Strategies of Voting in Stochastic Environment: Egoism and Collectivism

ISSN 0005-1179, Automation and R emote Control, 2006, V ol. 67, No. 2, p. 311. c  Pleiades Publishing, L td., 2006. Original Russian T ext c  Borzenko, L ezina, L o ginov, Tso dikova, C heb otarev, 2006, publishe d in Avtomatika i T elemekhanika, 2006, No. 2, 154–173. CONTR OL IN SOCIAL ECONOMIC SYSTEMS Strategies of V oting in Sto c has tic En vironmen t: Egoism and Collecti vism 1 V. I. Borzenk o, Z. M . Lezina, A. K. Logino v , Y a. Y u. Tso diko v a, and P . Y u. Chebo tarev T r ap eznikov Institute of Contr ol Scienc es, Russian A c ademy of Scienc es, Mosc ow, Russia Received F ebruary 1, 20 05 Abstract— Consideratio n was given to a mo del of so cia l dynamics con trolle d b y successive collective decisions ba sed on the threshold ma jority pro ce dur es. The current s y stem state is characterized by the vector of par ticipants’ capitals (utilities). At each step, the voters can e ither retain their status quo or acce pt the prop os a l which is a vector o f the a lgebraic incr ement s in the capitals of the par ticipant s. In this v ersion of the mo del, the vector is gener ated sto chastically . Comparative utility of t wo so cial attitudes—egoism a nd collectivism—w as ana ly zed. It w as established that, except for some sp ecial cases, the c o llectivists hav e adv antages, whic h makes realizable the following sce na rio: on the conditions of protecting the corp ora te interests, a group is c reated which is joined then b y the egoists attracted by its achiev ements. At that, group egois m appro aches altruism. Additionally , one of the c onsidered v ariants of co llectivism handicaps manipulation of voting by the organizers. P ACS num ber s: 89.65.-s, 89.65.E f DOI: 10.11 3 4/S000 5117906020093 1. INTR ODUCTION As was sho w n by A.V. Malishevskii [1, p p. 92– 95] at a turn of the 1970’s, for any d istribution of r ic hes in so ciet y one can form ulate a num b er of prop osals such that eac h of them enric hes 99% mem b ers of the so ciet y , b u t as a result of successive acceptance of these prop osals eac h mem b er of the so ciet y b ecomes very p o or. Bluntl y sp eaking, these prop osals lie in su ccessiv e disp ossession of all p articipan ts one after another. A t that, part of the participan t’s prop erty is divid ed b et wee n the rest of them including the already disp ossessed ones, another p art is p assed to the p ers on form ulating th e pr op osal. This algorithm ca n b e u sed in a n umb er of v otings, pro vided that the participan ts v ote according to their v aluable in terests and completely ignore the int erests of the others. On e can readily see that the heart of the problem r emains unc h an ged if instead of defending their o wn int erests the v oters defend the inte rests of their electors, clans, indu stries, countries (in the UN), and so on. This is due to the fact that if the v oters are guided by an y particular inte rests dividing them into sm aller groups, then v oting b ecomes easily manipulatable by those who ha v e priorit y in form ulating the pr op osals s ubmitted to v oting (see also [2, 3]). Ho w to eliminate this manipu lation? It is difficult to restrict the p o wers of the “Presidium” an d the influ ence of the p olitical b ac kstage sp in do ctors. Th e p assage from shorthand ed p r otectio n b y the v oters of their pr iv ate int erests to the un iv ersal principle of “all for one and one for all” w ould b e a r eliable safeguard, but as a ru le it is unlike ly . First, the “vo ters will not understand.” S econd, “self lik es itsel f b est.” F rom the practical p oin t of view, the p articipant is int erested only in “all for one,” and o nly if this “one” is himself. Y et the p articipan t ma y hop e nev er to b e in th e role 1 This work was supp orted in part b y the Ru ssian F oundation for Basic Researc h , pro ject n o. 02-01-00614. 311 312 BORZENKO et al . of the “one” needing protection, esp ecially if he in tend s to r emain truth ful to the “Presidium.” Moreo v er, he un derstands that if sometimes he will b e in this p osition, the others ma y b etra y h im, no matter ho w arduously he defended their in terests in the past. Therefore, the immediate utilit y of the everyda y mission of “b eing for all” for wh ic h he will h a ve to pa y a prett y p enny is very doubtful for h im. Ho wev er, it is not alw ays the case that the collectiv e decisions are made in the conditions of selfish or malev olen t manip u lation by the prop osal mak ers. In essence, the agenda is often defined not by them, bu t rather by external—natural, man-caused, economical, p olitical, demographic, and so on—p henomena. In what concerns these phenomena, it is p ossible to d iscuss only the general trends, rather than the times of o ccurrence and the scop e of the future ev ents. In the first appro ximation this r ealit y ma y b e mo deled b y a sequence of sto c hastically generated pr op osals. Eac h prop osal is adv an tageous to some participant s and disadv an tageous to some other participan ts. In the s implistic mod el, this prop osal ma y be iden tified with a vec tor of alg ebraic incremen ts in some abstractly u ndersto o d capitals (utilities) of the p articipan ts. A mo del of this s ort is discussed b elo w. The main sub ject matter is represen ted by the so cial attitudes of the vot ing participants. The main attitudes are egoistic and collectivistic . Lik e in the Malishevskii mo del, the egoistic p articipan t supp orts any prop osal pro viding an in cremen t to its capital. There are also other participan ts making up (still one) group and v oting jointl y for the prop osals that are benefi cial, at least min imally , to the group as a whole. They sacrifice to some exten t their p riv ate in terests to the int erests of their group. This kind of collect ivism can b e justly called the group egoism. The pr esen t pap er is concerned mostly with the attit ude—egoistic or collectivistic—that is b eneficial to an individual participant. More sp ecifically , consideration is giv en to the dyn amics of the mean capital of the egoists and th e group mem b er s . The fol lo wing hyp othetical mec hanism is of sp ecial in terest. Let under certain conditions the collect ivistic att itude b e more adv an tageous. Then th e egoists h av e an incentiv e to join the group. If the group puts no obstacles in this w ay and grows by including more and more mem b ers, then its group egoism r esem bles more and more the act ions in the in terests of all, that is, the b asically altruistic interests. It deserv es noting that in this d ecision mo del the enti re s o ciet y participates in decision making, rather than th e “Parlia men t” that rep resen ts it. Under the assum ption of utilit y of th e collec tivistic attitude, this mec hanism enables the follo w- ing scenario based on the pragmatic in terest and not on the altruistic ideals o f the participan ts. Let some participan ts m ak e up a group taking u p on itself the resp onsibilit y to v ote not for their o wn in terests, but for those of their group. F ormulatio n of the latter is, of c ourse, a p roblem p er se . The group is op en to n ew memb ers who j oin it little b y li ttle as they see that the ca pital o f its participan ts on the a ve rage gro ws faster than that of the egoists. The conditions under whic h this scenario is feasible are the same as those under w hic h the collectivist attitude is pr eferable to the egoist one. These conditions which are defined by certain v alues o f the model inputs are the sub ject matter of our study . Wh y w e discuss only a single group whereas in practice we mostly observ e confronta tion of more than one group? This is du e to the f act that in the envi ronment wher e collectivism brings the b est results it is more b eneficial to the groups to u nite, rather than to comp ete. In th e p olitics, the m ain obstacle to this is represen ted b y the ideological differences, am bitions of the leaders, and the d esire to fin d its o wn p olitical “nic he” (see, for example, [4]). Someho w or other, but the mec hanisms of in teraction of sev eral groups within the framework of a mo d el undoubtedly deserv e inv estigation; and th is inv estigatio n is pro jected. W e also plan to consider so cially oriented groups w hic h supp ort their p o orest mem b ers in ord er to preven t their ruin, the mo del v ariants where the increments of capital depen d on their cur ren t v alues, t he v ariants using mechanisms for collection of taxes and “part y dues” within the groups, and so on. At the same time, we do not plan to mo del pur ely AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 STRA TEGIES OF VOTING 313 economic m ec hanisms of repro d uction and dr ain of capital b ecause we aim at analyzing the so cial, rather than economic phenomena. W e again rep eat that in the m o del the “capital” is mean t in the most general (like “utilit y”) and n ot economic sense. 2. MODEL OF SO CIAL DYNA MICS 2.1. Basic M o del There are n participants among which n e are egoists and n g = n − n e b elong to the group. It will b e con venien t in what follo ws to d efine the rela tion of the participan ts b y the parameter β = n e / 2 n , half of the fractio n of ego ists among all p articipan ts. The “so ciet y” coincides with the set of p articipan ts. Its state at eac h instan t is d escrib ed b y the n -dimen sional vecto r of capitals whose i th comp onent is a real num b er c haracterizing the capital of the i th p articipan t. Defined is the vec tor of in itial capitals; by the “prop osal of the en vironmen t” is m ean t the v ector ( d 1 , . . . , d n ) of capital incr ements, w here d i is the algebraic increment in th e capital of the i th participant according to the prop osal of th e en vironmen t. It is assumed b elo w as in [5–7] that the prop osals of the en vironm en t are generated sto c h astically and th eir distr ibution remains unchanged from step to step. Namely , we assume that d i is a normally distributed random v ariable whose mean v alue a nd v ariance are denoted, resp ectiv ely , b y µ and σ 2 . Th e v alues d 1 , . . . , d n are assumed to b e ind ep endent in the aggregate . W e denote by ξ e and ξ g the random v ariables represen ting, respectiv ely , the fractions of th ose egoists among all egoists and those group members among all group mem b ers to whom a rand om prop osal of the environmen t pro vides a p ositiv e increment of the capital. These random v ariables tak e v alues o ve r the interv al [0 , 1]. W e do n ot rule out the p ossibilit y of zero incremen t and at calculating ξ e and ξ g use the co efficient 0 . 5 to tak e in to accoun t the num b ers of participants getting the zero increment. A t eac h step th e participan ts lea rn the next prop osal an d v ote in line with their pr inciples. If the i th p articipant is egoist, he v otes for a p rop osal if and on ly if d i > 0; if d i < 0, he votes against; and if d i = 0, he abstains fr om vot ing (giv es half-v ote “for” and h alf-v ote “aga inst”). All group mem b ers v ote identica lly in co mpliance with the group principle of v oting. In the pr esen t p ap er, w e consider t w o principles, “A” and “B”. The gr oup votes “for” a pr op osal ( d 1 , . . . , d n ) if and only if. . . – Principle A : . . . as the r esult of ac c epting it the numb er of gr oup memb ers ge tting a p ositive incr ement in the c apital exc e e ds the numb er of gr oup memb ers getting the ne gative incr ement : ξ g > 0 . 5; – Principle B : . . . the sum of the inc r ements in the c apitals of gr oup memb ers is p ositive : P d i > 0 ( the sum is taken over the p articip ants of the gr oup ). W e denote by ξ the fraction of participant s voting for the giv en rand om prop osal. Since the group v otes joint ly , ξ = ( ( n e ξ e + n g ) /n if the group su pp orts the prop osal, n e ξ e /n, otherwise, (1) = ( 2 β ξ e + (1 − 2 β ) if the group su p p orts the prop osal, 2 β ξ e , otherwise. Decision making follo ws the “ α -ma jority” p ro cedure according to whic h a prop osal is acc epted if and only if ξ > α , wh ere α ∈ [0 , 1] is the de cision thr eshold . Consideration w ill b e giv en n ot only to the v oting thresholds α > 0 . 5, b ut also to α < 0 . 5 whic h are used in the p ractice of v oting for v arious initiativ es suc h as organizatio n of a n ew p arliamentary group, sending a letter of inquiry AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 314 BORZENKO et al . to the Cons titutional Court, initiating a referend um, putting a question on the agenda, and so on. Suc h in itiativ e decisions often pla y in the so cial life a role not s m aller than the m a j ority decisions. If a prop osal is accepted, then the corresp ond ing v ector of incremen ts ( d 1 , . . . , d n ) is added to the current ve ctor of capitals; otherwise, the latt er remains un c hanged. P assage is made to the next step w here a new prop osal is considered . T h u s, the v oting tr aje ctory is constructed. W e are going to consider the dynamics of the mean capital of the egoists and th e grou p on suc h tr a jectories and again emphasize that in the m o del at h and the environmen tal prop osals are random, that is, consideration is give n to the utili t y o f the egoistic a nd collec tivistic attitudes under sto c hastic uncertain t y , rather than to d elib erate manipu lation of v oting by its organizers. Nev ertheless, the problem of manipulatabilit y will not drop out of sigh t. So, one can note straigh tw a y that in fact the v oting principle A do es not con tribu te to solving this p roblem. Ind eed, if the group d emonstrates its efficiency and all egoists j oin it, then the s ituation will return to th e in itial one where no group existed at all: the group w ill conti n ue to vote join tly exactly for those d ecisions for whic h the simple ma jority would vote if they w ere egoists. It is namely this voti ng that is m ost readily manipulatable. Nev ertheless, it is of in terest to compare th e dynamics of the mean capitals of th e egoists a nd the group guided b y principle A under sto c h astic uncertain t y . Principle B resem bles in many resp ects principle A, but excludes—in the case where the group coincides with the ent ire so ciet y—manipu lation by the organizers as describ ed b y A.V. Malishevskii. Indeed, a ccording to principle B th e g roup sup p orts only those pr op osals that replenish its common stoc k. Therefore, after a series of su c h d ecisions the capitals of all p articipan ts cannot decrease. 2.2. A dditiona l Options of the Mo del It is of in terest to consider dynamics of participant’ s ruin. T o analyze this ph enomenon, pr o vided w as a v ariant of th e mo d el w h ere the p articipan t with the capital falling down to a n egativ e v alue “lea v es th e field,” is disregarded in the new prop osal of the e n vironmen t, and do es not vote . Additionally , th e mo del pr ovides as option the p ossibilit y for egoists to join or lea v e the group. Tw o t yp es of c onditions f or joining or leaving the group w ere c onsidered. (i) The ego ist is ready to join a group if its capital remains smaller than the mean capital of the group mem b er s dur ing s 1 successiv e steps, where s 1 is a p arameter. Corresp ondingly , a mem b er of the group is r eady to lea v e it if its capital is smaller than the mean capital of the egoists o ver s 2 mo ves; s 1 and s 2 can differ. (ii) The same comparison is carr ied out for the capital incr ements rather than for the capitals thems elves. F or the transitions to b e smo other and less “mec hanistic,” it is assu med that if the co ndition f or the fir s t or second type of transition is sa tisfied, then the transition does n ot o ccur of necessit y b ut with a certain pr obabilit y w hic h is th e mo del parameter. 3. SOME EXAMPLES The present pap er consider s the basic mo del which disregards r uin, joining the group , and exit from it. Some regularities of th ese p h enomena w ere d escrib ed in brief in [6]. In the follo wing examples the n umb er of participan ts is 200, the r ms deviation of the capital increment is σ = 10, and the group adheres to p rinciple B. W e consider fi rst the case of the neutral en vironment ( µ = 0), the group of 50% of all participan ts (2 β = 0 . 5), and the decision threshold 50% ( α = 0 . 5). Let the initial capital of all participan ts b e a = 700. The t ypical d ep endence of the capitals of th e group mem b ers and the egoists vs. the step n u m b er 2 on a v oting tra jectory is depicted in Fig. 1a. The mean capita l of the egoists is practically time-indep end en t, and that of the group gro ws uniformly . V ariation of the v oting threshold within rather wide limits do es not affect the p icture. Is 2 One should not assume th at hundreds of steps are alw a ys required for appreciable c h anges in th e capitals. The parameters are delib erately taken here in suc h a w ay that the c hanges are slow and the heigh t of the graph steps is insignificant on pu rp ose not to dim the general trend by random fluctu ations. AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 STRA TEGIES OF VOTING 315 Fig. 1. The mean capital of egoists and group members vs. the number of the step. Tw o h undred participants, σ = 10, principle B. a ) µ = 0; 2 β = 0 . 5; α = 0 . 5; b ) µ = − 1; 2 β = 0 . 92; α = 0 . 48; c ) µ = − 1; 2 β = 0 . 92; α = 0 . 5. Fig. 2. The mean capital of egoists and group members vs. the number of the step. Tw o h undred participants, σ = 10, principle B. a ) µ = − 1; 2 β = 0 . 08; α = 0 . 07; b ) µ = − 1; 2 β = 0 . 08; α = 0 . 04. it p ossible that sim ultaneously the capitals of one category of the participan ts grow and those of the other, diminish? This example is sho wn in Fig. 1b where the environmen t is unfav orable ( µ = − 1), the group is small (2 β = 0 . 92), and the v oting threshold is muc h h igher th an one h alf ( α = 0 . 48). Do es the fact that decisions ma y b e made agai nst the op in ion of ma jorit y play here the key role? No, an increase in the threshold up to α = 0 . 5 en tails no essen tial c han ges (Fig. 1c). One more example of similar dyn amics arises under an u nfa vorable environmen t ( µ = − 1), o v erw h elming ma jorit y of the group (2 β = 0 . 08), and extremely lo w decision th reshold α = 0 . 07 (Fig. 2a). A t first sight it is su rprising that ev en a greater reduction in the decision threshold ( α = 0 . 04) mak es the curv es pr actically change places, the capita l of t he egoists g ro w and that of the group decrease (Fig. 2b). Finally , w e discuss an example of high decision threshold. F or α = 0 . 97, v ery large group (2 β = 0 . 08) , and the fa v orable en vironment ( µ = 0 . 5), the egoists also are ahead (Fig. 3): the m ean capital of th e group m emb er gro ws m uc h slo wer than that of the ego ists. Consideration of these few examples con vinces us that the regularities of the so cial d ynamics corresp ondin g to the giv en mod el are not ob vious; th er efore, it is of interest to analyze them b y means of mathematical to ols. AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 316 BORZENKO et al . Fig. 3. The mean capital of egoists and group members vs. the number of the step. Tw o h undred participants, σ = 10, principle B. µ = 0 . 5; 2 β = 0 . 08; α = 0 . 97. 4. ON RANDOM V ARIABLES IN THE MODEL In the give n mo del, the prop osal of the en vir onmen t is the vect or ( d 1 , . . . , d n ) of the capital incremen ts of all participants, where d 1 , . . . , d n are indep endent r andom v ariables with the distri- bution N ( µ, σ 2 ). The corresp onding one-dimens ional densit y and the distrib ution fu n ction will b e denoted b y f µ,σ ( · ) and F µ,σ ( · ); f ( · ) and F ( · ) denote the densit y and th e distribution function of the n orm al distribution with th e cen ter 0 and v ariance 1; M ( η ) and σ ( η ) denote the exp ectatio n and the rm s deviation of any r andom v ariable η und er consideration. Eac h egoistic participant i vo tes for a prop osal if a nd only if d i > 0. The probabilit y of this ev ent is as follo ws : p = P { d i > 0 } = 1 − F µ,σ (0) = F  µ σ  ; (2) the probability of vot ing “against” is as follo ws: q = 1 − p = P { d i < 0 } = F µ,σ (0) = 1 − F  µ σ  = F  − µ σ  . (3) According to the mo del, the p robabilit y that the participan t abstains from voting is zero b ecause the normal d istr ibution is con tin uou s . Therefore, the vot ing of eac h egoistic participant is the Bernoulli test with the parameter p . Then, since the v alues d i are ind ep endent, the num b er of egoists v oting “for” is distributed binomial ly with the parameters n e and p . The mean v alue and the v ariance of this distribution are, resp ectiv ely , pn e and pq n e . No rmalization by dividin g by n e pro vides the aforemen tioned rand om v ariable ξ e , the fr action of e goists voting “for” . It h as the mean p and the v ariance n − 2 e n e pq = n − 1 e pq . The f raction ξ g of the group mem b ers getting a p ositiv e incremen t in ca pital acc ording t o the random prop osal of the en vir onmen t has the same distribution with the mean p and the v ariance n − 1 g pq , ξ e and ξ g b eing indep enden t. The confidence in terv al—usually sym metrical or cen tered at M ( η )—whic h the random v ariable η h its with the probabilit y 0 . 995 will b e called f or brevit y the c onc entr ation zone of η . In the case of normal r an d om v ariable with the parameters µ and σ 2 , this interv al lies inside the s egment [ µ − 3 σ , µ + 3 σ ] where ab out 99 . 73% of the normal d istribution are concen tr ated. Exit of the random v ariable fr om the concen tration zone will b e regarded as a highly improbable even t. In a series of, say , 1000 steps, such ev en ts happ en sev eral times, but mak e no appr eciable con trib ution to the mean indices, whic h ju s tifies their screening-out. The follo w ing sections will b e dev oted to analysis of the social d y n amics u nder v arious mo del parameters. AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 STRA TEGIES OF VOTING 317 5. CASE OF NEUTRAL ENVIRONMENT W e first assume that µ = 0, that is, the en vironmen t is neutral and the distribution of its prop osals is sym m etrical ab out 0. A t that, p = q = 0 . 5; the distributions of the random v ariables ξ e and ξ g are symmetrical ab out 0 . 5. W e assume for the time b eing, un less the cont rary is allo wed, that the group is guided b y the v oting principle A. T he effects of ruin and p assage of the participan t from one category to another are disregarded. 5.1. De cision Thr eshold α = 1 − β Let us assume that th e decision threshold i s set as α = 1 − β and that 2 β < 2 / 3. Then the v oices eve n of all eg oists will be insufficient to mak e decision. It is necessary and sufficient that the prop osal b e supp orted by the group and at leat one half of the egoists, that is, that the ev ents ξ e > 0 . 5 and ξ g > 0 . 5, b e realized (b y default th e group u s es the vo ting principle A). As was noted ab o ve, the probabilities of eac h of th ese ev ents are 0 . 5, and th e ev en ts are indep end en t. Therefore, their joint probabilit y is 0 . 25, and , therefore, th e asymptotic (for a great n u m b er of steps) v alue of the fraction of acc epted prop osals w ill b e 0 . 25. Let us consider n o w the dynamics of the mea n capital of egoists and the group memb er s . As it w as ju st established, the pr op osal is accepted if and only if it pro vides for a p ositiv e increment in capital for more than half of the egoists and more than h alf of the group members . T he m ean p ositive increment coincides with the me an magnitude σ p 2 /π of the deviation fr om the me an for th e normal distribu tion with the parameters µ and σ . Let us estimate th e numb er of these p ositiv e incremen ts. Th e v ariance of the binomially d istributed n um b er of egoists who v oted “for” is pq n e = 0 . 25 n e , and the rms deviation is 0 . 5 √ n e . As in the case of the p ositiv e incremen t, it is desired to determine the magnitude of me an devi ation . Sin ce the symmetrical b inomial distribution is w ell appro ximated by the normal d istribution ev en for a relativ ely small num b er of tests (u sually this app ro ximation is used for a num b er of Bernoulli tests exceeding 9( pq ) − 1 ), in order to pass from the rms deviation to the mean deviation magnitude w e make use of the same co efficien t p 2 /π as for the normal distr ib ution, that is, estimate the mean deviation magnitud e b y p 0 . 5 n e /π . Then, under the condition that more participan ts v oted “for,” th e mean deviation of the num b er of “fors” o ver the n umb er of “againsts” is estimated by p 2 n e /π . By multiplying it by the estimated p ositiv e incremen t (this v alue and the n um b er of p ositiv e incr ements are indep endent), we get that the total incremen t for the egoists is 2 σ π √ n e . Then, 2 σ π √ n e is the est imate of the capital increment of one egoist. Since on the whole a quarter of p rop osals is accepted, the mean incremen t for eac h egoist participan t in one step is estimate d as σ 2 π √ n e . If the total n u m b er of steps is s , then after this series σ s 2 π √ n e is the estimate of the exp ected capital incremen t of the egoist. F or example, if σ = 10 and n e = 50, then the estimate d incremen t for one participan t in 100 steps is ab out 22 . 5. S imilarly , the estimated mean capital in cremen t in a series of s mov es for a memb er of the group adhering to principle A is σ s 2 π √ n g . Therefore, the gro wth rate of the mean capital of eac h category is in in verse prop ortion to the ro ot of its quant it y (according to th e we ll-kno wn p seudo-scien tific aphorism , the sp eed of a human group is also in in v erse prop ortion to its size). In particular, for n g = n e the capital d ynamics of the group and egoists is th e same, and for n g 6 = n e the smaller category is in a b etter p osition. Ho wev er, one must b ear in min d that the considered d ecision thr esh old α = 1 − β itself dep ends on the relatio n b et wee n the quantitie s. Let no w the group adhere to principle B and give all its vo tes for the prop osal of the en viron- men t if and only if the tota l increment in the capitals of its memb er s is p ositiv e. F or a neutral en vir onmen t, this condition, as that of pr inciple A, is satisfied with the probabilit y 0 . 5. Sin ce the AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 318 BORZENKO et al . incremen ts in the capitals of the group and th e egoists are indep end en t, a quarter of th e prop osals is accepted asymp totica lly as b efore. Ho w the group dynamics c hanges a t that? F or an arbitrary prop osal of the environmen t, th e mean incremen t in the capital of the group member is normally distributed with the mean 0 and v ariance σ 2 /n g . T he mean in cremen t in the capital of the group mem b er, pro vided that it is p ositive as required by prin ciple B, is equal to the me an magnitude of deviation, that is, d iffers from the r ms deviation by the co efficient p 2 /π and is equal to σ s 2 π n g . By estimating again the fract ion of the accepted prop osals as one quarter, we obtain for a series of s mo v es the mean incremen t of the group m em b er equal to σ s  p 8 π n g . The ratio of this v alue to that obtained f or the group using principle A is p π / 2 ≈ 1 . 25. This 25% gain can b e attributed to the f act that in b oth cases the group supp orts half of the prop osals, b ut among the sup p orted prop osals there are s u c h that, although providing a p ositiv e increment to more th an h alf of its mem b ers, they nev erth eless p ro vid e a negativ e mean increment. Th erefore, in the case o f prin ci- ple A the mean capital gro ws slo w er than in the case of principle B guaran teeing supp ort of the group ev en to the p rop osals whic h provi de p ositiv e total incr ement for the group and increase the capital only of the minority . Note 1. The assu mption of 2 β < 2 3 made at the b eginning of this section can b e r elaxed using the notion of concen tration zone of the random v ariable ξ e . In deed, to d isable the ego ists to mak e a decision b y th eir o wn forces w ithout approv al b y the group, it is su fficien t to the accuracy of a h ighly improbable ev ent that the threshold α = 1 − β b e ab o ve the right b oun dary of the concen tration zone of 2 β ξ e . Th is b ound ary is estimated by 2 β ( M ( ξ e ) + 3 σ ( ξ e )), wh ere M ( ξ e ) = 0 . 5, σ ( ξ e ) = 0 . 5 / √ n e . By solving the inequalit y 2 β  0 . 5 + 3 · 0 . 5 / √ n e  < α = 1 − β , w e get 2 β < 1 1 + 3 σ ( ξ e ) = √ n e √ n e + 1 . 5 . (4) F or example, for n e = 225 this condition pro vid es 2 β < 10 / 11, th us relaxing th e initial constrain t 2 β < 2 / 3. 5.2. De cision Thr eshold α = β No w w e consider the “mirror” case of v oting with the th r eshold α = β and assu me for a start that 2 β < 2 / 3. A t that, su fficien t is not only app ro v al of the p rop osal b y the ma j ority of egoists, but also b y the group. If in the ab ov e case a conjunction of b oth conditions (supp ort by the group and the m a jority of egoists) wa s required, here it suffices to satisfy their disjunction. Since the en vir onmen t is neu tr al, the p r obabilit y of none of the ev ents is 1/4; consequent ly , the d isjunction of the conditions is satisfied in 3/4 of cases. These 75% are divided into 50% where the ma jority of the egoists are “for” and 25% wher e the m a j ority are “aga inst.” T hese 25% are fully symmetrical to half of the first of 50% and “balance” them in the sens e of the m ean capital in cremen t (the sum of t wo means is zero). W e apply to the remaining half of 50% the same reasoning as in the case of the threshold α = 1 − β and obtain the s ame resu lt: the mean capital increment of the egoist is σ s 2 π √ n e o ver a series of s steps. The same results are ob tained for the group: the mean capital increment is σ s 2 π √ n g or σ s p 8 π n g dep endin g on whic h principle, A or B, is used. Th e on ly difference of the general dynamics lies in th e fact that the presence of “coun terbalancing” decisions, that is, p ro vid ing the mean total zero of the “quarters” (fractions of 25%), increases the spread as compared to the case of the threshold α = 1 − β where they are absen t. No w , b oth in the group and among the egoists a stronger s tratificatio n in incomes is observ ed. Therefore, th e v oting thresholds α = 1 − β and α = β that are symmetrical ab out 0 . 5 pr o vide d ifferen t (b y the factor of three) n um b ers of the AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 STRA TEGIES OF VOTING 319 accepted p r op osals and different spreads of the incomes in the group and among the eg oists, but the same dynamics of the mean v alues. As will b e shown b elo w, in the case of neutral en vironment the situation with the mean v alues is alw a ys the same f or α = α ′ and α = α ′′ if α ′ + α ′′ = 1. Note 2. As in the ab o v e section, we relax the initial constr aint 2 β < 2 / 3 . No w we need that the appro v al b y the group should suffice (to within a highly impr obable even t) for accepting the prop osal of the environmen t. By assuming that sup p ort of the prop osal by the egoists d o es n ot diminish the left b oundary of the concen tration zone of ξ e whic h is equ al to M ( ξ e ) − 3 σ ( ξ e ), we obtain that the minimal total su pp ort of the prop osal as ap p ro ved b y the group is 2 β ( M ( ξ e ) − 3 σ ( ξ e )) + (1 − 2 β ) b ecause 1 − 2 β is th e fraction of the group among the p articipan ts. Therefore, the condition for sufficiency of su pp ort b y the group for approv al of the prop osal looks lik e 2 β ( M ( ξ e ) − 3 σ ( ξ e )) + (1 − 2 β ) > α = β . By substituting M ( ξ e ) = β and σ ( ξ e ) = 0 . 5 / √ n e , w e get th e same condition (4) as in the p receding case. 5.3. Other V alues of the De cision Thr eshold Since th e symmetrical binomial distribution is well approximat ed by the norm al distribu tion ev en for a relativ ely small n um b er of tests, we still estimat e the concen tration zone of the sym- metrical random v ariable ξ e b y the in terv al with the b ou n daries ( M ( ξ e ) ± 3 σ ( ξ e )), where in this case M ( ξ e ) = p = 0 . 5 and σ ( ξ e ) = p pq /n e = 0 . 5 / √ n e , that is, the in terv al with the boun daries 0 . 5  1 ± 3 / √ n e  . Then, according to (1), it is unlike ly that th e random v ariable ξ misses the union of the interv al with the b oundaries β ± 3 β √ n e (if the group do es n ot supp ort the prop osal) (5) and the interv al with th e b oundaries 1 − β ± 3 β √ n e (if the group sup p orts the prop osal), (6) F or example, if n e = n g = 225, then β = 1 / 4 a nd the v alues of ξ are concen trated i n the domain [0 . 2; 0 . 3] ∪ [0 . 7; 0 . 8]. Let u s consider the five cases of lo calization of the decision thr esh olds α . Zone 1. α < β − 3 β √ n e . Here, almost alwa ys ξ > α , and practically all pr op osals of the en vir onmen t are acc epted. Since the en v ir onmen t is neutral, the exp ectation of the mean incremen t in capital in the series of s steps is extremely close to zero b oth for the egoists and the group mem b ers, and th e r ms deviation of this v alue is p s/n e σ for the egoists and q s/n g σ f or the group. Zone 2. α > 1 − β + 3 β √ n e . Then, almost alwa ys ξ < α , and actually the prop osals of the en vir onmen t are nev er accepted. A t that, th e capital of the participan ts d o es not v ary , b u t the extremely rare accepted prop osals increase the capitals of b oth the group mem b ers and the egoists (the increase of the latte r is greater b ecause they ha ve a h igher decision thr esh old than th e group). Zone 3. β + 3 β √ n e < α < 1 − β − 3 β √ n e . F or these decision thr esholds, ξ > α is satisfied to an accuracy of lo w-pr obable ev ents if and only if the group su p p orts th e prop osal. If the group uses the voting principles A or B, then this h app ens on the a v erage in one half of cases, whic h is t wice as frequent as in the case discussed in Sec. 5.1 (page 317), the rest b eing the same b ecause the exp ected v alue of the increment in the capital of th e group memb er in th e s -step series is σs π √ n g for principle A and σs √ 2 π n g for principle B. The egoists do not a ffect the v oting; therefore, for them AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 320 BORZENKO et al . Fig. 4. The capitals a vera ged by th e categories and steps vs. the t hreshold α . One realization, 450 participants, µ = 0, σ = 10, principle B, 2 β = 0 . 5. the p r op osals of the en vironment are neutral on the a v erage and the exp ected incremen t in th eir mean ca pital in an s -step series is slightl y grea ter than zero (nev ertheless, sligh tly greate r o wing to the rare improbable ev en ts) and h as the rms deviation p s/ 2 n e σ . T h is zone of v ariations of the threshold α exists if the condition β + 3 β √ n e < 1 − β − 3 β √ n e , whic h is equiv alent to the condition 2 β < √ n e √ n e +3 , is satisfied. F or example, for n e = 225 the zone exists for β < 5 / 6. Zone 4. 1 − β − 3 β √ n e 6 α 6 1 − β + 3 β √ n e . With an increase of the decision threshold α from 1 − β − 3 β √ n e to 1 − β (zone “4a”), the egoists exerts more and more infl uence on the decisions and consisten tly reject more and more prop osals that are unadv an tageous for them on th e av erage. At that, the mean increment in the capital (in what f ollo ws, simply “mean incremen t”) of an egoist in an s -step series gro ws from zero (see case 3) to σs 2 π √ n e (Sec. 5.1, page 31 7). A t the same time, the in cremen t of the group mem b er in an s -step series is halve d from σs π √ n g (zone 3) to σs 2 π √ n g (for principle A) or from σs √ 2 π n g to σs 2 √ 2 π n g (for prin ciple B). With further gro wth of the threshold from 1 − β to 1 − β + 3 β √ n e (zone “4b”), the mean incremen t of the group mem b er conti n ues to v anish b ecause the n umb er of accepted prop osals v anishes. F or the egoists, the mean incremen t also diminishes b ecause no w n ot all bu t only th e most adv ant ageous prop osals are accepted. An in teresting effe ct is observe d here. If in zone 4b n e = n g (2 β = 0 . 5) and th e group adh eres to p rinciple A, then the mean in cremen t is higher for th e egoists rather than the group mem b ers b ecause the condition ξ g > 0 . 5 suffices to appro ve the pr op osal by the group, whereas the fraction of the ego ists’ vot es required for making a decision is higher. Therefore, the “utilit y threshold” of the accepte d prop osals is h igher for th e egoists than for the group. Th is effect w as men tioned when discussin g zone 2, and one can readily s ee that it is retained also wh en the group adheres to pr inciple B. Zone 5. β − 3 β √ n e 6 α ≤ β + 3 β √ n e . Here, the dynamics of the m ean incremen ts is a mirror reflection of that observ ed in zone 4. When α v aries from β − 3 β √ n e to β (zone “5a”), less and less prop osals for whic h ξ ∈ ( β − 3 β √ n e , β ) are accepted. They are n ot supp orted either b y the g roup or b y the ma jorit y of the egoists. Therefore, in w hat co ncerns the exp ected mean incremen ts in the capital, they are u nfa vorable to all, and a reduction in the num b er of suc h accepted pr op osals leads to higher mean incremen ts b oth for the group (p rinciples A an d B) and the egoists. Similar to the case of the “mirror” zone 4b, here also for n g = n e (2 β = 0 . 5) the mean capital increments AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 STRA TEGIES OF VOTING 321 Fig. 5. The capitals a vera ged by th e categories and steps vs. the t hreshold α . One realization, 450 participants, µ = 0, σ = 10, principle B. are higher for the ego ists than for the group, whic h is accoun ted for by the fact that the rejected prop osals here are just unfav orable for the group ( ξ g 6 0 . 5) and esp e cial ly unfavor able for the egoists f or which the stricter condition ξ e 6 α/β < 0 . 5 is satisfied. T h erefore, th e egoists gain more from their r ejection. With further in cr ease of α from β to β + 3 β √ n e (zone “5b”), the set of the a ccepted p rop osals con tracts ev en m ore (their exp ected fraction decreases from 75% to 50%) at the exp ense of the prop osals rejected by the group but sup p orted b y the ma jorit y of the egoists ( ξ > β , consequen tly , ξ e > 0 . 5). Th erefore, the mean incremen t for the group cont in ues to gro w from th e v alues reac hed at α = β (Sec. 5.2, p age 318) to the v alue of zone 3, and the mean increment for th e egoists decreases fr om σs 2 π √ n e (Sec. 5.2, page 318) to zero. The results of the computer-aided mo deling are illus trated in Figs. 4 and 5 wh ere the mean capitals of the egoists and the group m emb ers in a series of s = 10 00 s teps corresp on d ing to v arious v alues of the decision thr eshold are laid off on the v ertical axis. The initial capital of eac h participan t is 3000, the en vironm en t is neutral ( µ = 0), σ = 10, n = 450; the ruin and passage of the egoists to the group and exit from the group are disregarded. Averagi ng is carried out b oth o ve r the participan ts and steps. By virtue of stationarit y of the d istribution of the capital incremen ts, the exp ectation of the step me an differs fr om the initial v alue half as many as th e exp ectation of the capital after the s th step. AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 322 BORZENKO et al . These and th e follo wing figur es d epict the m ean v alues for one realization of the pr o cess wh ic h also r eflect the spread in v alues ab out the exp ectatio n, rather th an the c apital exp ectatio ns (the data f or it are p resen ted ab ov e, th e general analytical expressions will b e pub lished later). F or example, if the v oting threshold lies in zone 3, the exp ectati on of the capital of egoists almost do es not differ f rom their initial capital, but their mean capital is appreciably (b y 10 u nits) smaller b ecause of the sp read in the graph (Fig. 4). Figure 4 sho ws the case of as many egoists as there are memb ers in the group, n e = n g (2 β = 0 . 5). The first graph of Fig. 5 refers to the case of small group, 2 β = 0 . 92, in which case the condition of “Case 3” cannot b e met and no charac teristic horizont al segmen t exists on the curv es of the mean capital of the ego ists and the group. In the example a t hand, the v alue of β is sligh tly higher than the threshold of Note s 1 and 2 in whic h case for α = 1 − β a very small fraction of decisions is made only b y the egoists without appro v al of the group an d for α = β the p r op osal is not accepted despite supp ort of the group in a small fractio n of ca ses. The second graph of Fig. 5 sho ws the opp osite case of a v er y large group, 2 β = 0 . 08. Here, the greater part of the enti re r ange of α lies in zone 3. Since n e < n g , for α = β and α = 1 − β , the mean capitals of the egoists are higher th an those of the group mem b ers. Additionally , o wing to a lo w v alue of n e , the capital s of the egoists ha v e a substantia l sp read for the v alues of α related to zones 1 and 3. Th is sp read accoun ts for the noticeable difference b etw een the observ ed mean capital of the egoists in zones 1 and 3 an d the exp ected capital coinciding with the initial capital (300 0 un its). 5.4. Some Mor e Ab out the Zones wher e the E g oists H ave A dvantages In the case of neutral environmen t n e = n g , the ab o v e analysis shows that in what concerns the exp ected incremen t in th e capital the egoists ha ve adv anta ge o ve r the grou p mem b ers only in zones 4b and 5a. In zone 4b, of all pr op osals adv an tageous for the group the ego ists tak e only those that are most adv an tageous to them. In the “mirror” c ase 5a, the egoists blo c k appr ov al of the prop osals that are most disadv antag eous for th em, whereas the group cannot d o so. W e note that for n e = n g in these t w o zones the group can easily bring to nothing the egoists’ adv an tage by using the follo w ing pro cedure. W e defi ne for the group a s p ecial “in ternal v oting threshold” α ′ dep endin g on α and β . α ′ =                  1 2 − δ 2 β , where δ = β − α if α < β ; 1 2 + δ 2 β , where δ = α − (1 − β ) if α > 1 − β ; 1 2 if β 6 α 6 1 − β = (7) =              α 2 β if α < β ; 1 − 1 − α 2 β if α > 1 − β ; 1 2 if β 6 α 6 1 − β . Let u s consider principle A ′ . Principle A ′ . The gr oup votes for the pr op osal of the envir onment if and only i f in the c ase of its appr oval the fr action ξ g of its memb ers ge tting a p ositive incr ement in c apital e xc e e ds the thr eshold α ′ define d b y ( 7 ). V oting by prin ciple A ′ offers to the group the same p ossibilities of influencing the decisions made in zones 4 b and 5a as enjo yed b y the egoists. In deed, in zone 4b α = 1 − β + δ . In the c ase of AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 STRA TEGIES OF VOTING 323 accepting the prop osal of the environmen t, (1 − 2 β ) n vo tes are giv en by the group; consequent ly , it is necessary that among all p articipan ts the fr action of egoists supp orting the p r op osal exceed α − (1 − 2 β ) = β + δ . Consequently , the condition ξ e > β + δ 2 β = 1 2 + δ 2 β m u st b e satisfied f or approv al of the decision. If the group establishes for itself the same threshold of voting, that is, decision, it will b e in the same p osition as th e egoists: they will hav e identica l “utilit y thresholds” for t he supp orted pr op osals and , th erefore, iden tical exp ected dyn amics of the capital . Similarly , in zone 5a α = β − δ . T o approv e a p r op osal by the efforts of egoists, that is, without participation of the group, it is necessary that they provide more than αn = ( β − δ ) n v otes. Therefore, among all egoists the fraction of those vo ting “for” m ust exceed β − δ 2 β = 1 2 − δ 2 β . E s tablishmen t of the same “in ternal threshold” smal ler than 0 . 5 puts t he group in the same conditions as the egoists: if the v otes collected in the group exceed this threshold, then the prop osal will b e accepted f or s ure, and this will hap p en as fr equen tly as the v otes of th e egoists exceed the same thresh old. As the result, the exp ected dynamics of capital in th e group and among the egoists again will b e the same. If the n u m b er o f egoists is smaller than one half, then under the ab o v e changes in the in tra- group thresh old they will b e again ahead in zones 5a and 4b, bu t their adv an tage will decrease substanti ally . F urther r ed uction of the intrag roup threshold in zone 5a and in crease in zone 4b 3 will offer adv an tage to th e group ov er the egoists. How ev er, this adv an tage will b e relativ e, that is, the group really d ecreases its in cr ement in capital almost for all v alues of the thr eshold α , b u t the reduction o f th e egoi sts will b e even greater. If f or the g roup to “liv e b etter than the egoist s” is preferable ju st to just “living b etter,” th en it can reac h this aim. 6. CASES OF F A V OR ABLE AND UNF A V ORABLE ENVIR ONMENT The case of µ > 0 corresp onds to th e fa vorable en vironment , the case of µ < 0, to the unfa v orable en vir onmen t. Reasoning p ro vid in g conclusions ab out the nature of the dynamics of mean capitals are in this case similar to those ab o ve . Figure 6 illustrates the case of n e = n g (2 β = 0 . 5), the rest of the p arameters b eing th e same as b efore. F or small deviatio ns of µ from 0 (Fig. 6a,b), the graphs ha ve fiv e zones lik e those considered ab o ve. In zone 2 (numeration as ab o ve) , that is, f or the d ecision thresholds close to 1, the capital is equal to th e initial capital b ecause the pr op osals are rejected. On the cont rary , in zone 1 almost all prop osals are accepted. Therefore, the mean increment in capital afte r an s -step series is µs (w e recall t hat the c apitals of the participan ts are a ve raged on the graphs also o ve r the steps, therefore t he v alues sho wn there are half as man y) for the rms deviation √ s σ . In zone 3, only those decisions are made th at are supp orted by the group. Let us estimate their frequency assumin g that th e group adheres to principle B. The mean in cr ement in capital of a group mem b er in one step e d G has the distribu tion N ( µ, ( σ ′ ) 2 ), wh ere σ ′ = σ √ n g . The exp ectation of the fr equ ency of decisions made in zone 3 is equal to the pr obabilit y of p ositiv eness of this v alue, that is, P n e d G > 0 o = F  µ σ ′  = F ( µ ′ ) , (8) 3 This may b e done in a most natural wa y by establishing an intrag roup th reshold equal to the decision t h reshold α . AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 324 BORZENKO et al . Fig. 6. Dependence of th e capitals av eraged o ver the categorie s and steps on the th reshold α . One realiza tion, 450 participants, σ = 10, 2 β = 0 . 5, principle B. where as b efore F ( · ) is the standard normal distribution fun ction and µ ′ stands for µ/σ ′ . Let us determine the exp ectation of e d G , provided that it is p ositiv e. After in tegratio n w e get M  e d G | e d G > 0  =  P n e d G > 0 o − 1 ∞ Z 0 xf µ,σ ′ ( x ) dx = µ + σ ′ f ( µ ′ ) F ( µ ′ ) . (9) Since the un cond itional exp ectatio n of e d G is as follo ws: M  e d G  = P n e d G > 0 o M  e d G | e d G > 0  = σ ′ f ( µ ′ ) + µF ( µ ′ ) , (10) the exp ectation of an incremen t in the capital of a group mem b er after s steps is as follo ws: sM  e d G  = s  σ ′ f ( µ ′ ) + µF ( µ ′ )  . (11) The in cremen t of the capital of an egoist in zone 3 after s s teps is estimated by sM  e d E  = sP n e d G > 0 o M  e d E | e d G > 0  = sµF ( µ ′ ) . (12) AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 STRA TEGIES OF VOTING 325 Fig. 7. Dep enden ce of the capitals av eraged in categories and steps vs. the t h reshold α . One realization, 450 participants, σ = 10, principle B. Therefore, in zone 3 the estimated exp ectation of the difference in the capitals of the group mem b er and the egoist after s s teps is as follo ws: s  M  e d G  − M  e d E  = sσ ′ f ( µ ′ ) . (13) As can b e seen in Fig. 6, this v alue and f ( µ ′ ) rapidly decrease with increase in | µ | : as compared to the case of | µ | = 0 . 2, for | µ | = 1 the graph s of the mean capital of the group memb ers and the egoists app roac h closely eac h other and actually fus e for | µ | = 2. The case of n e 6 = n g is illustrated in Fig. 7. The cases of β = 0 . 92 and β = 0 . 08 are shown here for con venience of comparing with Fig. 5 . It go es without saying that the regularities are the same: with an increase in µ > 0, to the righ t of zone 3 the graphs mak e a h igher and higher “step” and a lo wer and low er ste p to it s left. On the contrary , with a decrease of µ < 0, the graphs ha ve a higher “step” to th e left of zone 3 and a lo w er one to the right of it. With in crease in | µ | , the graphs of the mean capitals of the egoist s and the group mem b ers draw together u n til almost complete fusion. AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 326 BORZENKO et al . 7. SUMMAR Y Although the presen t pap er models the “capital dynamics,” the term “c apital” should not b e misleading. I t is not an economic mod el that is considered ab o ve, but that concerning only the so cial and p olitical sciences. Indeed, it do es not cov er repro du ction of capital by in v esting into pro du ction or using the fin ancial to ols, as we ll as c h anges in the capital that are related with other factors indep enden t of collectiv e decisio ns. All this is unnecessary b ecause it is required here to stu dy the dynamics d efi ned by the m ec hanism of demo cratic voting with the built-in basic so cial attitudes of egoism and collectivi sm, rather than the economic or un controlla ble random mec hanism s . Therefore th e term “capital” refers her e to an y coun table resour ce (utilit y) con trolled b y the collectiv e decisions and having nothing to do with sp ecial regularities of repro duction or w aste. In w h at follo ws, w e su mmarize some conclusions obtained b y analyzing the m o d el. 1. The v oting-dep enden t s o cial dyn amics is d efined basically by the d ecision thresh old. F or higher th resholds (zone 2), the prop osals are not accepted and the status quo is retained. F or lo we r v alues (zone 1), actually all pr op osals are acce pted, the dynamics b eing iden tical b oth for the group and the ego ists and defined by the mean and th e v ariance of distribution of the prop osals of the en vir onmen t. The zone b oundaries dep end on th e n umb er of egoists and the group mem b ers, as w ell as on the p arameters of the environmen t. With increase in the total n umb er of the participan t and the fr action of egoists, zones 1 and 2 widen; for lo w er v alues of these parameter they con tract. In s ide the middle (zone 3), w hic h exists if the group is n ot to o small, the dynamics is in d ep endent of the decision threshold and the group has sup eriorit y ov er th e egoists. F or the neutral or a mo derately unfa v orable en vironment, it manages to realize decisions th at on the a v erage are adv an tageous for its mem b ers. A t that, th e egoists do not influence the decisions, and f or them the accepted prop osals do not differ fr om a rand om sample of th e pr op osals of the environmen t. There are tw o zones on either side of zone 3 (zone 5 to the left and z one 4 to the right) where th e ego ists influen ce the decisions. There are t wo p eaks (maxima) of the increment of the egoists’ capital whic h , ho wev er, v anish in t he case of a pparent ly fa vo rable and apparen tly unfa vo rable en vironmen ts. The group c haracteristics within these zones v ary monotonica lly from the v alues in the extreme zones 1 and 2 to the v alue in the middle zone. F or the neutral environmen t, the exp ected in cr ement in the capital reac hed b y the egoists at the t wo maxima—for α = 1 − β and α = β —are th e same as for a group of the sa me size for the same thresholds; for different sizes, the a dv an tage b elongs to the sm aller catego ry . If the egoists make 20% of the total num b er of participants, the increment in their capital reac hes at the maxima the v alue which the group has in th e midd le zone in the case of v oting b y principle A. If the egoists mak e ab out 14% of the participan ts, then at the maxima th ey r eac h the v alue whic h the group has in the middle zone in the case of the v oting principle B. If the n um b er of egoists is still smaller, then f or the neutral en vir on m en t, th e in cremen t in their c apital at the maxima exceeds a ll increments reac hab le b y the group. F or a small group, zones 4 and 5 merge, and zone 3 degenerates. At the in terfaces of zones 4 and 5 b oth the group and the egoists ha v e maximal increments in the capital, but the group maxim um is h igher. 2. As for the scenario describ ed in the Introd uction—egoists join the group and th us approac h the group egoism to altruism,—it is abs olutely realistic. The group is esp ecially attractiv e for the mean v alues of the decision thr eshold to whic h the sim p le-ma jority threshold α = 0 . 5 b elongs. F or suc h thr esholds, the group retains its attractiv eness eve n if it includes the m a jority of the participan ts; the egoists can hav e adv antag e only for h igh and lo w thresholds. 3. As w as noted in Item 1 of this S ummary , a smaller cate gory of the participan ts has adv an tage in zones 4 and 5. Explanation of this curious phenomenon is related with th e law of large num b ers: since the v ariance of the sample mean decreases with sample volume, th e p rop osals that are “v ery go o d ” in the sense of the m ean increment of capital are less frequent for the larger cat egory . The AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 STRA TEGIES OF VOTING 327 “v ery bad” prop osals are less frequent as w ell, bu t this fact do es n ot affect the dynamics b ecause these prop osals are r ejected b y v oting. 4. Principle B whic h b etter protects the group against the manipulations of the o rganizers is preferable to principle A also in the sense of the mean increment in capital, whic h can b e readily explained by the f act that it i s n amely the p ositiv eness of the mea n (total) incremen t in capital, rather than satisfact ion of the ma jorit y of the group ( as it is the case with principle A) , that is declared b y principle B as the group utilit y . F or example, in the case of the neutral en vir on m en t, the ratio of the incremen ts in capitals of the groups adhering to principles B and A is p π / 2 ≈ 1 . 25. 5. F or c hanges in µ and σ th at retain their ratio µ/σ , the graphs of the capital increments extend/con tract alo ng the Y -axis in prop ortion to σ . Therefore, the impact of th e parameter σ on the increments in capital is defined by µ . Namely , the passage from σ to σ ′ = ρσ ( ρ > 0) provides graphs extended b y the factor of ρ th at are ob tained b y p assing from µ to µ/ρ . 6. Th e maxima of the increment in the ego ists’ capitals lie in the middles of zones 4 and 5. In the more distan t, “external” parts of th ese zones, the egoists ha v e adv an tage o v er the group f or smaller, equal, and ev en somewhat higher their n um b er. The group can reduce or, for a su fficien tly high relativ e num b er of the egoists, ev en brin g to nothing this effect by passing to the v oting principle A ′ . By c han ging its “in ternal v oting threshold,” the group can ev en ac hiev e adv an tage o ver the egoists. At the same time, in the ma jority of cases it reduces its mean incremen t in capital, but the mean in cr ement in the egoists’ capital r educes ev en more. Is it adv an tageous to the group? There is n o unam biguous answ er. In the so cial practice, the question “W hat is mo re att ractiv e, to live b etter than b efore or to liv e b etter then the rest?” alwa ys remains op en. It is only clear that as a d isciplined unit th e group can c h o ose an answer, whereas the egoists do n ot ha ve such a p ossibilit y . 8. CONCLUSIO NS The pap er analyzed the m o del of so cial dynamics defin ed by voti ng in the s tationary sto c hastic en vir onmen t. Time un iformit y of the environmen t p arameter d efines the sp ecificit y of the r esults and distinguishes the case at hand from the situatio n of v oting whic h the organizers try to ma- nipulate. As w as sho w n b y a nalysis, for wider domains in t he parameter sp ace, th e g roup has a b etter dynamics of capital th an the egoists, whic h mak es realistic the scenario where the ego ists join the group and the group egoism approac hes altruism. Th e narro w domains where the ego ists ha ve adv an tage ov er the group were id entified. T h e main results of analysis w ere in terpreted in terms of the s o cial sciences. REFERENCES 1. Mirkin, B.G., Pr oblema grupp ovo go vyb or a (Problem of Group Choice), Mosc ow: Nauk a (Fizmatlit), 19 74. 2. Aizerman, M.A., Dynamic Asp ects of the V oting Theo ry (Review of the Problem), Avtom. T elemekh. , 1981, no. 12, pp. 103–11 8. 3. Chebotare v, P .Y u., Some Prop erties o f T r a jectories in the Dynamic Problem o f V o ting , Avtom. T elemekh. , 1986, no. 1, pp. 133–1 38. 4. Alesk er ov, F.T. a nd Or teshu k, P ., Vyb ory. Golosovanie. Partii (Elections. V o ting. Parties), Mos c ow: Ak ademiya, 19 9 5. 5. Borzenko, B.I., Lezina, Z.M., L ezina, I.B., et al. , Mo del of So cia l Dynamics Defined by Collective Decisio ns and Ra ndom E n vironmental Changes, in: II Mezhdunar. konf. p o pr obl. upr. T ez. dokl. (II In t, Co nf Control, Abstracts), Moscow: IPU RAN, 2003, p. 120. 6. Chebotare v, P .Y u., Borzenko, V.I., Le z ina, Z.M., et al. , Mo del o f So cia l Dynamics Controlled by Colle c tiv e Decisions, in: T r. In-t a pr obl. upr. RAN. (Pro c. Int. Cont rol Pr obl.), vol. XXI I I, Moskv a , 20 0 4, pp. 102– 109. AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006 328 BORZENKO et al . 7. Chebotare v, P .Ju., Bo rzenko, V.I., Lezina, Z.M., et al. , A mo del of So cia l Dynamics Gov erned b y Collec- tive Decisio ns, in: Pr o c. Int. Conf. “Math. Mo del ling of So cial and Ec onomic Dynamics” (MMSED-2004), June 23–25, 2004 , Mos cow: RSSU, 200 4, pp. 8 0–83. This p ap er was r e c ommende d for public ation by F.T. Alesker ov, a memb er of the Editorial Bo ar d AUTOMA TION AN D REMOTE CONTROL V ol. 67 No. 2 2006