Phase transition and information cascade in a voting model
aa r X i v : . [ phy s i c s . d a t a - a n ] J un Phase transition and information cascadeina voting model
M Hisakado ∗ and S Mori † August 29, 2018 *Standard and Poor’s, Marunouchi 1-6-5, Chiyoda-ku, Tokyo 100-0005, Japan † Department of Physics, School of Science, Kitasato University, Kitasato 1-15-1Sagamihara, Kanagawa 252-0373, Japan
Abstract
We introduce a voting model that is similar to a Keynesian beauty contest and analyze itfrom a mathematical point of view. There are two types of voters-copycat and independent-andtwo candidates. Our voting model is a binomial distribution (independent voters) doped in abeta binomial distribution (copycat voters). We find that the phase transition in this system isat the upper limit of t , where t is the time (or the number of the votes). Our model containsthree phases. If copycats constitute a majority or even half of the total voters, the voting rateconverges more slowly than it would in a binomial distribution. If independents constitutethe majority of voters, the voting rate converges at the same rate as it would in a binomialdistribution. We also study why it is difficult to estimate the conclusion of a Keynesian beautycontest when there is an information cascade. ∗ [1] masato [email protected] † [2] [email protected] Introduction
A Keynesian beauty contest is a popular concept used to explain price fluctuations in equity markets.[1]Keynes described the action of rational agents in a market using an analogy based on a fictional news-paper contest. In the contest, entrants are asked to choose a set of the six most beautiful faces fromamong photographs of different women. Those entrants who would select the most popular facewould be then eligible for a prize. A naive strategy would be to choose the most beautiful faceaccording to the opinion of the entrant. Entrants are known as employing such a strategy indepen-dent voters. A more sophisticated entrant, aiming to maximize his/her chances of winning a prize,would try to deduce the majority’s perception of beauty. This implies that the entrant would make aselection on the basis of some inference from his/her knowledge of public perception. Such voters areknown as copycats. To estimate public perception, people observe the actions of other individuals;then, they make a choice similar to that of others. Because it is usually sensible to do what otherpeople are doing, the phenomenon is assumed to be the result of a rational choice. Nevertheless, thisapproach can sometimes lead to arbitrary or even erroneous decisions. This phenomenon is calledan information cascade. [2]Collective herding phenomena in general pose quite interesting problems in statistical physics.To name a few examples, anomalous fluctuations in the financial market [3],[4] and opinion dynamics[5] have been related to percolation and random field Ising model. A recent agent-based modelproposed by Curty and Marsili [6] focused on the limitations that herding imposed on the efficiency ofinformation aggregation. Specifically, it was shown that when the fraction of herders in a populationof agents increases, the probability that herding produces the correct forecast (i.e. that individualinformation bits are correctly aggregated) undergoes a transition to a state in which either all herdersforecast rightly or no herder does.We can observe super-diffusive behaviour in the sense that variance D ( L ) grows asymptoticallyfaster than L (where L is the long memory) in several fields.[7],[8],[9],[10], [11] It is characterized bythe variance D ∼ L α when α >
1. When α = 1, the diffusion of the variance becomes a standardBrownian motion. For example, in the case of daily financial data, L represents the time seriesof data. The past price affects the present price, and the diffusion becomes faster than Brownianmotion. Such phenomena can be attributed to long-range positive correlations. We may observedynamical phase transition (from normal to super-diffusive behaviour).[9] In such a phase transition,correlation plays an important role. Further our voting model shows a similar transition. The herdersmake long-range correlations and display super-diffusive behaviour. Therefore, a majority of votersreach the wrong conclusion.In this paper, we discuss a voting model with two candidates C and C . As mentioned above,we set two types of voters-independent and copycat. Independent voters’ voting is based on theirfundamental values; on the other hand, copycat voters’ voting is based on the number of votes. Inour previous paper, we investigated the case wherein all the voters are copycats.[12] In such a case,the process is a P´olya process, and the voting rate converges to a beta distribution in a large timelimit.[13] Our present model exhibits a scale-invariant behaviour. This behaviour is observed in themixing of the binary candidates. Furthermore, the power law holds over the entire range in a doublescaling limit. This paper is an extension of our previous works.Although our model is very simple, it contains three phases. We believe that it is as adequate asthe percolation and random field Ising models, and that it is useful for understanding phase transitionin several fields. We discuss two specific issues: one is the distribution in votes that appears for amixture of independent and copycat voters and the other is the change in the vote distributions overtime. On the basis of these above mentioned points, we discuss phase transition for information2ascade.The organization of this paper is as follows. In section 2, we introduce our voting model anddefine the two types of voters-independent and copycat-mathematically. In section 3, we calculate thedistribution functions strictly for the special cases-independent voters always vote for either of thetwo candidates; their behavior is not probabilistic. Then, we obtain a solution that is an extensionof the solution given in [14]; in this case, there is no phase transition. In section 4, we discuss moregeneral cases. We use a stochastic differential equation, the Fokker-Planck equation, and a numericalsimulation. In this model, we can observe phase transition at the ratio of copycats to independentsthrough the variance of the distributions. There are three phases. If copycats constitute a majorityor number half of the total number of voters, the voting rate converges more slowly than it wouldin a binomial distribution. If independents constitute the majority of voters, the voting converges atthe same rate as it would in a binomial distribution. This implies that the proportion of copycatsinfluences the results of the voting. The last section presents the conclusions. Figure 1: Demonstration of model.We model the voting of two candidates, C and C . At time t , each candidate has c ( t ) and c ( t ) votes. At the beginning ( t = 1), the two candidates, C and C , have c (1) and c (1) votes,respectively. Hereafter, we omit the time for the initial votes ( c ≡ c (1) and c ≡ c (1)) and define c ≡ c + c . At each time step, one voter votes for one candidate. Voters are allowed see the numberof votes for each candidate when they vote so that they have knowledge of public perception.There are two types of voters-independent and copycat. Independent voters vote for C and C with probabilities 1 − q and q , respectively. Their votes are independent of others’ vote and dependon what they think their fundamental value is. Copycat voters vote for each candidate with the3robabilities that are proportional to the candidates’ votes. If the number of votes are c ( t ) and c ( t )at time t , a copycat voter votes with probability c ( t ) / ( c ( t ) + c ( t )) for C and c ( t ) / ( c ( t ) + c ( t ))for C . Copycat voters’ votes are based on the number of votes.Here, we set the ratio of independent voters to copycat voters as 1 − p and p , respectively. If weset p = 1, this system becomes a P´olya model with c starting elements of type C and c startingelements of type C . In this case, it is well known that the distribution of the voting rate is a betadistribution. As such, this system is a P´olya process doped with a binomial distribution.The evolution equation for a candidate C is P ( k, t ) = p k − c + c + t − P ( k − , t −
1) + p (1 − kc + c + t − P ( k, t − − p ) qP ( k, t −
1) + (1 − p )(1 − q ) P ( k − , t − . (1) P ( k, t ) is the distribution of the number of votes k at time t for candidate C . The first and secondterms of (1) denote the votes of the copycat voters; the third and fourth terms denote the votes ofthe independent voters. If we set Q ( k, t ) as the distribution of the number of votes k at time t forcandidate C , we have the relation Q ( k, t ) = 1 − P ( c + c + t − − k, t ) . (2)The initial condition is P ( c ,
1) = 1. This is the relation between the back and front. q = 1 (or q = 0 ) In this section, we study the exact solution of (1) for a special case. For q = 1, we obtain the followingmaster equation: P ( k, t ) = p k − c + t − P ( k − , t −
1) + p (1 − kc + t − P ( k, t − − p ) P ( k, t − . (3)At this limit, independent voters always vote for only one candidate, C (if we set q = 0, independentvoters vote only for C ). The master equation has a simpler form: P ( k, t ) = P ( k, t − − kpc + t − P ( k, t −
1) + pc + t − k − P ( k − , t − . (4)When we substitute k = c in the above equation, the last term of the RHS vanishes, and thus, theprobability P ( k, t ) can be calculated easily: P ( c , t ) = (1 − c pc )(1 − c pc + 1 ) · · · (1 − c pc + t − . (5)For k > C , we can prove (see Appendix A) that the following equality holds: P ( k ′ , t ) = k ′ − c +1 X l =1 ( − l − ( c ) k ′ − c (1) k ′ − c − l +1 (1) l − ( c − ( l + c − p ) t − ( c ) t − . (6)This is the distribution of the votes for the special case wherein the independent voters always votefor only one candidate, C . 4f we set p = 1, all voters are copycats, and we obtain the following reduction: P ( k ′ , t ) = ( c ) k ′ − c ( c ) t − (1) k ′ − c k ′ − c +1 X l =1 ( − l − k ′ − c l − ! ( c − l + 1) t − , = t − k ′ − c ! ( c ) k ′ − c ( c ) t − − k ′ + c ( c ) t − . (7)This is a beta binomial distribution. At the limit t → ∞ , the above equation becomes a betadistribution. Note that to obtain (7), we use the identity k ′ − c +1 X l =1 ( − l − k ′ − c l − ! ( c − l + 1) t − = (1) t − (1) t − − k ′ + c ( c ) t − − k ′ + c . In [12], we discussed the physical characteristic of this model. In the limit t → ∞ and c , c → α = c /c fixed, the scale invariance holds over the entire range.Here, we can calculate the momentum of these distributions to analyze them. The momentum isgiven by µ r ( t ) = c + t − X k = c k r P ( k, t ) . (8)We introduce quasi-momentum asˆ µ r ( t ) ≡ c + t − X k = c k ( k + 1) · · · ( k + r − P ( k, t ) . (9)We can prove (see Appendix B) that the quasi-momentum can have the following form:ˆ µ r ( t ) = t X l =1 ( − l − ( c ) t + r (1) l − (1) t − l ( l + c + r −
1) ( c − ( l + c − p ) t − ( c ) t − . (10)If we set r = 1, we get the average voteˆ µ ( t ) = t X l =1 ( − l − ( c ) t +1 (1) l − (1) t − l ( l + c ) ( c − ( l + c − p ) t − ( c ) t − . (11)We study t → ∞ . The coefficients of master equation (3) do not contain the initial votes for C ; given by c . If we set t >> c = c + c , the master equation does not depend on the initialconditions. Therefore, for a large t limit, the behaviour of the moment does not depend on the initialconditions c , and c . We can also observe this in (17) and (18) in the next section. Here, we set c = c = 1 for the representative case. Direct calculation using (10) is difficult. Hence, we study P ( k ′ , t ) as t → ∞ . In the above case, distribution (6) becomes a constant distribution with cut-off k ∗ . The cut-off implies a fast decay for larger values ( k ′ > k ∗ ). If we set p = 1, we get a constantdistribution. Using (6), we getlim t →∞ P ( k ′ , t ) = k ′ X l =1 ( − l − (1) k ′ − (1) k ′ − l (1) l − t − p . (12)5or large time values, the only time dependent term is t − lp . In the case of t >> k ′ , we can assumethat only the first term of the summation is non-negligible. Therefore,lim t →∞ P ( k ′ , t ) ∼ t − p . (13)Cut-off k ∗ is the inflection point of P ( k ′ , t ). Using (12), we can obtain k ∗ = t p + 2. Then, themomentum is µ r ( t ) ∼ k ∗ X k =1 k r P ( k, t ) ∼ t rp . (14)(14) is a continuous function of p . Hence, there is no phase transition throughout. In the nextsection, we study the general case in the continuous limit. To investigate long-ranged correlations, we analyze in the limit t → ∞ . We can rewrite (3) as c ( t ) = k → k + 1 : P ( k, t ) = kpc + t − − p )(1 − q )= p k − ( c + t − c + t − − p )(1 − q ) . (15)We define ∆ t = 2 c ( t ) − ( c + t −
1) with the initial condition ∆ = c − c = 2 c − c . We changethe notation from k to ∆ t for convenience. Then, we have | ∆ t | = | k − ( c + t − | < c + t −
1. Thesupport for the law of ∆ t is thus { ∆ − ( t − , ∆ + ( t − } . Given ∆ t = s , we obtain a randomwalk model ∆ t = s → s + 1 : P s + c + t − ,t = p ( s + c + t − c + t −
1) + (1 − p )(1 − q ) , ∆ n = s → s − Q s + c + t − ,t = 1 − P s + c + t − ,t . Let ǫ = 1 /c →
0. We now consider X τ = ǫ ∆ [ t/ǫ ] ,P ( x, τ ) = ǫP (∆ t /ǫ, t/ǫ ) , (16)where τ = t/ǫ and x = ∆ t /ǫ . Approaching the continuous limit, we can obtain the Fokker-Plankdiffusion equation for this process (see Appendix C): ∂P∂τ = 12 ∂ P∂x − pτ + 1 ∂ ( xP ) ∂x − (1 − p )(1 − q ) ∂P∂x . (17)We can also obtain X τ such that it obeys a diffusion equation with small additive noise:d X τ = [(1 − p )(1 − q ) + pxτ + 1 ]d τ + √ ǫ d B τ , X = c − c c . (18)Though (17) and (18) are equivalent, hereafter, we only deal with (18) for simplicity. Assume c israndom or deterministic. Let σ ≡ σ ( Y ) = 4 ǫ σ ( s ) (19)6e the variance of X . If X is Gaussian ( X ∼ c( y , σ )) or deterministic ( X ∼ δ x ), the law of X τ ensures that the Gaussian is in accordance with density p τ ( x ) ∼ √ πσ τ e − ( x − x τ ) / σ τ , (20)where x τ = E( X τ ) is the expected value of X τ and σ τ ≡ v τ is its variance. If Φ τ ( λ ) = log(ei λX τ ) isthe logarithm of the characteristic function of the law of X τ , we have ∂ τ Φ τ ( λ ) = p τ λ∂ λ Φ τ ( λ ) + i(1 − p )(1 − q ) λ − ǫ λ (21)and Φ τ ( λ ) = i λx τ − λ v τ . (22)Identifying the real and imaginary parts of Φ τ ( λ ), we obtain the dynamics of the mean of X τ as˙ x τ = p τ x τ + (1 − p )(1 − q ) . (23)The solution for x τ is x τ = ( x + 2 q − τ ) p + (1 − q )(1 + τ ) ∼ ( x + 2 q − τ p + (1 − q ) τ. (24)Since we are interested in the voting rate obtained, we introduce a new scaled variable:˜ x τ ≡ x τ τ . The solution for ˜ x τ is ˜ x τ ∼ ( x + 2 q − τ ( p − + (1 − q ) . (25)When p = 1, ˜ x τ ∼ − q . This implies that the average percentage of C ’s votes againt the totalpoll is 1 − q . When p = 1, ˜ x τ ∼ x . This agrees with our assertion that the scaled distribution ofvotes becomes a beta distribution when τ is large. In this case, the mean value does not change.From the above discussion, we can infer that the distribution becomes similar to a delta function.The question of how this distribution converges to a delta function constitutes the next problem.To investigate this, we analyze the dynamics of the variance. The dynamics of v τ are given by theRiccati equation ˙ v τ = 2 p τ v τ + ǫ. (26)If p = 1 /
2, we get v τ = v + Z τ ( 1 + τ r ) p ( 2 p r v + ǫ )d r = v (1 + τ ) p + ǫ Z τ ( 1 + τ r ) p d r = v (1 + τ ) p + ǫ − p (1 + τ ) p ((1 + τ ) − p − . (27)If p = 1 /
2, we get v τ = v + Z τ ( 1 + τ r )( 11 + r v + ǫ )d r = v (1 + τ ) + ǫ (1 + τ )log(1 + τ ) . (28)7ow, we can summarize the temporal behaviour of the variance as v τ ∼ ǫ − p τ if p < , (29) v τ ∼ ( v + ǫ p − τ p if p > , (30) v τ ∼ ǫτ log( τ ) if p = 12 . (31)Here, we introduce rescaled variables ˜ v τ ≡ v τ τ . The solution for ˜ v τ is ˜ v τ ∼ ǫ − p τ − if p < , (32)˜ v τ ∼ ( v + ǫ p − τ p − if p > , (33)˜ v τ ∼ ǫ log( τ ) τ if p = 12 . (34)If p = 1, ˜ v τ becomes v . This agrees with our assertion that the distribution of votes becomes abeta distribution. If p > / p = 1 /
2, candidate C gathers 1 − q of all the votes in the scaleddistributions, but the voting rate converges more slowly than that in a binomial distribution. If0 < p < /
2, the voting rate becomes 1 − q , and the distribution converges as it would in a binomialdistribution. Hence, if independent voters form a majority, the distribution of votes becomes similarto a delta function and the convergence is at the same rate as that in a binomial distribution. Ifcopycat voters form the majority, the distribution remains the same but the convergence is at a rateslower than that in a binomial distribution. In this phase, it is difficult to ascertain the causes for thedelay of the convergence. Similar phenomena can be seen in several fields. In daily financial data,the motion of the price does not represent a Markov process, and it is difficult to forecast the futureprice. [10] In fact, it has been pointed out that the motion of price is super-diffusive behaviour andthe stochastic differential equation for price is similar to (18).[9] When all voters are copycats, thedistribution becomes a beta distribution and does not converges.Curty and Marsili [6] recently introduced a model about information cascade. Their model isbased on game theory. They showed that when the fraction of herders in a population of agentsincreases, the probability that herding produces the correct forecast undergoes a transition to a statein which either all herders forecast rightly or no herder does. Their model is similar to the limitationof our model in the case wherein voters are unable to see the votes of all the voters but can onlysee the votes of previous voters. However, there is a significant difference between our model andtheir model with respect to the behaviour of copy cats. In their model, copycats always select themajority of votes, which is visible to them. Thus, the behaviour becomes digital (discontinuous). Weaim to carry out an analysis of the influence of this behaviour in the future.We now consider the correlation. For a beta binominal distribution, we can define the parameter ρ ≡ / ( c + 1).[13] This parameter represents the strength of following a decision. If we set ρ = 1,everyone votes for the candidate who received the first voter’s vote. On the other hand, when ρ = 0,copycats become independent. It should be noted that our conclusion does not depend on ρ exceptwhen ρ = 0. For large t , convergence is not related to ρ , but is related to p , the appearance probabilityof independent voters. 8ere we discuss the solution in the previous section. If we set q = 0 ( q = 1 is the same as rela-tion(2)), (15) becomes 0 for large t . This is because independent voters’ votes become deterministic.Hence, in (17), the diffusion term disappears. In (26), the noise term ǫ disappears, and the dynamicsof v τ are given by ˙ v τ = 2 p τ v τ . (35)Then, phase transition disappears, and the behaviour of v is continuous˜ v τ ∼ ( v ) τ p − for all p. (36)This result is acceptable, following the discussion in the previous section and that (10) is continuouswith respect to p . (See (14).) If there is a consensus about the fundamental value, copycat votersaffect the convergence in proportion to their ratio. Further, in this case, the convergence does notdepend on correlation ρ .In order to confirm the analytical results pertaining to the asymptotic behaviour, we performnumerical simulations. We use the master equation (4) directly.Figures 2 and 3 display the deformation of the distribution of votes for C over time t . Figure2 is the case wherein the independent voters vote for C with the probability 1 − q , and Figure 3is the case wherein the independent voters always vote for C . We can see that the distributionconverges to a delta function for the votes of the independent voters. If all voters are copycats( p = 1), the distribution becomes a beta binomial distribution. Because of the doped binomialdistribution (independent voters), the distribution is deformed. For the case q = 0, we can obtainan exact solution in section 3. p ( x ,t ) x p=0.9,q=1/3p=0.3,q=1/3 Figure 2: Asymptotic behaviour of the distributionof votes for C at t = 100 → q = 1 / p = 0 . , . p ( x ,t ) x p=0.9,q=0.0p=0.3,q=0.0 Figure 3: Asymptotic behaviour of the distributionof votes for C at t = 100 → q = 0 and p = 0 . , . p . The distribution convergesto a delta function over time. However, there are differences between Figures 2 and 3 at p ≥ . v .In the general case ( q = 0 , p > .
5, thevariance converges at the same rate as that in a binomial distribution (slope = − p < . v τ τ p=0.9,q=0.5p=0.7,q=0.5p=0.5,q=0.5p=0.3,q=0.5p=0.1,q=0.5 τ -0.2 τ -0.6 τ -1.0 Figure 4: Asymptotic behaviour of the scaled vari-ance ˜ v τ for q = 0 . p = 0 . , . , . , . , . v τ τ p=0.9,q=0.0p=0.7,q=0.0p=0.5,q=0.0p=0.3,q=0.0p=0.1,q=0.0 τ -0.2 τ -0.6 τ -1.0 τ -1.4 τ -1.8 Figure 5: Asymptotic behaviour of the scaled vari-ance ˜ v τ for q = 0 and p = 0 . , . , . , . , . p = 0 .
5, super-diffusive behaviour is exhibited, and convergence is slower than that in a binomialdistribution (slope > − p < . p = 0 . q = 0 , We investigated a voting model that is similar to a Keynesian beauty contest. Mathematically, ourmodel is a binomial distribution (independent voters) doped in a beta binomial distribution (copycatvoters). We calculated the exact solution for special cases and analyzed the general case using astochastic differential equation. In the special cases, there is no phase transition. We will extendthis function to the general q in the future. We believe that the obtained solution is a useful clue tounderstand phase transitions clearly.In general, q , the correlation structure, exhibits a dramatic change at a critical value of thedoping. If copycats constitutes a majority or number half of the total number of voters, the varianceconverges slower than it would in a binomial distributions. This implies that our conclusion isextremely volatile because the fundamental value becomes irrelevant, and it is difficult to estimatethe conclusion of the vote.We observed phase transition in the limit t → ∞ . However, the long memory is finite; therefore,this gives rise to the question of whether we can observe the phase transition. This question arisesbecause in this case, voters are unable to see the votes of all the voters but can only see the votes ofprevious voters. This model is useful to understand the model introduced by Curty and Marsili.[6]We intend to address this issue in the future. Acknowledgment
This work was supported by Grant-in-Aid for Challenging Exploratory Research 21654054 (SM).10 ppendix A
We prove the assumption (42). Multiplying (3) by ( − k − c ( l − / { ( k − k − · · · c ( l − k + c − } = ( − k − c (1) l − / ( c ) k − c (1) l − k + c − and summing over k = c , c + 1 , · · · , l + c − l + c − X k = c ( − k − c (1) l − ( c ) k − c (1) l − k + c − P ( k, t )= l + c − X k = c ( − k − c (1) l − ( c ) k − c (1) l − k + c − P ( k, t − − pc + t − × l + c − X k = c ( − k − c (1) l − ( c ) k − c (1) l − k + c − [ kP ( k, t − − ( k − P ( k − , t )]] , (37)where ( z ) i = z · ( z + 1) · · · ( z + i − . We call the second and third terms of the RHS without thecoefficient p/ ( c + c + t − A and B , respectively. We can rewrite A as A = l + c − X k = c ( − k − c (1) l − ( c ) k − c (1) l − k + c − kP ( k, t − − l − (1) l − ( l + c − c ) l − P ( l + c − , t − . (38)We can rewrite B as B = l + c − X k = c +1 ( − k − c (1) l − ( c ) k − c − (1) l − k + c − P ( k − , t − − l + c − X k = c ( − k − c (1) l − ( l − k + c − c ) k − c (1) l − k + c − P ( k, t − . (39)Then, A − B is given by A − B = ( l + c − l − X k = c ( − k − c (1) l − ( c ) k − c (1) l − k + c − P ( k, t − . (40)Substituting (40) in (37), we can obtain the time evolution of the summation: l − k − X k = c ( − k − c (1) l − ( c ) k − c (1) l − k + c − P ( k, t )= c + t − − ( l + c − pc + t − l − k − X k = c ( − k − c (1) l − ( c ) k − c (1) l − k + c − P ( k, t − . (41)For k > C , we can prove that the following equality holds: l + c − X k = c ( − k − c (1) l − ( c ) k − c (1) l − k + c − P ( k, t ) = ( c − ( l + c − p ) t − ( c ) t − . (42)11he analytic form can be obtained by multiplying both sides with ( − l − ( k ′ − k ′ − · · · s/ [( k ′ − l − c + 1)!( l − − l − ( c ) k ′ − c − / [(1) k ′ − l − c − (1) l − ] and summing over l = 1 , · · · , k ′ − c + 1 P ( k ′ , t ) = k ′ − c +1 X l =1 ( − l − ( c ) k ′ − c (1) k ′ − c − l +1 (1) l − ( c − ( l + c − p ) t − ( c ) t − . (43) Appendix B
Replacing the analytical form (9), we can obtainˆ µ r ( t ) = c + t − X k ′ = c ( k ′ ) r k ′ − c +1 X l =1 ( − l − ( c ) k ′ − c (1) k ′ − l − c +1 (1) l − ( c − ( l + c − p ) t − ( c ) t − . (44)Further, t + c − X k ′ = c k ′ − c +1 X l =1 = t X l =1 t + c − X k ′ = l + c − (45)and t + c − X k ′ = l + c − ( k ) r ( c ) k − c (1) k ′ − l − c +1 (1) l − = ( c ) r + t (1) l − (1) t − l ( l + c + r − . (46)the quasi-momentum can have the following form:ˆ µ r ( t ) = t X l =1 ( − l − ( c ) t + r (1) l − (1) t − l ( l + c + r −
1) ( c − ( l + c − p ) t − ( c ) t − . (47) Appendix C
We use δX τ = X τ + ǫ − X τ and ζ τ , a standard iid Gaussian sequence; our objective is to identify thedrift f τ and variance g τ such that δX τ = f τ ( X τ ) ǫ + √ ǫg τ ( X τ ) ζ τ + ǫ . (48)Given X τ = x , using the transition probabilities of ∆ n , we getE( δX τ ) = ǫ E(∆ [ τ/ǫ ]+1 − ∆ [ τ/ǫ ] ) = ǫ (2 p [ l/ǫ + c + τ/ǫ − ] ,τ/ǫ −
1) = ǫ [(1 − p )(1 − q ) + pxτ + 1 ] . (49)Then, the drift term is f τ ( x ) = (1 − p )(1 − q ) + px/ ( τ + 1). Moreover, σ ( δX τ ) = ǫ [1 p [ l/ǫ + c + τ/ǫ − ] ,τ/ǫ + ( − (1 − p [ l/ǫ + c + τ/ǫ − ] ,τ/ǫ )] = ǫ , (50)such that g ǫ,τ ( x ) = √ ǫ. In the continuous limit, we can obtain the Fokker-Plank diffusion equationfor this process: ∂P∂τ = 12 ∂ P∂x − pτ + 1 ∂ ( xP ) ∂x − (1 − p )(1 − q ) ∂P∂x . (51)12 eferences [1] Keynes J M 1936 General Theory of Employment Interest and Money [2] Bikhchandani S, Hirshleifer D and Welch I 1992
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