The captain won the toss. He elected to bat first.
Having won the toss the captain elected to bat first.
Winning the toss the captain elected to bat first.
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Having won the toss, the captain elected to bat first.
Is correct because the captain cannot elect to bat until (s)he has won the toss. You should place a comma before a conjunction introducing an independent clause.
The Captain may 'want' to bat first, but (s)he can only make that decision after (s)he has won the toss. The other team may win the toss and take the decision to bat...
Winning the toss, the captain elected to bat first.
This sentence as-is is wrong, and becomes clearer if you try to separate the two clauses, only the second makes sense in isolation: "Winning the toss...(?)" "The Captain elected to bat first." The dependent clause is incomplete.
It could be correct if phrased like this:
Winning the toss enabled (or allowed) the captain to bat first. [no comma...]
The act of winning the toss gave the Captain the opportunity to choose.