In the National Geographic series "Jesus: Rise To Power", it seemed as if some Christians actively sought out martyrdom. One anecdote it mentioned, featuring Arrius Antoninus, is also mentioned in Wikipedia:
Some early Christians sought out and welcomed martyrdom. Roman authorities tried hard to avoid Christians because they "goaded, chided, belittled and insulted the crowds until they demanded their death." According to Droge and Tabor, "in 185 the proconsul of Asia, Arrius Antoninus, was approached by a group of Christians demanding to be executed. The proconsul obliged some of them and then sent the rest away, saying that if they wanted to kill themselves there was plenty of rope available or cliffs they could jump off." Such seeking after death is found in Tertullian's Scorpiace or in the letters of Saint Ignatius of Antioch but was certainly not the only view of martyrdom in the Christian church. Both Polycarp and Cyprian, bishops in Smyrna and Carthage respectively, attempted to avoid martyrdom. (emphasis added)
It also mentioned a woman who was supposed to be killed by a gladiator, but the gladiator refused to kill her, so she pulled the sword on her neck.
I know that martyrdom is a concept in Christianity, but I didn't think that Christians sought out martyrdom. Is there any evidence showing that Christians did seek out martyrdom during the Roman empire?