ТрВ №21 (40N): Математика выборов

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Газета

Троицкий Вариант

Газета Троицкий Вариант # 40 (27_10_2009)

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Газета

Троицкий Вариант

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2010-01-29 nmd20100129144851-010 1.1

ТрВ №21 (40N): Математика выборов

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МАТЕМАТИКА ВЫБОРОВ

Статистическое исследование результатов российских выборов 2007 — 2009 гг.

С. Шпилькин

Прошедшие выборы в Мосгордуму породили волну альтернативного анализа их результатов. «ТрВ-Наука» предлагает один из вариантов статистического анализа итогов выборов.

Игровой подход к определению предела

Леонид Костюков

Многие математические понятия не всегда легко усвоить. В статье рассматривается определение предела с довольно неожиданной — игровой — точки зрения.

ФЦП «Кадры» — гладко было на бумаге...

Евгений Онищенко

«Троицкий вариант» продолжает отслеживать ход конкурсов Федеральной целевой программы «Научные и научно-педагогические кадры инновационной России».

Юбилей классика

Борис Штерн

В Институте ядерных исследований 23 октября, в связи с 75-летним юбилеем, чествовали академика РАН, лауреата Ленинской и прочих премий Владимира Михайловича Лобашева.

Экстравагантные консерваторы и консервативные эксцентрики

Сергей Попов

Насколько консервативна наука? Чем гипотеза отличается от «веры»? Конкретные примеры стандартных гипотез и моделей и альтернатив им позволяют разобраться в этих непростых вопросах.

Любимый храм Тараса Шевченко

Алексей Паевский

В статье рассказывается о Михайловском храме XI века в Выдубическом монастыре в Киеве.

Отпугивающая морфология

Георгий Любарский

Красота гусениц одновременно завораживает и пугает. Яркая окраска сообщает хищнику: «Лучше меня не есть!»

Начало «Одиссеи». Американская школа глазами дедушки

Борис Докторов

«Троицкий вариант» публикует статью об одной из школ в штате Калифорния, США. В основу очерка легли как собственные наблюдения автора, так и рассказы его внучки, которая учится в этой школе.

Механизмы Чебышёва обретают новую жизнь

Наталия Демина

Заметка рассказывает о новом проекте фонда «Математические этюды».

Еще раз к вопросу о корчевании

П. П. Федоров

Рассказ о борьбе за качество публикаций в еще одном научном журнале и рассуждения о различных аспектах «корчевания».

О раскопках на Охте, судьбе охтинского мыса и попугае Джона Сильвера

Сергей Белецкий

Чем важны археологические раскопки на Охте?

АВТОРСКИЕ КОЛОНКИ:

Смысл образования

Анастасия Казанцева

Если вашей статьи нет в Интернете...

Ревекка Фрумкина

Настольная эволюция

Ирина Якутенко

We shall overcome

Ирина Левонтина

Лохи из Петербурга

Лев Клейн

Итоги выборов, или Горе от ума

Иван Экономов

Математика выборов

Политика не относится к основным сферам, которые призван освещать «Троицкий вариант». У газеты совсем другие точки приложения сил, например просветительство, борьба с невежеством и мракобесием. Однако политика зачастую вторгается именно в эти сферы со своими грязными башмаками, ложью и невежеством. Естественно, эти случаи являются предметом нашего дотошного анализа.

«Призывы к изменению предусмотренных законом процедур можно рассматривать как совершение действий, подпадающих под признаки преступления, — воспрепятствование осуществлению избирательных прав или работе избирательных комиссий, поскольку данные призывы препятствуют деятельности избирательных комиссий по подведению итогов голосования и определению результатов».

Владимир Чуров,

председатель Центральной избирательной комиссии РФ (ИТАР-ТАСС)

«Люди, услышав о каком-то необыкновенном явлении, начинают предлагать для их объяснения малоправдоподобные гипотезы. Прежде всего, рассмотрите простейшее объяснение — что все это — враньё».

Лев Ландау

(по БЛ. Иоффе, «Без ретуши. Портреты физиков на фоне эпохи»)

Именно таким случаем являются выборы последних лет, причем год от года результаты выглядят все более вопиющими. Человеку, имеющему хотя бы общее представление о статистике и обладающему здравым смыслом, достаточно посмотреть на график в центре этой страницы, чтобы понять: на выборах этого года в Москве имела место массовая беззастенчивая фальсификация (более чем на двух третях участков), а на некоторых участках фальсификация была чудовищной. При этом официальные лица, глазом не моргнув, заявляют, что выборы были честными, а все сомнения в этом — на руку врагам. Конечно, эти самые официальные лица понимают, что им не верят не только образованные люди, но и огромное число простых людей. Им хорошо бы получить поддержку титулованных ученых — чтобы было чем заткнуть всяких там математиков, биологов и физиков, по роду деятельности хорошо знающих статистику и измывающихся над ЦИК. Видимо, именно за этим Владимир Чуров обратился в Президиум РАН, выступив там с докладом «Избирательный процесс: теория, практика и научное сотрудничество».

К чести членов Президиума, восторженной ответной реакции докладчик не получил. При обсуждении доклада академик Александр Дмитриевич Некипелов сказал, что «в докладе определены очень интересные направления сотрудничества с РАН и сферы, организующие выборную систему в стране. Всего за 15 лет проделана очень большая работа и достигнуты впечатляющие результаты. Есть проблема доверия населения к институту выборов. В этом направлении делается очень много, но все эти действия не слишком убеждают. Например, в Москве примерно в 16.00 была явка 5%, а к концу голосования — около 35%. Институт выборов необычайно важен в условиях демократической системы. В связи с этим очень важна роль общественных наук. Владимир Евгеньевич и его коллеги многое делают, чтобы поднять институт выборов на должную высоту. Хотелось бы, чтобы последовали дальнейшие контакты с представителями общественных наук, чтобы определить конкретные темы нашего сотрудничества».

Президент РАН академик Юрий Сергеевич Осипов поблагодарил автора доклада и сообщил, что «в структуре РАН имеется институт, который занимается разработкой системы выборов с высокой степенью защиты информации. РАН имеет отношение и к разработке системы ГЛОНАСС, здесь мы тоже можем вам помочь. Для того, чтобы повысить уровень доверия населения, необходимо привлекать высокообразованных социологов и статистиков».

Генеральный директор ВЦИОМ Валерий Федоров в комментарии ER.RU в субботу заметил, что уже сам термин «сложные математические вычисления» выглядит неубедительным, поскольку метод, которым пользовались исследователи, так и не раскрыт.

Официальный сайт партии «Единая Россия»

«Это на руку тем силам, которые хотят дестабилизировать обстановку в России, которые хотят поставить Россию на колени».

Борис Грызлов,

председатель Государственной Думы РФ (Полит.ру)

«Не надо математикам со своими моделями лезть в политику, анализировать выборы, придумывать какие-то свои версии. Это просто смешно, когда математик рассуждает о выборах. Давайте еще биолога спросим! Или узнаем, что по поводу результатов выборов думает физик-ядерщик!»

Виталий Иванов,

вице-президент Центра политической конъюнктуры России (Полит.ру)

Следуя указаниям президента РАН, ТрВ публикует статью математика Сергея Шпилькина 26 о статистическом анализе результатов выборов.

Статистическое исследование результатов российских выборов 2007 — 2009 гг.

Избирательная система РФ в цифрах

Общее количество зарегистрированных в РФ избирателей составляет около 100 млн человек (величина варьирует от одних выборов к другим, причем не всегда понятным образом).

Общее количество избирательных участков в стране составляет немного менее 100 тыс. Таким образом, в среднем на один избирательный участок приходится примерно 1000 человек. Типичный размер избирательного участка в городах — 1500–2500 избирателей, в сельской местности — несколько сотен. Существуют «закрытые» участки (военные части, больницы, тюрьмы, суда торгового флота), в которых обычно немного избирателей и высокая (до 100%) явка.

Участковые избирательные комиссии (УИК) подчинены территориальным избирательным комиссиям (ТИК); их в РФ примерно 2750. При проведении голосований по мажоритарной системе формируются избирательные округа: в небольших субъектах Федерации — по одному, в крупных — по несколько; на выборах в Госдуму 2007 г. их было 153 в РФ, в том числе 10 в Москве, на выборах в Мосгордуму 2009 г. — 17 (в Москве).

Начиная примерно с 2003 года (с выборов в Государственную Думу 4-го созыва) все результаты выборов с детализацией вплоть до избирательных участков публикуются на сайте ЦИК www.izbirkom.ru 56 .

Введение

Тема статистического анализа результатов выборов вновь стала популярной в связи с недавними выборами в Мосгордуму. Как и после федеральных выборов 2007–2008 года, отношение к самой идее статистического анализа выборов неоднозначное. Некоторые эксперты вообще высказываются в том духе, что неспециалистам в эту тему соваться не стоит. Так, вице-президент Центра политической конъюнктуры Виталий Иванов говорит: «Не надо математикам со своими моделями лезть в политику, анализировать выборы, придумывать какие-то свои версии. Это просто смешно, когда математик рассуждает о выборах. Давайте еще биолога спросим! Или узнаем, что по поводу результатов выборов думает физик-ядерщик! У них, наверное, тоже найдутся идеи, как высчитать, какой на самом деле должен быть результат. Я считаю, всерьез обсуждать здесь нечего!» ( www.polit.ru/news/2009/10/19/mathematics.html 57 ).

У автора по этому вопросу другое мнение. Пока для результатов выборов не введена какая-либо специальная суверенная арифметика, любой человек имеет право анализировать эти результаты и делать свои выводы. На самом деле статистические подходы к анализу результатов выборов в России используются с начала 1990-х годов, с тех самых пор, как появились свободные выборы. Среди основополагающих работ на эту тему (большая подборка ссылок на различные материалы приведена в (1)) можно отметить исследования А.А. Собянина и В.Г.Суховольского (2). В настоящее время исследованиями на эту тему занимаются М. Мягков, П. Ордешук и Д. Шакин (3), У. Мебейн (4), А. Любарев и А.Бузин (5). После федеральных выборов 20072008 года также вышла статья председателя ЦИК В. Чурова с соавторами (6), оппонирующая книге Мягкова, Ордешука и Шакина и публикациям автора настоящей статьи в Livejournal. Хороший обзор результатов недавних выборов с точки зрения теории вероятностей написал А. Шень (7).

В этой статье я хотел бы предложить вниманию читателей результаты своих исследований статистики выборов, начатых после выборов в Государственную Думу РФ в декабре 2007 г. Материалом для исследования являются исключительно данные голосований по избирательным участкам, территориальным комиссиям, округам и субъектам Федерации, полученные с сайта Центризбиркома (см. врезку). По сути каждые выборы представляют собой гигантский эксперимент, дающий огромный (N~106 чисел для федеральных выборов, ~105 чисел для выборов в Москве) полный (без пропусков) структурированный массив данных. При этом количество отсчетов данных N настолько велико, что этот массив можно разбивать на большое количество подкатегорий, и каждая из этих категорий будет по-прежнему содержать статистически значимое количество отсчетов данных. Эта работа не строгое статистическое исследование, а скорее поиск «срезов» данных, позволяющих выявить интересные особенности выборов.

Статистические особенности результатов российских выборов 2007–2009 гг.

Первой и наиболее заметной особенностью выборов последних лет является зависимость результатов голосования от явки.

Для наглядности начнем сразу с конкретных примеров.

Рис. 1. Голосование избирателей за партии на выборах в Государственную Думу РФ 2007 г. Данные по избирательным округам субъектов РФ.

Рис. 2. То же самое голосование, данные по территориальным избирательным комиссиям.

Рис. 3. Голосование избирателей за кандидатов на выборах Президента РФ 2008 г. Данные по субъектам Федерации.

Рис. 4. Голосование избирателей за партии на выборах в Московскую городскую думу 2009 г. Данные по избирательным участкам.

На рис. 1–4 приведены диаграммы распределения голосов за различные кандидатуры (партии — на выборах в представительные органы, кандидатов — на президентских выборах) на думских выборах 2007 г., президентских выборах 2008 г. и выборах в Мосгордуму 2009 г. на различных уровнях детализации данных. На всех графиках по оси абсцисс отложена явка в процентах, а по оси ординат — процентная доля избирателей, проголосовавших за данную кандидатуру, от общего числа зарегистрированных избирателей (списочного состава) на избирательных участках с такой явкой (явка округляется вниз до целого числа процентов).

Все четыре диаграммы, несмотря на то, что относятся к разным выборам и имеют разную степень детализации, обладают общими особенностями.

При малых явках имеется более или менее выраженный участок, где результаты разных кандидатов растут пропорционально друг другу (точки ложатся на прямые, проходящие через начало координат). Это означает, что при изменении явки количество голосующих за все кандидатуры растет пропорционально, а процентные доли сторонников разных партий среди пришедших на выборы остаются постоянными.

При дальнейшем росте явки количество голосующих за все кандидатуры, кроме кандидатуры власти, остается постоянным, а все дополнительные голоса, возникающие от прироста явки, отходят к кандидатуре власти.

Наконец, при высоких явках доли голосов за все кандидатуры, кроме кандидатуры власти, начинают падать, а все потерянные ими голоса вкупе с дополнительными голосами от роста явки отходят к кандидатуре власти.

Поведение a) соответствует так называемому «правилу Собянина-Суховольского», предложенному еще в 1990-е годы в качестве критерия честного подсчета голосов на выборах (см. (1)). Оно возникает, например, если вероятность прихода избирателей на участок не зависит от их политических предпочтений.

Поведение b) можно объяснить различными способами. Одно возможное объяснение: избиратели оппозиционных кандидатов демонстрируют на выборах высокую активность и в большинстве случаев приходят на выборы все до одного, а избиратели кандидатуры власти пассивны, и их доля среди пришедших голосовать меняется в широких пределах. Другое объяснение: начиная с некоторой явки, весь прирост явки и все дополнительные голоса, поданные за партию власти, являются результатом вброса, приписок или административного давления (последний вариант в некотором смысле смыкается с первым объяснением — если избиратели кандидатуры власти ходят на выборы из-под административной палки).

Наконец, поведение с) может быть объяснено, например, тем, что избирательные участки с высокой явкой находятся в местностях или местах, где за кандидатуру власти голосуют охотно, а за остальных кандидатов — мало и редко. Другое возможное объяснение — отъем голосов у «прочих» кандидатов и приписки в пользу кандидатуры власти.

Чтобы понять, какие из этих механизмов больше соответствуют действительности, обратимся к другим особенностям статистики голосований.

Распределение избирательных участков по явке

Хотя явка не является официальным отчетным параметром и не фигурирует в итогах выборов, публикуемых избирательными комиссиями, неформально к ней относятся очень внимательно. Это дает порой весьма неожиданные результаты.

На рис. 5 приведено распределение избирательных участков по явке на думских выборах 2007 г. По оси абсцисс отложена явка в процентах, по оси ординат — количество избирательных участков, показавших такую явку (с округлением вниз, до ближайшего целого числа процентов).

Рис. 5. Выборы в Государственную Думу РФ 2007 г. Распределение избирательных участков по явке.

Первое, что бросается в глаза, — это необычная форма распределения в целом. Помимо основного максимума в районе 55%, распределение имеет «плечо» со стороны больших явок и резкий пик вблизи 100%. Хотя пик у 100% в основном обусловлен небольшими избирательными участками, общее число избирателей, проголосовавших на участках со 100%-ной явкой, составляет ни много ни мало 1,5 млн человек! Для объяснения столь необычной формы распределения избирательных участков по явке обычно используется тезис, что страна неоднородна и существуют территории, для которых характерны высокая явка и одновременно высокий уровень голосования за кандидатуру власти (например, сельская местность). Такая мысль проводится, например, в статье В. Чурова с соавторами (6); там же фактически признается наличие «регионов особой электоральной культуры» (читай — несвободного волеизъявления), где явка и процент голосования за властную кандидатуру могут доходить до 100%. Насколько справедлива эта аргументация — вопрос спорный; для сравнения приведем распределение избирательных участков по явке на выборах в немаленькой стране Польше (рис. 6). Здесь распределение разительно отличается от российского и гораздо больше соответствует здравому смыслу.

Рис. 6. 2-й тур выборов Президента Польши, 2005 г. Распределение избирательных участков по явке.

Однако у распределения на рис. 5 есть и вторая замечательная особенность: пики на значениях явки 60, 70, 75, 80, 85, 90, 95%. Появление таких пиков нелегко объяснить статистическими механизмами, но очень просто — человеческой психологией и стремлением показать «красивые» цифры в отчете. Это означает, что явка на выборах была предметом манипуляций и отчетным параметром, а сами результаты выборов по крайней мере частично сформированы административным воздействием. Более того, манипуляции не возникают сами по себе «снизу» — значит, было как минимум «молчаливое согласие» между теми, кто их непосредственно реализовал (на уровне избирательных участков), и теми, кто их принимал (на уровне территориальных избирательных комиссий и выше).

Рис. 7. Выборы Президента РФ 2008 г. Распределение избирательных участков по явке.

На президентских выборах 2008 г. странности распределения избирательных участков по явке только усилились (рис. 7). Начальный пик вообще пропал, а «плечо» превратилось в «плато» с мощными пиками на красивых значениях явки.

Чтобы лучше понять механизм появления подобных распределений, сделаем следующее. Разделим явку на две составляющие: явку избирателей, голосующих за кандидатуру власти, и явку избирателей, голосующих за других кандидатов, и посмотрим на распределения избирательных участков по этим двум показателям. Если вклад обеих составляющих явки в «странности» распределения по полной явке одинаков, мы можем ожидать, что распределения по обеим составляющим будут подобны, если нет — будут различия. На рис. 8 приведены соответствующие графики для президентских выборов 2008 г. (распределение по полной явке то же, что и на рис. 7).

Итак, распределение участков по явке избирателей, проголосовавших за «другие» кандидатуры, похоже на обычное нормальное распределение, за исключением добавочного «плеча» на малых явках, которое естественно связать с вышеупомянутыми «регионами особой электоральной культуры». В то же время распределение по явке избирателей, проголосовавших за Дмитрия Медведева, выглядит в высшей степени нетривиально. Возникает ощущение, что механизмы, управляющие явкой избирателей Дмитрия Медведева, с одной стороны, и избирателей других партий — с другой, радикально различаются. Подобная картина наблюдается и на других выборах. Для примера, на рис. 9 приведены аналогичные распределения для выборов 2009 г. в Мосгордуму. Картина та же: распределение по явке избирателей «остальных» партий похоже на обычное гауссово, а распределение по явке избирателей «Единой России» не похоже вообще ни на что разумное.

Рис. 8. Выборы Президента РФ 2008 г. Распределения избирательных участков по явке избирателей, проголосовавших за Дмитрия Медведева, по явке избирателей, проголосовавших за другие кандидатуры, и по общей явке.

Рис. 9. Выборы в Московскую городскую думу 2009 г. Распределения избирательных участков по явке избирателей, проголосовавших за «Единую Россию», по явке избирателей, проголосовавших за другие кандидатуры, и по общей явке.

Зависимость статистических характеристик выборов от способа голосования

Предшествующее рассмотрение наводит на мысль, что голосование за кандидатуру власти и голосование за остальные кандидатуры управляются разными механизмами. К счастью, наша избирательная система дает возможность практически «чистой» проверки этого предположения. Эта возможность связана с комплексами автоматической обработки избирательных бюллетеней (КОИБ), которыми оборудуется часть избирательных участков. Очень удобна в этом смысле Москва, где КОИБами на выборах последних лет оснащается около 30% участков. При этом КОИБы распределяются по районам города достаточно широко и бессистемно и каждый раз по-новому, так что можно считать, что контингенты избирателей на участках с КОИБ и без них одинаковые (единственное исключение — «закрытые» участки, где КОИБы не ставятся, но в Москве на таких участках избирателей немного).

Сравним результаты голосований в Москве на участках с КОИБ и без. На рис. 10 приведены распределения избирательных участков Москвы по явке на выборах в Государственную Думу РФ 2007 г.

Рис. 10. Выборы в Государственную Думу РФ 2007 г. Распределение избирательных участков г. Москвы по явке.

Распределения схожи между собой и похожи на нормальное (гауссово), хотя для участков без КОИБ есть «хвост» в сторону высоких явок. Однако численные результаты выборов ощутимо различаются:

Явка

Доля голосов за «Единую Россию»

Участки с КОИБ

52,6%

49,7%

Участки без КОИБ

56,3%

56,1%

Таким образом, наличие КОИБ снижает явку и почему-то долю голосов за «Единую Россию».

На президентских выборах 2008 г. различие между двумя типами участков значительно усилилось.

Рис. 11. Выборы Президента РФ 2008 г. Распределение избирательных участков г. Москвы по явке.

Распределения выглядят так, будто вообще относятся к разным территориям. Распределение участков с КОИБ еще сохраняет колоколообразную форму, хотя и обзавелось изрядным «хвостом» к высоким явкам. Участки же без КОИБ распределены по какому-то самобытному закону. При низких явках просматривается сходство с участками без КОИБ, но затем начинаются замечательные пики на «круглых» и «полукруглых» значениях с кульминацией на 80%-ной явке. Результаты голосования тоже различаются существенно:

Явка

Доля голосов за Д.Медведева

Участки с КОИБ

56,1%

65,9%

Участки без КОИБ

70,1%

73,6%

Рис. 12. Выборы в Московскую городскую думу 2009 г. Распределение избирательных участков Москвы по явке.

И в довершение картины — выборы в Мосгордуму в 2009 г. Распределения участков с КОИБ и без опять радикально различаются. Результаты голосования тоже различаются, но не так сильно, как на предыдущих выборах. Такое ощущение, что КОИБы «приручили»:

Явка

Доля голосов за «Единую Россию»

Участки с КОИБ

33,2%

67,4%

Участки без КОИБ

36,1%

67,5%

Количественная оценка аномалий

Из предыдущих двух глав видно, что с точки зрения статистики выборов избиратели кандидатуры власти и избиратели всех остальных кандидатов на российских выборах последних лет ведут себя совершенно по-разному: приходят на избирательные участки не по закону нормального распределения, а по своим особым законам, имеют особую склонность к голосованию на участках с высокой явкой, по-разному голосуют на участках с автоматическими сканерами бюллетеней и без них.

Вообще говоря, науке известны подобные явления. Самый известный пример — жидкий гелий, ведущий себя при температуре ниже температуры фазового перехода как смесь двух компонент — нормальной и сверхтекучей, которая отличается совершенно экзотическими свойствами: способна протекать через мельчайшие щели, утекать из сосуда по капиллярной пленке и т.п. Избиратели кандидатур власти своими экзотическими привычками во многих отношениях напоминают сверхтекучую компоненту жидкого гелия: например, склонность голосовать при высокой явке является несомненным аналогом бозе-конденсации. Этот вопрос несомненно заслуживает дополнительного изучения, хотя и выходит за рамки данной статьи.

Если же отбросить фантастические гипотезы, самым разумным объяснением такого поведения избирателей, голосующих за кандидатуры власти, представляется то, что часть голосов за такие кандидатуры получена в результате манипуляций — от административного давления вроде «не будете голосовать как надо — отключим газ (не привезем дров, закроем магазин, уволим по статье)» до вбросов и приписок. Тогда возникает вопрос: можно ли, опираясь только на статистические данные, определить, чему равна эта часть голосов (назовем ее аномальной составляющей голосов за кандидатуру власти, в противоположность нормальной, полученной в результате свободного волеизъявления граждан)?

Понятно, что в общем случае задача неразрешима: если статистические данные выборов подделаны полностью и не имеют никакого отношения к реальным результатам голосования, восстановить реальные результаты невозможно. С другой стороны, некоторые признаки (постоянство долей голосов за «прочие» партии при изменении явки на рис. 1–4, подобие распределений участков при низких явках на рис. 11) показывают, что в официальных результатах голосования есть реальная составляющая. Попробуем определить «нормальную» и «аномальную» части голосов за кандидатуры власти, используя эти реальные данные как эталон.

В связи с этим можно высказать несколько наводящих соображений. Мы видели, что голосование за разные партии существенно зависит от явки. Кроме того, очевидно, что простое вбрасывание/приписывание голосов за кандидатуру власти (самый простой вид манипуляции) на конкретном избирательном участке приводит к смещению этого избирательного участка вместе со всеми его данными (и реальными, и фальсифицированными) в сторону более высоких явок. В результате на низких явках должны оставаться участки с нефальсифицированными данными. Далее, можно предполагать, что голоса вбрасываются только за кандидатуру власти, а у остальных кандидатур они остаются неизменными либо, при более злостных манипуляциях, отбираются. Поэтому можно попробовать использовать в качестве эталона данные участков с невысокой явкой, а также данные голосований за «прочие» кандидатуры.

В соответствии с этими соображениями исследуем распределения голосов за разные кандидатуры в зависимости от явки на избирательном участке. На рис. 13 показано такое распределение для думских выборов 2007 г. По оси абсцисс отложена явка в процентах, по оси ординат — суммарное количество голосов за различные кандидатуры на участках с данной явкой (явка округляется вниз, до ближайшего целого, т.е. голоса суммируются по интервалам в 1%). Дополнительно показано распределение суммы голосов за все партии, кроме «Единой России».

Может ли российский электорат состоять из двух частей, настолько отличающихся по поведению, как избиратели кандидатов от власти и прочих кандидатов? Вообще говоря, науке известны подобные явления. Самый известный пример -жидкий гелий, ведущий себя при температуре ниже температуры фазового перехода как смесь двух компонент — нормальной и сверхтекучей, которая отличается совершенно экзотическими свойствами: способна протекать без трения через мельчайшие щели, утекать из сосуда по капиллярной пленке и т.п. Избиратели кандидатур власти своими экзотическими привычками во многих отношениях напоминают сверхтекучую компоненту жидкого гелия — например, склонность голосовать при высокой явке является несомненным аналогом бозе-конденсации. Этот вопрос несомненно заслуживает дополнительного изучения.

Из рис. 13 видно, что распределения по явке голосов за все партии, кроме «Единой России», с хорошей точностью подобны (это подтверждается, если нормировать их на суммарное распределение голосов «прочих» партий: полученные нормированные распределения практически постоянны в широком диапазоне явок), и только распределение голосов за ЕР ведет себя по-другому. При этом при невысоких явках (где-то до 55%) оно также подобно распределению голосов за «прочие» партии и лишь затем начинает отклоняться вверх от общей тенденции. Естественно предположить, что это отклонение и представляет собой «аномальную» часть голосов за ЕР, а «нормальная» часть голосов за ЕР должна быть распределена подобно распределениям голосов за все остальные партии. Сформулируем это предположение математически. Представим распределение голосов за ЕР в виде: (все голоса за ЕР) = С * (голоса за все партии без ЕР) + (аномальная часть голосов за ЕР), где С — подгоночный коэффициент, подбираемый так, чтобы при невысоких явках (где, как говорилось выше, должны оставаться только «честные» участки) аномальная часть была по возможности близка к нулю. Эмпирическим путем выясняется, что такое представление действительно возможно.

Рис. 13. Выборы в Государственную Думу РФ 2007 г. Распределение голосов за партии по явке.

Рис. 14. Выборы в Государственную Думу РФ 2007 г. Разделение голосов за «Единую Россию» на «нормальную» и «аномальную» части.

Видно, что голоса за ЕР действительно удается разделить так, что «нормальная» часть подобна распределению голосов за другие партии, а «аномальная» часть близка к нулю ниже определенного значения явки (примерно 52%). Выше этой пороговой явки аномальная часть начинает резко и устойчиво расти. Понятно, что выбор подгоночного коэффициента С сопряжен с некоторым произволом; в данном случае он выбран так, чтобы аномальная часть была везде положительна.

Рис. 15. Выборы в Государственную Думу РФ 2007 г. Разделение голосов за КПРФ на «нормальную» и «аномальную» части.

Для контроля методики полезно попытаться разделить на «нормальную» и «аномальную» части голоса какой-либо другой партии, которую мы не подозреваем в применении административного ресурса. На рис. 15 приведено такое разделение для голосов за КПРФ на тех же думских выборах 2007 г. Видно, что «аномальная» составляющая голосов, если и присутствует, интегрально значительно меньше, чем у ЕР, что задним числом оправдывает наш подход.

Теперь, имея разделение голосов за «Единую Россию» на «нормальную» и «аномальную» части, можно задаться вопросом: какими были бы результаты думских выборов 2007 года в отсутствие «аномального» вклада в голосование за ЕР? Для показанного выше разделения количество «нормальных» голосов за «Единую Россию» составляет 30,7 млн, «аномальных» — 13,8 млн. Отбросив «аномальные» голоса за ЕР, получим следующие гипотетические результаты выборов:

Официально

С коррекцией

Явка

63,7%

51,0%

КПРФ

11,6%

(57 депутатов)

14,5%

(72 депутата)

ЛДПР

8,1%

(40 депутатов)

10,1%

(51–52 депутата)

СпРос

7,7%

(38 депутатов)

9,7%

(48–49 депутатов)

ЕР

64,3%

(315 депутатов)

55,7%

(278 депутатов)

Таким образом, «Единая Россия» при гипотетическом «скорректированном» голосовании по-прежнему имела бы в парламенте большинство, но уже не конституционное. Естественно, скорректированные результаты зависят от выбора подгоночного коэффициента в разделении на «нормальную» и «аномальную» части, однако варьирование этого коэффициента в разумных пределах, при которых «аномальная» часть голосов на рис. 13 остается близкой к нулю при явках ниже порога, изменяет количество мандатов ЕР в Думе не более чем на ±1%, так что оценку можно считать корректной.

Таким же образом можно разделить на «нормальную» и «аномальную» части голо са за Дмитрия Медведева на президентских выборах 2008 г. (рис. 16):

Рис. 16. Выборы Президента РФ 2008 г. Разделение голосов за Д. Медведева на «нормальную» и «аномальную» часть.

Такое разделение дает 37,8 млн «нормальных» и 14,8 млн «аномальных» голосов за Медведева; гипотетические скорректированные результаты выборов выглядят следующим образом:

Официально

С коррекцией

Явка

69,7%

55,9%

Богданов

1,3%

1,6%

Жириновский

9,35%

11,65%

Зюганов

17,7%

22,1%

Медведев

70,28%

62,96%

Недействительные бюллетени

1,36%

1,69%

Наконец, разделим на «нормальную» и «аномальную» части голоса за список «Единой России» на недавних выборах в Мосгордуму:

Рис. 17. Выборы в Мосгордуму 2009 г. Разделение голосов за «Единую Россию» на «нормальную» и «аномальную» части.

Скорректированные итоги выборов выглядят следующим образом:

Официально

С коррекцией

Явка

35,3%

22,0%

ЛДПР

6,13%

9,82%

Патриоты России

1,81%

2,90%

Единая Россия

66,24%

45,95%

КПРФ

13,30%

21,29%

Справедливая Россия

5,33%

8,54%

Яблоко

4,71%

7,54%

Недействительные бюллетени

2,48%

3,97%

Можно отметить, что скорректированные оценки явки и процента голосов за «Единую Россию» лучше согласуются с числами, которые дают различные наблюдатели, чем официальные результаты.

Заключение

При всех недостатках современной российской выборной системы у нее есть несомненное достоинство — открытость данных. Надеюсь, эта статья и изложенные в ней подходы привлекут к анализу выборной статистики общественный интерес.

Автор хотел бы выразить благодарность всем читателям блога podmoskovnik.livejournal.com 75 за многочисленные плодотворные обсуждения, а также А.Е. Любареву и А.Ю. Бузину, посвятившим автора-дилетанта в историю вопроса.

Литература

1. www.vibory.ru/disc-mat.htm 76

2. Собянин А.А., Суховольский В.Г. Демократия, ограниченная фальсификациями: Выборы и референдумы в России в 1991–1993 гг. М., 1995 www. hrights.ru/text/sob/ 77

3. M. Myagkov, P. Ordeshook, D. Shakin, The Forensics of Election Fraud: Russia and Ukraine, Cambridge University Press, 2008; см. также http://vote.caltech.edu/drupal/files/working_paper/vtp_wp63.pdf 78

4. Mebane, Walter R., Jr., and Kirill Kalinin 2009. «Comparative Election Fraud Detection». www.umich.edu/~wmebane/apsa09.pdf 79

5. Бузин А.Ю., Любарев А.Е. Преступление без наказания. Административные технологии федеральных выборов 2007–2008 годов. Группа Компаний «Никколо М», Панорама, 2008

6. Чуров В.Е., Арлазаров В.Л., Соловьев А.В. Итоги выборов. Анализ электоральных предпочтений. www.cikrf.ru/newsite/illuziya/itogi_160908.jsp 80

7. Шень А. Выборы и статистика: казус «Единой России» (2009). http://alexander.shen.free.fr/elections.pdf 81 Некоторые массивы выборных данных, использованные в статье, размещены на сайте Независимого института выборов, на странице www.vibory.ru/elects/UIK.htm 82

С автором статьи можно связаться по электронной почте podmoskovnik@gmail.com 83 или через его блог podmoskovnik.livejournal.com 84

P.S.

Когда верстался номер, пришло сообщение о том, что на участке № 192 в Хамовниках, где голосовал глава «Яблока» Сергей Митрохин с семьей и где по официальным результатам не было подано ни одного голоса за «Яблоко», в соответствии с решением Хамовни-ческого суда был произведен пересчет бюллетеней. В результате пересчета среди 87 бюллетеней, ранее засчитанных КПРФ, было найдено 16 бюллетеней за «Яблоко», 3 бюллетеня за ЛДПР и 1 за «Патриотов России». Среди 29 бюллетеней за «Справедливую Россию» было найдено 2 недействительных. Среди 904 бюллетеня за «Единую Россию» неправильно подсчитанных обнаружено не было.

Таким образом, среди 116 бюллетеней, поданных за оппозиционные партии, оказалось 22 неверно учтенных, а среди 904 бюллетеней за ЕР — ни одного. Нетрудно подсчитать, что вероятность такого события, в предположении, что при изначальном подсчете все бюллетени учитывались одинаково тщательно и пересчет выполнен точно, составляет (116!*998!)/(94!*1020!), т.е. примерно 2,5 на 10 в минус 22-й степени. Российская избирательная система еще раз подтвердила, что для нее нет непреодолимых препятствий в теории вероятностей.

С.Шпилькин, независимый исследователь

Игровой подход к определению предела

Любому человеку, имевшему опыт столкновения с математическим анализом, известно: тяжело вначале — потом довольно легко. Кажется, даже медведя можно обучить дифференцировать по заданным правилам. Искать пределы на практике тоже можно наловчиться. А вот дать точное определение предела последовательности или функции да еще объяснить его может лишь один студент технического вуза из пяти (о старшеклассниках и говорить не приходится). Действительно, банальная сходимость последовательности x(n) к 5 формально выглядит как

∀ ε>0 ∃ N(ε): ∀ n>N x(n) — 5 < ε.

(Для людей, не ежедневно соприкасающихся с математическим анализом, привожу вариант определения на русском языке: «Для любого, возможно очень маленького, но положительного числа эпсилон всегда найдется номер, зависящий от этого эпсилон такой, что все члены последовательности с номерами больше этого отличаются от 5 менее чем на эпсилон».)

Возникает искушение говорить о предмете неформально. Например, определение вектора как «направленного отрезка» ненаучно, но на практике не приводит ни к каким бедам. Такой неформальный, разговорный подход к теме можно уподобить черному ходу в дом. Но в нашем случае черный ход (правдоподобные разговоры об «очень маленьких числах») ведет в другой дом. Напомним вкратце историю предмета, впрочем, хорошо известную специалистам.

Анализ бесконечно малых возник в головах Ньютона и Лейбница в XVII в. Эти два гения, не нуждаясь в формализации своих интуитивных озарений, делали верные умозаключения. Но даже великие математики того времени, полагаясь на интуитивное полупонимание предмета, совершали нелепые и комические ошибки. И лишь 200 лет спустя Коши и Адамар сочинили «язык е — А», на котором можно доказательно рассуждать о бесконечно малых. Так что неформальной альтернативы ему нет, и, если пропустить этот этап обучения, все здание матанализа будет лишено твердого фундамента, а все навыки студента сведутся к непонятным по сути обычаям и пассам.

Так что остается вдохнуть и выдохнуть — и все же одним учить определение предела, а другим — его (по возможности доходчиво) преподавать.

Есть предел последовательности, есть функции. Аргумент функции может стремиться к некоторому числу слева, справа или с обеих сторон, а может — к ±оо (плюс-минус бесконечности). Значение — к конечному или бесконечному пределу. Если различать +<» и -» (а формулы их различают), мы получим только в случае функций 15 конкретных определений предела. Понятно, что заучить их без понимания — непосильный труд. Они во многом однотипны; надо усвоить их внутреннюю логику.

Сложность для студента представляют три квантора, всегда идущие в одном порядке:

∀… ∃… ∀… (для любого ... существует ... для любого)

Возникает естественная идея: а что если исходить не из содержания каждого отдельного квантора, а из самой идеи их чередования? Пусть их будет вообще много:

∀… ∃…∀… ∃…∀… ∃…∀… ∃…∀… ∃…∀… ∃…∀… ∃…

На что это похоже?

Представим себе игру на двоих типа шахмат (чередование ходов), но для простоты — без возможности ничейного исхода. То есть неминуем мат белым (МБ) или мат черным (МЧ). Тогда нетрудно заметить, что из данной позиции мат черным в три хода записывается такой формулой (где б(к); ч(к) — соответственно к-е ходы белых и черных):

∃ б(1) ∀ ч(1) ∃б(2) ∀ ч(2) ∃б(3) МЧ

(Опять по-русски: Есть такой первый ход белых, что на любой ответ черных найдется — естественно, не универсальный, а зависящий от этого ответа второй ход белых, такой, что на любой ответ черных третьим ходом — белые ставят мат.)

Мат же белым предполагает другое начало — с квантора всеобщности: как бы ни пошли белые, они обречены.

Наша игра «предел» предполагает три хода — белых, черных и белых, потом — сверку полученного результата и непременный выигрыш черных. Вот как выглядит сходимость последовательности к 5 (исходная формула):

Первый играющий называет значение числа е. Например, 0,000000001.

Второй, подумав, выдает N = 16573. Первый вправе назвать любое число больше N. Он называет 16585.

Идет сверка. И х(16585) — 5 оказывается меньше 0,000000001.

Первый расплачивается. Азарт толкает его на повторение эксперимента. Теперь он выдвигает и вовсе микроскопическое s = 0,000000000000000000000000000000000000001.

Второй, подумав, выдает N=178539763. И, вне зависимости от следующего декоративного хода Первого, Второй выигрывает по итогам сверки.

Если Первый — не клинический идиот и не чересчур богат, после получаса игры он понимает две вещи:

1) второй выигрывает всегда (это и есть сходимость — квантор в квантор);

2) так как он выигрывает одним ходом, в нем заключена суть выигрыша.

Опыт показывает, что такая игровая модель наглядна и способствует лучшему усвоению материала.

Здесь же — и возможность немного неформального, нестрогого, но быстрого и конструктивного определения предела, того самого разумного упрощения без катастрофической потери смысла. В подавляющем большинстве случаев сходимость последовательности (функции) — это способ вычисления N (или А) по s. Иначе говоря, решающий ход. Заметим, что просто наблюдением за формулой предела выделить из нее решающее звено 3 N(s) не так-то легко. А в метафоре игры — мгновенно.

Всматривание в природу решающего хода наводит на еще одну модель. Давайте будем говорить о сходимости как о существовании машинки, перерабатывающей ε в N (или Δ). Вот такой: ε → ♦→ Δ. Тогда доказательства сходимости сводятся к конструированию и предъявлению таких машинок — либо «винтик за винтиком» (сходимость конкретных функций и последовательностей), либо с использованием уже существующих, данных машинок, неважно как, но действующих, типа запаянных микросхем (сумма, произведение, суперпозиция функций).

Здесь надо сказать, что этот (довольно очевидный) язык — с «распараллеливанием» s на две машинки, с черными ящичками и т.п. — оказывается весьма современным. Отвинтив боковую стенку компьютера, любой студент видит нечто подобное. Поэтому здесь идет оживление формулы, визуализация.

К сожалению, в большинстве вузов мальчики и девочки довольно бойко дифференцируют и вычисляют пределы, не зная, что такое производная и предел. По-моему, этот компромисс вреден со многих точек зрения. Об этом можно впоследствии поговорить отдельно.

Леонид Костюков

ФЦП «Кадры» — гладко было на бумаге...

110

«Троицкий вариант» продолжает отслеживать ход конкурсов Федеральной целевой программы «Научные и научно-педагогические кадры инновационной России»

Федеральная целевая программа (ФЦП) «Кадры» была задумана, чтобы поддержать лучшие научно-образовательные центры, научные группы и отдельных молодых исследователей. Пробив себе дорогу в серьезной конкуренции с проектами многих других федеральных целевых программ, ФЦП «Кадры», наконец, заработала в этом году, пусть первоначально планировавшиеся расходы на нее и были урезаны в связи с кризисом. Однако ей не суждено было оправдать надежд на качественное отличие в организации конкурсов и экспертизы от того, что как это устроено в уже действующих федеральных целевых программах научно-технической направленности. Виной тому в первую очередь Федеральный закон от 21.07.2005 № 94-ФЗ «О размещении заказов на поставки товаров, выполнение работ, оказание услуг для государственных и муниципальных нужд» и психология чиновников, не желающих отступать от наработанных шаблонов проведения конкурсов. Дело в том, что Федеральный закон 94-ФЗ устанавливает такие правила оценки заявок, подаваемых на конкурсы, которые практически не учитывают особенностей научно-образовательной сферы (см. выпуск «Троицкого варианта» от 17 марта 2009 г.), и, кроме того, прописывает процедуру подачи заявок, неудобную для нормальной организации научных конкурсов, а чиновники по своей инициативе делают процедуру подачи заявок еще более тяжелой и запутанной.

В случае конкурсов научно-образовательных центров (НОЦ), проводимых Роснаукой, у которой был определенный опыт проведения конкурсов научной направленности (ведущие научные школы и т.д.), конкурсная документация была неудобной. Но -насколько это позволяет Федеральный закон 94-ФЗ — она была хотя бы относительно адекватной и, если так можно выразиться, посильной для коллективов НОЦ. Поэтому основные содержательные потери были связаны с диктуемыми федеральным законом критериями оценки заявок (из числа победивших коллективов менее двух третей выиграли бы, если бы заявки оценивались только по научным критериям). А вот в конкурсах Рособразова-ния, никогда не проводившего массовых научных конкурсов, конкурсная документация оказалась крайне неудобна и громоздка для заявителей. В результате «основной тур» конкурсов для научных групп под руководством докторов и кандидатов наук, а также для молодых кандидатов наук и аспирантов, прошедший в мае — августе 2009 г., оказался без преувеличения провальным.

Во-первых, многие из тех, кто мог бы участвовать в конкурсе, сознательно отказались делать это, ознакомившись с комплектом конкурсной документации. Для групп под руководством докторов наук было подано 3,1 заявки на место, для групп под руководством кандидатов наук -2,2 заявки на место, для молодых кандидатов наук — 2,5 заявки на место, для аспирантов — ... 0,6 заявки на место. Для сравнения: для НОЦ средний конкурс составлял примерно 6 заявок на место, а в 2008 г. конкурс на гранты Президента РФ для молодых кандидатов наук составлял примерно 5,4 заявки на место. И это при том, что размер гранта Президента РФ для молодого кандидата наук составлял на тот момент 150 тыс. руб. в год, а цена госконтракта для аспиранта в рамках ФЦП «Кадры» может доходить до 500 тыс. руб. в год, а для молодого кандидата наук — до 1 млн руб. в год.

Во-вторых, мало подать заявку — необходимо еще, чтобы она была допущена к конкурсу. И аспиранты, и доктора наук показали себя в важнейшей бюрократической науке правильной подготовки заявок не лучшим образом: примерно 40% заявок не было допущено к конкурсу за несоблюдение требований к оформлению заявки. Таким образом, реальный конкурс оказался еще более низким, чем написано выше. По некоторым лотам были отвергнуты все (!) поданные заявки, по другим лотам существовавшая изначально конкуренция была сведена на нет. Так, по лоту «Создание электронной компонентной базы», предусматривавшему заключение 7 государственных контрактов, из 29 поданных заявок соответствующими требованиям было признано 8, а по лоту «Производство топлив и энергии из органического сырья», предусматривавшему заключение 8 государственных контрактов, из 24 заявок было допущено к конкурсу только 6. По сути, был проведен конкурс на знание всех бюрократических нюансов проведения конкурса.

Но, может быть, при всей уродливости процедуры подготовки бумаг и подачи заявок прошедшие бюрократическое сито заявки качественно оценивались хотя бы по научным критериям? Увы, нельзя сказать и этого, в чем можно убедиться, сравнивая оценки по научным критериям, полученные заявками от вузов и НИИ в двух «турах» конкурсов НОЦ. Дело в том, что многие учреждения, не выигравшие в «первом туре» конкурсов НОЦ (очереди I-VI), подали заявки на «второй тур» (очереди VII-X) по той же самой тематике. Изучая протоколы заседания конкурсных комиссий по оценке и сопоставлению заявок, можно найти множество случаев, когда заявки учреждений, в первом туре получавшие высокие оценки, во втором получали в полтора, два и даже более раз низкие оценки (и наоборот). Так, заявка Института физической химии и электрохимии им. А.Н. Фрумки-на РАН в первом туре по критерию «качество работ и квалификация участника» получила 44,45 баллов, а во втором туре — 19,8 балла. Заявка Института автоматики и электрометрии Сибирского отделения РАН в первом туре по критерию «качество работ и квалификация участника» получила 42,25 баллов, а во втором -23,55 балла. Из обратных примеров наиболее впечатляющим является, пожалуй, следующий: заявка Федерального государственного унитарного предприятия «Государственный ордена Трудового Красного Знамени научно-исследовательский институт химических реактивов и особо чистых химических веществ» в первом туре по критерию «качество работ и квалификация участника» набрала 9,93 балла, а во втором — 40,85 баллов.

Можно допустить, что в некоторых случаях столь значительное изменение оценки заявки связано с изменением самой заявки. Но очевидно (а в некоторых случаях — доподлинно известно), что учреждения, получившие высокие оценки по научным критериям, подали на конкурсы второго тура ровно те же самые заявки, подкорректировав только цену контракта и сроки выполнения работ. Каким образом при качественно организованной экспертизе, проводимой независимо двумя-тремя квалифицированными экспертами, в достаточно массовом порядке возможны столь значительные расхождения в оценке заявок — тайна великая есть.

Однако руководство Минобрнауки предпочитает сохранять олимпийское спокойствие и, более того, рапортует об успешной работе Программы. К примеру, в начале июля, когда уже было известно о провальном ходе конкурсов групп под руководством кандидатов наук и аспирантов, министр образования и науки А.А. Фурсенко докладывал премьер-министру В.В. Путину, что все идет хорошо и конкурсы ФЦП «Кадры» востребованы. В попытке создать благостное впечатление о ходе Программы министр порой откровенно лукавит. Так, в начале октября, говоря о грядущем скоро массовом возвращении работающих за рубежом российских ученых на Родину, А.А. Фурсенко сказал, что к реализуемой сейчас Минобрнау-ки программе, направленной на привлечение российских ученых, живущих за рубежом, к работе с российскими студентами и аспирантами, был проявлен большой интерес (конкурс составил 8 человек на место). Речь идет о мероприятии 1,5 ФЦП «Кадры», в рамках которого проводятся конкурсы для «приглашенных исследователей», и любой желающий может зайти на сайт ФЦП «Кадры» www.fcpk.ru 111 (там публикуется информация о мероприятиях Программы, курируемых Роснаукой) и убедиться, что по мероприятию 1,5 конкурс составил не 8, а примерно 3,5 заявки на один государственный контракт. Чиновники Роснауки также заявляют, что конкурсы проходят нормально, Федеральный закон 94-ФЗ не создает особых трудностей, а победителями становятся действительно достойные вузы и НИИ.

Говорят, что конкурсы будущего года ФЦП «Кадры» начнут объявлять уже в этом году. Будут ли внесены серьезные коррективы в процедуру подачи и оценки заявок? Похоже, позиция руководства Минобрнауки, а также подведомственных ему федеральных агентств — Роснауки и Рособразования, суть которой можно выразить в классической формуле «Все хорошо, прекрасная маркиза!», не оставляет шансов на это.

Евгений Онищенко

Дорогие коллеги!

Просмотр > Главная 1 / Анонсы 114 / Дорогие коллеги!

Дорогие коллеги!

27 Окт 2009 г.

Борис Штерн

Рубрика: Анонсы 114

Один комментарий 115

Я хочу попросить вас, тех, кто читает и любит «Троицкий вариант», при удобном случае поддержать газету необременительным пожертвованием, либо подписаться на нее. В данный момент, пока тянется оформление НКО для газеты, мы критически нуждаемся в деньгах и уже изрядно залезли в долги. Да и когда бюрократические проблемы будут решены, фонд «Династия» все равно будет обеспечивать лишь часть требуемых денег, и поддержка читателей жизненно необходима.

Удобный случай поддержать газету появляется каждый раз, как вы идете платить за телефон, или просто вам на пути попадается платежный терминал Qiwi (или любой другой терминал, принимающий «Яндекс-деньги»).

На примере Qiwi это делается так:

— Жмем верхнее меню, «Оплата услуг».

— Затем выбираем «электронную коммерцию».

— В новом меню жмем «Яндекс-деньги».

— Далее он спросит номер счета. Вот этот номер: 41001438067950 . (Кошелек оформлен на мое имя).

— Потом он спросит номер мобильника — это для качания прав, если что не так.

— Потом — ввод денег.

Комиссия — 5%

Если вы не просто жертвуете, а хотите подписаться на бумажную версию ТрВ (600р. В год), отправьте письмо со своим почтовым адресом Илье Мирмову: miily собака yandex.ru. Впрочем, даже если вы просто жертвуете на газету, все равно будем рады, если сообщите об этом Илье или мне.

Разумеется, перевести деньги можно и напрямую, с собственного счета «Яндекс-денег», если такой есть. Если удобнее заплатить через WebMoney, напишите, такая возможность тоже есть, перешлем номер кошелька WMZ или WMR.

Также будем рады, если вы перешлете это письмо своим знакомым.

Спасибо!

Борис Штерн

Notes

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