Ratings and Cooperative Information Transmission

Forthcoming at Management Science

The sender and receiver agree on a message function m:[0,1]->[0,1]. The sender privately observes q, which is uniformly distributed on [0,1]. She sends the receiver the message m(q). The receiver receives the message m_tilde = m(q)+e, where e is distributed on [-e_bar,e_bar] according to the PDF f. She then takes an action A(m_tilde). The sender and receiver incur the cost ((q-A(m_tilde))^2)*I(q) where I:[0,1]->[0,1] is the importance function.

We look at two dimensions: the importance function and the error distribution. In one set of analyses, we assume uniform importance and non-uniform error. In the other, we assume non-uniform importance and uniform error.

Non-Uniform Error, Uniform Importance

I(q)=1 and f is the PDF of a random variable that is distributed according to a quadratic distribution on [-e_bar,e_bar]. We require that (1) f is symmetric about zero and (2) f integrates to one on [-e_bar,e_bar], which leaves one free parameter we denote by b.

from constant_I import Message

B = [-.2,-.1,0.,1.,2.,4.]
while B:
    n = len(B)
    _b = B.pop()
    m = Message(M=100,N=4,b=_b)
    m.plot_msg("msg" + str(n) + ".pdf",title=False)
    m.plot_err("err" + str(n) + ".pdf",2.,6.,title=False)

Uniform Error, Non-Uniform Importance

f(e)=1/(2*e_bar) and I is specified below.

from constant_f import Message

# importance function(s)
I = {
    'i1' : lambda x: x**3., 
    'i2' : lambda x: x**(-1.5),
    'i3' : lambda x: (6.*(x-.5)**2.+.5)**3.
    }

# for each importance function, plot discrete messages of size 5 and 20
for i in I:
    for n in [5,20]:
        m = Message(n,I[i])
        m.plot_msg("msg" + i + str(n) + ".pdf",title=False)