Editing Cheap talk (section)

====Actions====
Since actions are functions of messages, actions are also constant over these intervals:
for ''t<sub>i-1</sub>(N) < t < t<sub>i</sub>(N)'', ''α(t) = α(m<sub>i</sub>) = a<sub>i</sub>''.

The action function is now indirectly characterized by the fact that each value ''a<sub>i</sub>'' optimizes return for the ''R'', knowing that ''t'' is between ''t<sub>1</sub>'' and ''t<sub>2</sub>''. Mathematically (assuming that ''t'' is uniformly distributed over [0, 1]),

<math>a_i = \bar{a}(t_{i-1}, t_i) = \mathrm{arg}\max_a \int_{t_{i-1}}^{t_i} U^R(a, t) dt</math>

→ ''Quadratic utilities:''

Given that ''R'' knows that ''t'' is between ''t<sub>i-1</sub>'' and ''t<sub>i</sub>'', and in the special case quadratic utility where ''R'' wants action ''a'' to be as close to ''t'' as possible, we can show that quite intuitively the optimal action is the middle of the interval:
<math display="block">a_i = \frac{t_{i-1} + t_i}{2}</math>