מי הוא השחקן? ישראל אומן סמינר משחקים אוניברסיטת בר אילן י"ב חשון תשע"ד

In applications of Game Theory, a “player” is a person, household (or family), team, country, persuasion (political, כיפה סרוגה, חילוני, חרדי) community, corporation, workplace (hospital, department, research center), species or other population, …

But, it isn’t really that way. Usually, this is understood as an idealization, a kind of “small worlds” principle (a la Savage). In games where, say, countries are modeled as players, the “real” players are the individual citizens, with their individual goals and individual decisions and individual free will. It’s only because this “true” game is too big and unwieldy to analyze that, it is held, game theorists model players as they do.

Today, I’m going to explore the idea that it IS “really” that way: that in large part, collectives are like individual people, and may be thought of as such. And, perhaps, not only in Game Theory.

Individual People composed of cells grouped in organs & limbs which are interdependent change over time; in particular, age and die have diseases and internal mechanisms to fight them have relationships with others – friendly or hostile have internal struggles make decisions – consciously or otherwise Collectives composed of people grouped in economic, social, political, … entities which are interdependent change over time; in particular, age and die crime, pollution, … police, courts, public awareness, … alliances, wars, economic cooperation and barriers.. have internal struggles make decisions – centrally or otherwise

Individual Will and Collective Will Individual “will” is heavily influenced by the collective: in the family (e.g., choice of careers by children) by and large, people don’t change persuasions by and large, people identify with the collectives to which they belong – and vice versa work ethic (hospitals, companies, research centers) shaped by communication and incentives

ויקרא כ"ד, י"א: ויקוב... את השם ויקלל... ושם אמו שלומית בת דברי למטה דן. רש"י על אתר: מגיד שהרשע גורם גנאי לו, גנאי לאביו, גנאי לשבטו. כיו"ב, (שמות ל"א, ו') אהליאב בן אחיסמך למטה דן – שבח לו, שבח לאביו, שבח לשבתו.

Personification in Science social insects population equilibrium = Nash equilibrium (Maynard Smith and Price, 1972) N-M stable sets: Hart’s theorem ( JET 7, 1974, p.453)

Social Insects With social insects (bees, ants, termites, wasps), there is a school of thought in entomology that thinks of the nest (or hive) as the “individual” – not the particular worker, soldier, or queen.

Population equilibrium = Nash equilibrium A Nash equilibrium of a game is a profile of (possibly mixed) strategies, one for each player, each of which is best possible for that player, given the others’ strategies. An equilibrium of populations (like species) is a profile of genotypes (or distributions of genotypes), one for each population, under which the population proportions do not change from generation to generation. Think of a “game” whose “players” are the populations, “strategies” – the possible genotypes, and “payoffs” to a profile of genotypes – changes in fitness of each individual when individuals with those genotypes meet. Thm. The population equilibria in the ecological story coincide with the Nash equilibria of the corresponding game.

N-M stable sets: Hart’s theorem A von Neumann-Morgenstern stable set of a coalitional game (like a market) is a collection of outcomes (payoff vectors) with certain stability properties (which we won’t specify here). In large markets with differentiated players (like employers and employees), the N-M stable set of the large market mimics the N-M stable set of the small market – that with only one representative of each type. For example, when there are only one employer and one employee, the N-M stable set provides that they can divide the surplus in any way they want. When there are many employers and many employees, the surplus can again be divided arbitrarily – but all employers must pay, and all employees get – the same wage.

האנשה בתורה שמות י"ד, י': וישאו בני ישראל את עיניהם, והנה מצרים נוסע אחריהם, וייראו מאד. רש"י: בלב אחד, כאיש אחד. Rashi: Of one mind, like one person. שמות י"ט, ב': ויבואו מדבר סיני... ויחן שם ישראל נגד ההר. רש"י: כאיש אחד, בלב אחד. Rashi: Like one person, of one mind.

Possible Implications “Full world” analysis of game situations. Moral judgment of collectives. The above viewpoint is a way of looking at things, which may be illuminating. I’m not advancing it as the “right” way. Dual approaches.

השלכות מוסריות אפשריות פיחות באחריות המוסרית של היחיד במסגרת הכלל פיחות באחריות המוסרית כלפי היחיד במסגרת הכלל מלחמות מדין, עמלק, שבעת העממים,... ויכוח אברהם אבינו עם הקב"ה

מלחמות מדין, עמלק, שבעת העממים,... עַתָּה לֵךְ וְהִכִּיתָה אֶת עֲמָלֵק וְהַחֲרַמְתֶּם אֶת כָּל אֲשֶׁר לוֹ וְלא תַחְמל עָלָיו וְהֵמַתָּה מֵאִישׁ עַד אִשָּׁה מֵעוֹלֵל וְעַד יוֹנֵק מִשּׁוֹר וְעַד שֶׂה מִגָּמָל וְעַד חֲמוֹר ( שמואל א', ט"ו, ג') רַק מֵעָרֵי הָעַמִּים הָאֵלֶּה אֲשֶׁר ה' אֱלקיךָ נתֵן לְךָ נַחֲלָה לא תְחַיֶּה כָּל נְשָׁמָה. כִּי הַחֲרֵם תַּחֲרִימֵם.... לְמַעַן אֲשֶׁר לא יְלַמְּדוּ אֶתְכֶם לַעֲשׂוֹת כְּכל תּוֹעֲבתָם... (דברים כ', ט"ז - י"ח)

ויכוח אברהם אבינו עם הקב"ה וַיִּגַּשׁ אַבְרָהָם וַיּאמַר הַאַף תִּסְפֶּה צַדִּיק עִם רָשָׁע.... חָלִלָה לְּךָ מֵעֲשׂת כַּדָּבָר הַזֶּה לְהָמִית צַדִּיק עִם רָשָׁע וְהָיָה כַצַּדִּיק כָּרָשָׁע חָלִלָה לָּךְ הֲשׁפֵט כָּל הָאָרֶץ לא יַעֲשֶׂה מִשְׁפָּט. ( בראשית י"ח, כ"ג - כ"ה)

תודה!