Is there a word or phrase to describe a state of mind within an arrogant person who has been overwhelmed in a debate by another person who is humble in his/her demeanor, but armed with the truth to the convincing of the majority present. I am looking for something more descriptive than simply embarrassment.
Who can really say what that person feels or even if they'd experience embarrassment if they are so arrogant? Unless they had a sudden epiphany that they were "bested" by someone, regardless of the victor's demeanor, they could be chagrined, dumbfounded, 'put in their place', impressed, surprised and even your word...embarrassed.
Sounds to me like they've been "brought down a peg or two" (slang).
to do something to show someone that they are not as good as they thought they were
OED 'peg' (n.1.II.3) has
a. The interval between two successive positions, such as could be marked by pegs; a step, a degree. Esp. in to take (a person) down a peg (or two) and variants: to lower (a person) in his or her own, or the general, estimation; to lower a person's view of his or her own status or ability; to humble, chasten, snub. Also to take (a person) a peg lower . Similarly occas. to come down a peg
In waiting for a further response to my question and reading none, I have decided to answer it myself having compiled information these past couple of days. First, I will pose an additional question to promote the idea that an observer can discern another's thinking in certain circumstances, namely: What is an outward appearance or reaction to an inward feeling? All of us are capable of many and can be discerned by those who can recognize them from experience. I have three examples hereafter:
The Trial of Socrates of 399 B.C. The main accuser was a man named Meletus and according to Plato's account of the trial, Socrates making defense for himself managed to make Meletus appear "dim-witted" or stupid and thusly his accuser would have demonstrated a surprised countenance and probably embarrassment before the court officials and 500 jurors.
Jesus of Nazareth facing his accusers who outwardly demonstrated their arrogance and hatred toward him yet as the Biblical account says they "marveled" on his intelligence and self-control in face of their efforts to trip him up. At his trial, Pilate also marveled as he witnessed Jesus under fire. To marvel is: to be struck with surprise, astonishment or wonder. Wonder is "the emotion excited by novelty, or the presentation of something new, strange extraordinary, or not well understood..." from Webster's New International English Dictionary, 1947.
The account of a prophet named Abinadi- "And they began to question him, that they might cross him, that thereby they might have wherewith to accuse him; but he answered them boldly, and withstood all their questions, yea, to their astonishment; for he did withstand them in all their questions, and did confound them in all their words." from Mosiah 12:19, The Book of Mormon. The key words here are astonishment and confound which are defined as "To stun; to render senseless; to stupefy the mind of, bewilder and lastly to overthrow, respectively. I would welcome other accounts from the academic community.
I think the adjective "humbled" could fit well here but really this all depends subtly on what you're trying to convey about the internal mindset of the loser. IF they still feel arrogant and just bested and have a feeling of anger or resentment or feel they've only been lowered in the eyes of the audience, then "embarrassed", "chagrined" and other options mentioned above would fit better. If as other's mention, they acknowledge the loss and reevaluate the opponent or themselves, then it could be they feel "humbled", "shamed", "chastened" etc.