A game has been invented that involves

“A game has been invented that involves a player’s directing a ball through four arches- 1,2,3, and 4 in that order.”
what is the meaning of the sentence above?

That sounds like a description of the game of croquet to me.
Which part of the sentence are you having a problem with?

“A game has been invented that involves a player’s directing a ball through four arches- 1,2,3, and 4 in that order.”
Underlined part.

Hmmm, yes, that’s just poor English. It should be one of these:
A game has been invented that involves players directing a ball through…
A game has been invented that involves a player directing a ball through…

Is that understandable now?

“A game has been invented that involves a player’s directing a ball through four arches- 1,2,3, and 4 in that order.”
What is the meaning here of the underlined word ?

A game has been invented that involves a player directing a ball through four arches.

directing - guiding (controlling the direction of the ball so that the ball passes through the arches in order).

In winning its bitter, protracted battle to acquire blue industries, Inc., Belle Industries has fulfilled its goal to lessen its reliance on tobacco holdings, while the tk 5.2 billion deal may spur more takeover activity in the insurance industry, analysts said.

What is the meaning of the above sentence?

Belle Industries entered into a long, drawn out bid to take over Blue Industries.
Eventually they managed to do so.
By taking over the other company they were able to be less reliant on tobacco holdings (presumably because Blue Industries were involved in tobacco holdings, so they no longer had to outsource).
Belle Industries paid out tk 5.2 billion to acquire Blue Industries.
Industry analysts now think that such a large deal may encourage other insurance companies to merge.

In winning its bitter, protracted battle to acquire blue industries, Inc., Belle Industries has fulfilled its goal to lessen its reliance on tobacco holdings, while the tk 5.2 billion deal may spur more takeover activity in the insurance industry, analysts said.

Here, Belle Industries relied on tobacco holding much than do now(after taking over blue Industry). So due to less reliant on tobacco company Belle Industry can spur more(More means same amount was paid out at the time of acquiring blue Industry) tk 5.2 billion to acquire another Industry.

Am I correct?

I don’t think so, no.

It just means that seeing how much Blue Industry sold for may encourage others to buy/sell different companies. The actual amount is not relevant.

What does “In winning its bitter” suggest?

You have parsed that incorrectly. ‘In winning its bitter’ is not the full phrase.
You need to consider:
“In winning its bitter, protracted battle”

‘Bitter’ and ‘protracted’ are adjectives describing the battle.

What is [b]Inc./b suggest? Through I know the meaning of the word I can’t find the synchronization with the other part of the sentence?

‘Blue Industries, Inc.’ is the full legal title of the company, ‘Blue Industries’. The ‘Inc’ is a legal addition to the name of an incorporated company.

Your question should read, ‘What does Inc. (Incorporated) suggest?’