The original method proposed by DeGroot [27] has some drawbacks. First, the agents might want to change the weights that they assign to their peers’ beliefs after learning their initial beliefs or after observing how much beliefs change from stage to stage. Further, beliefs and/or identities have to be disclosed to the whole group when agents are assigning weights. Hence, privacy is not preserved, a fact which might be troublesome when the underlying event is of a sensitive nature.
In order to tackle these problems, we derive weights that agents assign to the beliefs by interpreting each weight as a measure of distance between two beliefs. We start by
making the assumption that agents prefer beliefs that are close to their own beliefs, where closeness is measured by the following distance function:
D(qi, qj) =
r Pn
k=1(qi,k − qj,k)2
n (6.3)
i.e., it is the root-mean-square deviation between two beliefs qi and qj. Given the above
assumption, one can estimate the weight that agent i assigns to agent j’s belief at a given time t, for t ≥ 1, as follows:
wi,j(t) = α
(t) i
+ Dq(t−1)i , q(t−1)j
(6.4)
where α(t)i normalizes the weights so that they sum to one, and is a small, positive constant used to avoid division by zero. We set q(0)i = qi, i.e., it is the original belief reported by
agent i. There are some important points regarding equation (6.4). First, the distance between two beliefs is always non-negative. Hence, the constant ensures that every single weight is strictly greater than 0 and strictly less than 1. Further, the closer the beliefs q(t−1)i and q(t−1)j are, the higher the resulting weight w(t)i,j will be. Since D
q(t−1)i , q(t−1)i
= 0, the weight that each agent assigns to his own belief is always greater than or equal to the weights that he assigns to his peers’ beliefs. In spirit, the underlying learning model can be seen as a model of anchoring [85] in a sense that the belief of an agent is an “anchor”, and subsequent updates are biased towards beliefs close to the anchor.
Now, we can redefine equation (6.2) so as to allow the agents to update their weights based on the most recent beliefs. After t revisions, for t ≥ 1, we have that Q(t) =
W(t)Q(t−1) = W(t)W(t−1). . . W(1)Q(0), where each element of each matrix W(k) is com-
puted according to equation (6.4):
W(k) = w1,1(k) w(k)1,2 · · · w1,z(k) w2,1(k) w(k)2,2 · · · w2,z(k) .. . ... . .. ... wz,1(k) w(k)z,2 · · · wz,z(k)
The belief of each agent i at time t then becomes q(t)i = Pz
j=1w (t) i,jq
(t−1)
j . Algorithm 1
Algorithm 1 Algorithmic description of the proposed method to find a consensual belief. Require: z probability vectors q(0)1 , . . . , q(0)z .
Require: recalibration factor . Require: number of rounds τ .
1: for t = 1 to τ do 2: for i = 1 to z do 3: for j = 1 to z do 4: w(t)i,j = α (t) i +D q(t−1)i ,q(t−1)j 5: end for 6: q(t)i =Pz j=1w (t) i,jq (t−1) j 7: end for 8: end for
In order to prove that all beliefs converge towards a consensual belief when using the proposed method, consider the following functions:
δ (U) = 1 2maxi,j n X k=1 |ui,k− uj,k| γ(U) = min i,j n X k=1 min(ui,k, uj,k)
where 0 ≤ δ (U) , γ(U) ≤ 1, and U is a stochastic matrix. δ (U) computes the maximum absolute difference between two rows of a stochastic matrix U. Thus, when δ Q(t) = 0, all rows of Q(t) are the same, i.e., a consensus is reached. We use the following results in our proof [63]:
Proposition 3. Given two stochastic matrices U and V, δ(UV) < δ(U)δ(V). Proposition 4. Given a stochastic matrix U, then δ(U) = 1 − γ(U).
We state our main result below.
Proposition 5. When t → ∞, q(t)i = q(t)j , for every agent i and j.
Proof. Recall that Q(t) is the stochastic matrix representing the agents’ beliefs after t
δ Q(0) , δ Q(1) , . . . , δ Q(t)
We are interested in the behavior of this sequence when t → ∞. First, we show that such a sequence is monotonically decreasing:
δ Q(t) = δ W(t)Q(t−1) < δ W(t) δ Q(t−1) = 1 − γ W(t) δ Q(t−1) ≤ δ Q(t−1)
The second and third lines follow, respectively, from Propositions 3 and 4. Since δ (U) ≥ 0, for every stochastic matrix U, then the aforementioned sequence is a bounded decreasing sequence. Hence, we can apply the standard monotone convergence theorem [7] and δ Q(∞) = 0. Consequently, all rows of the stochastic matrix Q(∞) are the same.
To summarize, the sequence δ Q(0) , δ Q(1) , . . . , δ Q(t)
is strictly decreasing, which means that, according to the monotone convergence theorem, it will hit its tight lower bound after a potential infinite number of rounds. Given that the tight lower bound of δ(·) is 0, then the sequence converges to 0.
In words, a consensus is always reached under the proposed method, and this does not depend on the reported beliefs. A straightforward corollary of Proposition 5 is that all revised weights converge to the same value.
Corollary 1. When t → ∞, w(t)i,j = 1z, for every agent i and j.
Hence, the proposed method works as if agents were continuously exchanging informa- tion so that their individual knowledge becomes group knowledge and all beliefs are equally weighted. Since we derive weights from the reported beliefs, we are then able to avoid some problems that might arise when eliciting these weights directly, e.g., beliefs do not need to be disclosed to others in order for them to assign weights, thus preserving privacy.
The proposed method works very similarly to the behavioral aggregation method known as the Delphi method [48], where agents disclose and update their beliefs in a sequence of rounds in order to achieve consensus. In practice, however, consensus is not always
achieved under the Delphi method. By mimicking the Delphi method, our proposed algo- rithm always achieves consensus without requiring agents to explicitly participate in the consensus-reaching process. Hence, our proposed method is of great value whenever a re- quester is interested in a consensual belief, but achieving such a consensus might be costly in practice.
The resulting consensual belief can be represented as an instance of the linear opin- ion pool. Recall that q(t)i = Pz
j=1w (t) i,jq (t−1) j = Pz j=1w (t) i,j Pz k=1w (t−1) j,k q (t−2) k = · · · = Pz j=1βjq (0)
j , where β = (β1, β2, . . . , βz) is a probability vector that incorporates all the
previous weights. Hence, another interpretation of the proposed method is that agents reach a consensus regarding the weights in equation (6.1).
6.2.3
Numerical Example
A numerical example may clarify the mechanics of the proposed method. Consider three agents (z = 3) with the following beliefs: q1 = (0.9, 0.1), q2 = (0.05, 0.95), and q3 =
(0.2, 0.8). According to (6.3), the initial distance between, say, q1 and q2 is:
D(q1, q2) =
r
(0.9 − 0.05)2+ (0.1 − 0.95)2
2 = 0.85
Similarly, we have that D(q1, q1) = 0 and D(q1, q3) = 0.7. Using equation (6.4), we
can then derive the weights that each agent assigns to the reported beliefs. Focusing on agent 1 at time t = 1 and setting = 0.01, we obtain w(1)1,1 = α1(1)/0.01, w(1)1,2 = α(1)1 /0.86, and w(1)1,3 = α(1)1 /0.71. Since these weights must sum to one, we have α(1)1 ≈ 0.00975 and, consequently, w1,1(1) ≈ 0.975, w(1)1,2 ≈ 0.011, and w(1)1,3 ≈ 0.014. Repeating the same procedure for all agents, we obtain the matrix:
W(1) = 0.975 0.011 0.014 0.011 0.931 0.058 0.013 0.058 0.929
The updated belief of agent 1 is then q(1)1 =P3
j=1w (1)
1,jqj ≈ (0.8809, 0.1191). By repeat-
ing the above procedure, when t → ∞, W(t) converges to a matrix where all the elements
are equal to 1/3. Moreover, all agents’ beliefs converge to the belief (0.3175, 0.6825). An interesting point to note is that the resulting belief would be (0.3833, 0.6167) if we had
taken the average of the reported beliefs, i.e., agent 1, who has a very different belief, would have more influence on the aggregate belief.