand on it.

All of the games involve an element of chance. Red Light, Green Light, the old man might stumble. In the second game, his cookie could break. In both of these, a faked death would look just that, fake. Given the blood and brains visible from legitimate kills, the lack of such (i.e. being shot with a blank) would be obvious.

Keep in mind, his team in Tug Of War had five weak links; 3 women, himself and Ali, who was missing fingers on one hand. In a 100% legitimate world he was definitely taking a chance, but his strategy was the real winner. Even with an overwhelming number of weak links, his strategy put them in a position to win. On almost any other team, his strategy provides an easy win.

It's possible/probable that after that near miss, he called on the Marble game, since it was much easier to control his destiny in that game than any other. Perhaps after the adrenaline rush of almost losing Tug Of War, he decided it was time to call it quits.