n
v
Peak in Figure 3.2
20
0.8126
0.8185
50
0.8763
0.8822
200
0.9358
0.9326
500
0.9588
0.9599
Table 3.1: Comparing
v
() peak occurring points in histograms to the values obtained
from Claim 1
Figure 3.3, for different number of agents. We observe that the mean-field approximation
is quite accurate. We next consider p
V CG
().
Claim 2. Under the assumptions in Claim 1, the mean-field approximation for p
V CG
is
p
V CG
n(n - 1)/4 -2 log (n - 1) +
2
(n - 1)
+2 log (n) -
2
(n) ,
and lim
n
p
V CG
() = 1 almost surely.
Proof. We first note that
p
V CG
() =
iN j=i
j
log(1 + a
-i,j
(
-i
)
- (n - 1)
v
().
(3.7)
We may write
v,-i
() =
j=i
j
log(1 + a
-i,j
(
-i
)),
so that when n is large,
v,-i
takes the value of
v
, but with n - 1 agents, i.e.,
v,-i
=
n - 1
4
-2 log (n - 1) - 1 +
2
(n - 1) ,
(3.8)
26
Summary :
(3.7) We may write v,-i () = j=i j log(1 + a -i,j ( -i )), so that when n is large, v,-i takes the value of v , but with n - 1 agents, i.e., v,-i = n - 1 4 -2 log (n - 1) - 1 + 2 (n - 1) , (3.8) 26
Tags :
claim,log,approximation,log1,meanfield,figure,peak,agents,write,quite,takes,09358,different