We define a hierarchy of normal fine measures \mu_\alpha on some set
X_\alpha and discuss the consistency strength of the theory (T_\alpha) =``AD^+ + there
is a normal fine measure \mu_\alpha on X_\alpha." These measures arise naturally from
AD_R, which implies the determinacy of real games of fixed countable length. We
discuss the construction of measures \mu_\alpha on X_\alpha from AD_R (in this
context, \mu_0 is known as the Solovay measure). The theory (T_\alpha) is strictly
weaker than AD_R in terms of consistency strength. However, we show that (T_\alpha) is
equivalent to the determinacy of a certain class of long games with
\utilde{\Pi^1_1}-payoff (and <\omega^2-\utilde{\Pi^1_1}-payoff).