A suicidal subject's ambivalence is comprised of many elements, including doubt, hope, fear, the self-preservation instinct, guilt, moral conflict, religious proscription, meaningful attachment, worry about survivor(s), unfinished business, and other concerns. Addressing the subject's need to be acknowledged—that he "counts"—may be a fruitful avenue to follow. Consider this approach. "That [name] cares for you means that you count, that you matter to someone, and, when you care for someone, it means that they matter. Caring and being cared about matters. Who cares about you? Who do you care about?"
Negotiators should address the subject's suicidal ideation (thoughts, images, ideas) directly and determine his or her intent to act on those thoughts. A frank airing of the subject's self-destructive ideas and intent may help in the release of feelings and the exposure of ambivalence. Voicing concerns aloud to someone who is willing to listen may, in itself, highlight the subject's ambivalent feelings and shift the balance from a desire to die to a will to live. Also, the use of ginger phrases and "walking on eggs" has long been rejected, as they inadvertently communicate the negotiator's discomfort and fear.
The willingness of a negotiator to listen and to fearlessly explore the subject's despair and suicidality with them is rapport building and becomes the foundation for the next phase: problem solving. Nudging the subject into problem solving means moving him from what is largely an irrational and highly emotional plane to a more rational one. Herein lies the potential opening for a trust-based alliance between the negotiator and the healthy part of the subject to be sparked.
The most desirable and plausible outcome for a negotiator dealing with a specific subject on a specific day is to get the subject to choose to live for the rest of the day. A "contract for safety" between the negotiator and the subject can help by getting the subject to promise to wait until the next day, until after Christmas, until he can talk to his wife or priest, or until some other fixed point. Once this is achieved, a more lasting solution can then be sought for the suicidal subject through professional mental health services and social and family supports.
Of course some subjects will still be lost, despite such intervention. But that's really not the concern for law enforcement. Your goal as a police negotiator is to prevent the subject from using a fellow officer as a tool for killing himself during a specific incident. Succeeding in that effort is not easy. Negotiators can do everything right, and the subject may still force officers to shoot him.