Given an automaton \(A\), consider the following two games:

**Game I.** At each turn \(i\):

- Adam plays a letter \(a_i\)
- Eve plays a transition \(r_i\) over \(a_i\)

A play is the infinite word \(w=a_0a_1\ldots\) and the sequence of transitions \(r=r_0r_1\ldots\) Eve wins if either \(w\) is not in \(L(A)\) or \(r\) is an accepting run over \(w\) in \(A\).

**Game II.** At each turn turn \(i\):

- Adam plays a letter \(a_i\)
- Eve plays a transition \(r_i\) over \(a_i\)
- Adam plays a pair of transitions \(s_i\) and \(t_i\)

A play is the triple of sequences of runs \(r=r_0r_1\ldots, s=s_0s_1\ldots\) and \(t=t_0t_1\ldots\) Eve wins if either \(r\) is an accepting run of \(A\) or both \(s\) and \(t\) are not accepting runs of \(A\).