Fujimoto, Takao

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Consensus Formation between Two Experts:More Theorems and a Discrete CaseTakao Fujimoto1. IntroductionDe Groot(1974) obtained a convergence theorem among n expertsusing a linear homogeneous model. And this result on averaging processesis the starting point of the following researches on consensus formation.Based on a linear inhomogeneous model, Chatterjee(1975) and Chatterjeeand Seneta(1977) generalized the result due to De Groot. Recently Krause(1999) employed nonlinear inhomogeneous models, and established someconvergence theorems. In his proofs, a sort of contraction mappingtheorem is used. In a sense, Krause(1999) showed that the condition,which assures less discrepancy in each step of negotiation, can guaranteea path converging to consensus.Fujimoto(1999) took up a nonlinear inhomogeneous model, which issimilar to a nonlinear Leontiefmodel discussed in Fujimoto(1986). A pathof bargaining is either monotonically increasing or decreasing, and it iseasy to show the convergence given a certain limit. Then Ekuni andFujimoto(2000) considered a model of consensus formation between twoexperts. It is shown that under a set of very weak conditions, two expertscan reach a quasi-consensus state, thanks to the Poincare-Bendixson-193-688theorem.In this note, we give some more theorems using the same model as inEkuni and Fujimoto(2000), and also discuss the discrete adjustment casefor a model of two experts. In Section 2, the continuous version of ourmodel is presented, then some new propositions are presented. Section 3explains a model in which adjustment is made in a discrete way. A mainresult for the discrete case is stated in Section 3. In the final Section 4,some remarks are given.2. The Continuous AdjustmentWe repeat the definitions and assumptions in Ekuni and Fujimoto(2000). The symbol R 2 means the Euclidean space of dimension two, andR] the nonnegative orthant of R 2. The evaluations of two experts areexpressed by a vector in R]. The adjustment of their opinion is made astime goes on following the differential equation:. ~ fi 2x=-=f(x) orxER+,dtwhere t shows time, and a given function f(x) == (f1(x),/2(x»' maps x E R]to a vector in R 2 , and is possibly nonlinear. (A prime indicates thetransposition of a vector concerned.) The function f(x) is continuouslydifferentiable on R] with respect to Xl and X2 of x == (Xl, X2)'. Besides wemake the following assumptions.Assumption I. IfXl :f: X2 for X E R], then f(x):f: O.Assumption 2. If Xi = 0, i = 1 or 2, then fi (x) ? O.Assumption 3. There exists a positive scalar N such that ifo~ Xl =X2 < N for X E R], then f(x) = o.-194-Consensus Formation between Two Experts: More Theorems 689and a Discrete CaseAssumption 4. There exists a positive scalar M such that if Xi > M,i = 1 or 2 then fi (x) < O.Here are made two more definitions.Definition 1. The set of equilibrium points, E, is {x If(x) = o,x E Rl}.Definition 2. Let e be an arbitrary positive scalar. The set of e-equilibrium, QE, is defined as {x I 3y E E such that Ix - y 1< e, x E Rl}.In this section, we use one more symbol :s == {x IXl < M, X2 < M ,x E Rl}.Our Assumption 3 guarantees the nonemptiness of E, whileAssumption 1 requires that at least one expert modifies his / her opinionwhen the two have different estimates, prohibiting the existence ofequilibrium on the off-diagonal points. In Assumption 1, no compromiseis postulated, but simply a revision.What is shown in Ekuni and Fujimoto(2000) isProposition 1. Given Assumptions 1-4, the adjustment process with itsinitial vector anywhere in Rl reaches a point ofthe e-equilibrium set.This tells us that after a certain time the adjustment process entersthe set S, and QE as well, but may leave the latter later. We can,however, make a supplementary rule that the adjustment process shall beended when the discrepancy of the two is less than e.=dx = {/(X) for x E (R;- QE),dt 0 for x E QEThe problem associated with the above convergence to the set QE isthat we do not know how long it takes to reach there. Otherwise thisProposition is rather interesting because two experts do not have to make-195-690concessions in each step, i. e., they need not try to make the differencesmaller.Now we can state other results by adding more assumptions. TheJacobian matrix at each point of E has two eigenvalues associated withtwo eigenvectors. One of them is zero with its associated eigenvector(1, I)'. The second eigenvalue may be positive, zero, or negative. FromProposition 1, it is clear that not all the points in E have the secondeigenvalue positive. Otherwise Proposition 1 is impossible. Thenwe haveProposition 2. Given Assumptions 1-4, if all the equilibrium points havethe second eigenvalue negative, then the adjustment process converges toan equilibrium point.Proof. Since the set E is compact, the scalar e can be chosen so that ineach e-neighbourhood, the second eigenvalue with its eigenvector isforcing the adjustment process to the diagonal. The distance with thediagonal is 'strictly decreasing'. By Proposition 1, after a certain time, theadjustment process enters the set QE. Hence, the process converges to anequilibrium point. (We need a more detailed discussion at the end pointsof E.) QED.Proposition 2 amounts to saying that if each equilibrium point islocally stable, the equilibrium set E as a whole is globally stable.As a conjecture, we stateProposition 3. Given Assumptions 1-4, the adjustment processconverges to an equilibrium point.As a subset of QE , we consider the set SQE defined asSQE == {x Ix E QE, the second eigenvalue is negative or zero}.Using an argument similar to that used in the proof of Poincare--196-Consensus Formation between Two Experts: More Theorems 691and a Discrete CaseBendixson, we find that the direction of movement is the same near thediagonal in the set E. The direction along the diagonal becomesimpossible as the orbit approaches the diagonal in E. The remainingpossibility is a saddle point at an equilibrium where the secondeigenvalue is also zero. This is, however, also impossible because the firsteigenvalue is zero.3. The Discrete AdjustmentIn this section we consider a discrete adjustment process:x(t + 1) = max(x(t) +f(x(t »,0).The function f(x) is continuous on R1. We make the same Assumptions 1,2, and 3 as in Section 2. Without losing generality we suppose the point(l/2, 1/2)' is in the equilibrium set E. Two more assumptions are madetogether with two symbols.Assumption 4 D. The function f(x) is homogeneous, i. e.,f(kx) = s(k) -j(x),where k is a positive scalar, and s(k) is a continuous function of k forwhich there exists a positive scalar M such thats (k) > 0 for k < M, and s (k) < 0 for k > M.Definition 3. The map P is the projection from R1- {O} to the unitsimplex-197-692defined asAssumption 5. The function f(x) satisfies the condition that if theprocess jumps over the diagonal, then the distance between the diagonaland Px(t) becomes smaller.Proposition 4. Given Assumptions 1-5 with Assumption 4 replaced by 4D, the adjustment process converges to an equilibrium point.Proof. When we project any movement to the unit simplex, thatmovement should be directed toward the diagonal. Otherwise, therewould be a point on U such that the movement along the simplexvanishes because of Assumption 2. By Assumption 4 D, this implies theexistence of equilibrium on the off-diagonal area, contradictingAssumption 1. Thus thanks to Assumption 5, the distance between thediagonal and the point projected Px(t) monotonically decrease irrespectiveofjumps over the diagonal. QED.4. DiscussionWhat is shown in Ekuni and Fujimoto(2000), here presented asProposition 1, is that when at least one of the two experts is willing tochange his / her opinion if two have different views, the continuousadjustment process will bring the two to a quasi-equilibrium point, i.e., apoint where the discrepancy between two estimates can be smaller thanany preassigned magnitude. The trouble with this result is that later inthis process the discrepancy gets larger again, and we do not know how-198-Consensus Formation between Two Experts: More Theorems 693and a Discrete Caselong it takes for the process to go into the quasi-equilibrium set.Proposition 2 is of interest when we examine the adjustment processin the previous results for n experts. When the situation is limited to twoexperts, we can calculate the second eigenvalue of the Jacobian matrix onthe equilibrium set.Proposition 3 seems to be valid as it stands. A counter-example, ifany, should include an unnatural adjustment process as a negotiationbehaviour, and can be excluded by an additional 'plausible' assumption.The discrete cases present many challenging problems, which I willdiscuss in the future. The one that is dealt with in this note seemsuninteresting because a kind of homogeneity is assumed, and Assumption5 can be restrictive. This homogeneity can, however, be weakened to aconsiderable degree. Moreover, Assumption 5 may be natural in manysituations. The relative evaluation between two experts can be reversedonly when their relative difference gets decreased.One more case we should take up is the one where the space itself isdiscrete: for example, only the lattice points in R1 are the points of statevectors of which the estimates ofthe two experts can take the value.References[I] Chatterjee, S.(1975). "Reaching a Consensus: Some Limit Theorems", Proc. Int.Statist. Inst., pp.159-164.[2] Chatterjee, S. and E. Seneta(1977). ''Toward Consensus: Some ConvergenceTheorems on Repeated Averaging", J. Appl. Prob. 14, pp.89-97.[3] De Groot, M. H.(1974). "Reaching a Consensus", J. Amer. Statist. Assoc. 69, pp.118-121.[4] Ekuni, Fumiko and T. Fujimoto(2000). "Consensus Formation between TwoExperts", Theses, Shikoku-Gakuin, pp.167-172.[5] Fujimoto, T.(1986). "Non-Linear Leontief Models in Abstract Spaces", J. Math.Econ. 15, pp.151-156.-199-694[6] Fujimoto, T.(1999). "A Simple Model of Consensus Formation", Okayama EconomicReview, 31, pp.95-100.[7] Krause, U.(1999). "A Discrete Nonlinear and Non-Autonomous Model of ConsensusFormation", mimeo. Bremen University.[8] Seneta, E.(1980). Non-Negative Matrices and Markov Chains, 2 nd ed., Springer-Verlag, Berlin.-200-

Consensus Formation between Two Experts: More Theorems and a Discrete Case

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