What role does pride play in the deaths of Mercutio and Tybalt?

Pride is one of the major reasons for the deaths of Mercutio and Tybalt. In Act I, Scene 5, Tybalt's pride is hurt when Lord Capulet stops him from challenging Romeo at the party. Tybalt overhears Romeo, who is wearing a mask, talking about Juliet, and becomes enraged, calling for his sword. Lord Capulet intercepts him and refuses to allow him to engage Romeo and even says that Verona speaks of Romeo as a "well-governed youth." This obviously does not satisfy Tybalt, who is not a patient man and believes that the "intrusion" will only cause him to become angry again later:

Patience perforce with willful choler meeting
Makes my flesh tremble in their different greeting.
I will withdraw, but this intrusion shall,
Now seeming sweet, convert to bitt’rest gall.

Later in Act II, Benvolio reports that Tybalt has sent a letter to Romeo's house demanding satisfaction and challenging Romeo. Meanwhile, Romeo is preparing to marry Juliet, Tybalt's cousin, setting up the events of Act III, Scene 1.

In this scene it is Mercutio whose pride works against him. Despite Benvolio's warnings, Mercutio remains in the street when the Capulets enter. Tybalt asks about Romeo, which only draws insults from Mercutio as he flourishes his sword. When Romeo shows up, because he has just been married, he immediately backs down to Tybalt and even tells him that he loves him:

I do protest I never injured thee
But love thee better than thou canst devise
Till thou shalt know the reason of my love.
And so, good Capulet, which name I tender
As dearly as mine own, be satisfied.

Mercutio cannot take it. His pride is hurt as he believes that his best friend is backing down to a hated Capulet. He promptly steps into the fray and, noting Romeo's seeming cowardice, challenges Tybalt to a fight:

O calm, dishonorable, vile submission!
Alla stoccato carries it away. [He draws.]
Tybalt, you ratcatcher, will you walk?

As the two fight, Romeo jumps into the middle and Mercutio is fatally wounded. Even the wound, however, does not stop Mercutio from demonstrating his ability with words and his pride as he condemns the Montagues and Capulets:

I am hurt.
A plague o’ both your houses! I am sped.

Following the stabbing, Tybalt flees, but pride seemingly brings him back to the scene and he and Romeo fight, with Tybalt falling. Had Mercutio and Tybalt listened to Benvolio's warning to either withdraw or "reason coldly," they would have survived. Instead, they let their masculine pride overcome common sense.