  In this paper, we discuss a voting model in which voters can obtain
information from a finite number of previous voters. There exist three groups
of voters:
  (i) digital herders and independent voters, (ii) analog herders and
independent voters, and (iii) $\tanh$-type herders.
  In our previous paper [1], we used the mean field approximation for case (i).
In that study, if the reference number $r$ is above three, phase transition
occurs and the solution converges to one of the equilibria. However, the
conclusion is different from mean field approximation. In this paper, we show
that the solution oscillates between the two states. A good (bad) equilibrium
is where a majority of $r$ select the correct (wrong) candidate. In this paper,
we show that there is no phase transition when $r$ is finite. If the annealing
schedule is adequately slow from finite $r$ to infinite $r$, the voting rate
converges only to the good equilibrium.
  In case (ii), the state of reference votes is equivalent to that of Kirman's
ant colony model, and it follows beta binomial distribution.
  In case (iii), we show that the model is equivalent to the finite-size
kinetic Ising model. If the voters are rational, a simple herding experiment of
information cascade is conducted. Information cascade results from the
quenching of the kinetic Ising model.As case (i) is the limit of case (iii)
when $\tanh$ function becomes a step function, the phase transition can be
observed in infinite size limit. We can confirm that there is no phase
transition when the reference number $r$ is finite.

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