Why did David allow Ziba to inherit Saul's possession after Ziba apparently had lied to him?

Samuel 16:

3 And the king said, And where is thy master's son? And Ziba said unto the king, Behold, he abideth at Jerusalem: for he said, To day shall the house of Israel restore me the kingdom of my father.

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It makes no logical sense that Mephibosheth would say that. Absolom would not just hand the kingdom over to Mephibosheth without a fight and Mephibosheth wasn't in a position to fight.

4 Then said the king to Ziba, Behold, thine are all that pertained unto Mephibosheth.

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Despite the nonsense, David believed him. David likely had known something about Mephibosheth's character that we were not told. E.g., Mephibosheth had some tendency to utter nonsense at times without thinking.

And Ziba said, I humbly beseech thee that I may find grace in thy sight, my lord, O king.

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Ziba was taking a risk here. Absalom could have taken over the kingdom. So, there was some good in what Ziba did in the eyes of David.

After Absalom was defeated, 2 Samuel 19:

24 And Mephibosheth the son of Saul came down to meet the king, and had neither dressed his feet, nor trimmed his beard, nor washed his clothes, from the day the king departed until the day he came again in peace.

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The above is likely to be true. Still, Mephibosheth could have shown up with Ziba in the first place if he had insisted. Mephibosheth might have worried about David being killed by Absolom.

25 And it came to pass, when he was come to Jerusalem to meet the king, that the king said unto him, Wherefore wentest not thou with me, Mephibosheth? 26 And he answered, My lord, O king, my servant deceived me: for thy servant said, I will saddle me an ass, that I may ride thereon, and go to the king; because thy servant is lame. 27 And he hath slandered thy servant unto my lord the king;

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To some extent, the above is true. It wasn't the complete truth.

but my lord the king is as an angel of God: do therefore what is good in thine eyes. 28 For all of my father's house were but dead men before my lord the king: yet didst thou set thy servant among them that did eat at thine own table. What right therefore have I yet to cry any more unto the king? 29 And the king said unto him, Why speakest thou any more of thy matters? I have said, Thou and Ziba divide the land.

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David was a smart guy. He judged that both Ziba and Mephibosheth had told some truth and lies. He wasn't interested in exactly what portions. He had enough data to arrive at the judgment without further investigation. He was thinking in probabilistic terms.

Why did David allow Ziba to inherit Saul's possession after Ziba apparently had lied to him in 2 Samuel 16:3-4?

David ordered Saul's possession to be divided between Ziba and Mephibosheth because both of them had lied to David to some extent and Ziba had shown more loyalty to David but David had made a solemn promise to Jonathan, father of Mephibosheth. Taking all these factors into consideration, David had made a wise judgment.