using System;

using System.Collections.Generic;

using System.Linq;

public class Program

{

    public static void Main()

    {

        var duplidictionary = new Dictionary<string, List<string>>

        {

            { "first", new List<string> { "1", "2", "3", "4" } },

            { "second", new List<string> { "2", "3", "4", "3" } }

        };

        var uniq = duplidictionary.GroupBy(x => x.Value)

                          .Select(y => y.First())

                          .ToDictionary( x => x.Key, y => y.Value);

        Console.WriteLine(Newtonsoft.Json.JsonConvert.SerializeObject(uniq));

    }

}