When a randomized algorithm elects a leader on anonymous networks, initial information (which is called in general initial condition in this paper) of some sort is always needed. In this paper, we study common properties of initial conditions that enable a randomized algorithm to elect a leader. In the previous papers, the author introduced the notion of transformation between initial conditions using distributed algorithms. By using this notion of transformation, we investigate the property of initial conditions for the leader election. We define that an initial condition C is p(N)-complete if there exists some randomized algorithm that elects a leader with probability p(N) on any size N network satisfying C. We show that we can divide p(N)-completeness into four types as follows. 1. p(N)=1: For any 1-complete initial conditions, there exists a deterministic distributed algorithm that can compute the size of the network for any initial information satisfying the initial condition. 2. inf p(N) >0: For any p(N)-complete initial conditions with inf p(N) >0, there exists a deterministic distributed algorithm that can compute an upper-bound for the size of the network for any initial information satisfying the initial condition. 3. inf p(N) converges to 0: The set of p(N)-complete initial conditions varies depending on the decrease rate of p(N). 4. p(N) decreases exponentially: Any initial condition is regarded as p(N)-complete.

We call a network an anonymous network, if each vertex of the network is given no ID's. For distributed algorithms for anonymous networks, solvable problems depend strongly on the given initial conditions. In the past, initial conditions have been investigated, for example, by computation given the number of vertices as the initial condition, and in terms of what initial condition is needed to elect a leader. In this paper, we study the relations among initial conditions. To achieve this task, we define the relation between initial conditions A and B (denoted by A B) as the relation that some distributed algorithm can compute B on any network satisfying A. Then we show the following property of this relation among initial conditions. The relation is a partial order with respect to equivalence classes. Moreover, over initial conditions, it induces a lattice which has maxima and minima, and contains an infinite number of elements. On the other hand, we give new initial conditions k-LEADER and k-COLOR. k-LEADER denotes the initial condition that gives special condition only to k vertices. k-COLOR denotes the initial condition that divides the vertices into k groups. Then we investigate the property of the relation among these initial conditions.