Psalm 24:1 "The earth is the Lord's and everything in it," which nothing for Caesar.
The design of the trap is explained in the account of the incident in the Gospel of Luke (Ch. 20), but elided from the accounts in Matthew (Ch. 20) and Mark (Ch. 12). The spies were sent by the chief priests to "entangle Jesus in speech so as to hand him over to the authority of the governor (viz., Pilate)". The Pharisees and chief priests had received extensive knowledge of Jesus' words and deeds from their network of spies and were convinced that Jesus would condemn paying tribute (Caesar's tax). They didn't think Jesus would endorse Caesar's tax, which would have given them no cause to hand him over to Pilate, who was responsible for the collection of Rome's taxes in Judea. Jesus had previously persuaded tax collectors (i.e., Matthew, Levi, and Zaccheaus) to leave their duties to follow him. He had also equated the business of collecting taxes to prostitution and also told Peter that "the sons are exempt" (Matthew 17:26) from taxes.
Therefore, neither his disciples nor his enemies thought that Jesus would tell them to pay the tax. It would be inconsistent with the character of Jesus to claim that he avoided a simple affirmative response because it would cost him follower. He was no stranger to abandonment (Jn. 6:66) and his message was not one that was intended to pander to the powers that be.
Indeed, Jesus was also wise, spoke the truth, and, many times, left the audience to sit in his sayings. In this instance, if he said, "No, don't give one red cent of tribute (tax) to that thieving, murdering, war mongering, enslaving pretender to my Father's divinity (Tiberius Caesar claimed he was the son of the Divine Augustus) and usurper of God's prerogative as man's only law maker, he would have been made the fool by the chief priests, which was a state of affairs Jesus never would abide.
Jesus response to the question, "should we pay Caesar's tax?" turned the tables on his questioners. Give Caesar what is Caesar's but give God what is God's required them to try to determine who owns what. Those who know Scripture have no doubt, but they obviously didn't know Scripture or they would have seen that Jesus' response condemned Caesar's tax just as their bosses had expected he would but they were dumbfounded and befuddled. When they returned to the authorities and told them what Jesus had said, the chief priests knew he had condemned Caesar's tax with his render-unto-Caesar quip. So they chose to remove Jesus by force. When they took him before Pilate they translated Jesus' response for him. "We found this man perverting our nation (Rome). He forbids us to pay taxes to Caesar...He has been stirring up crowds from Galilee all the way here." So, Pilate crucified him."