What does it mean to persuade, to advocate, in mediation?
Who are you persuading? The other party.
A mediation brief is a unique opportunity to reach a litigant directly and to lay out the reasons why they ought to seriously consider a negotiated resolution. It is a valuable opportunity to tie together all the threads of a long-running case.
That is why it is critical to deliver your brief with enough time for the other party(ies) to read it before the mediation, and make sure your client reads the other side's brief too. This allows everyone to arrive at mediation ready to explore the opportunities for settlement, rather than spending the first half of the mediation being outraged by one another's positions.
When drafting a mediation brief, think about the litigant on the other side. What might they need to hear? What new information has arisen since the litigation began? Is there evidence on the record that will be very helpful or harmful to one side's case (a hyperlink to a transcript reference is absolute gold here)?
Writing for a person with a personal stake in litigation is different than persuading a trier of fact. Accept that you probably will not convince the other side of your position. I rarely hear a party say "yes, they have a point, I see that I might be wrong" after hours of back and forth. Remember this quotation by Dale Carnegie: "Those convinced against their will are of the same opinion still." That is why a mediation brief should also contain some reference to what a party is willing to discuss, or do, for the purposes of finding a resolution.
If you can use your mediation brief to persuade the other side, while leaving the door open to compromise, you will be well on your way to successfully advocating at mediation.
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