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Nash

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Lecturas Matem´ticas a Volumen 24 (2003), p´ginas 137–149 a

John Nash y la teor´ de juegos ıa
Sergio Monsalve Universidad Nacional de Colombia, Bogot´ a Al profesor y acad´mico Don Jairo Charris Casta˜ eda e n In memoriam

Abstract.

In the last twenty years, game theory has become the dominant model in economic theory and has made significant contributions to political science, biology, and international security studies. The central role of game theory in economic theory was recognized by the awarding of the Nobel Price in Economic Science in 1994 to John C. Harsanyi, John F. Nash, & Reinhard Selten. The fundamental works in game theory of John F. Nash together with a brief exposition of them are included in this article.

Key words and phrases. John Nash, History of Mathematics, Game Theory 1991 Mathematics Subject Classification. Primary 01A70. Secondary 91A12. Resumen. En los ultimos veinte a˜os, la teor´ de juegos se ha ´ n ıa convertido en el modelo dominante en la teor´ econ´mica y ha ıa o contribuido significativamente a la ciencia pol´ ıtica, a la biolog´ ıa y a estudios de seguridad nacional. El papel central de la teor´ ıa de juegos en teor´ econ´mica fue reconocido con el premio Nobel ıa o en Econom´ otorgado a John C. Harsanyi, John F. Nash & ıa Reinhard Selten en 1994. Se presentan los aportes de John Nash a la teor´ de juegos conjuntamente con una exposici´n ıa o elemental de ellos.

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1. Introduction
La Real Academia Sueca para las Ciencias le otorg´ el premio Nobel o en Ciencias Econ´miacs del a˜ o 1994 a los economistas John C. Haro n sanyi, Reinhard Selten y al matem´tico John F. Nash, debido a su a “an´lisis pionero de equilibrios en la teor´ de juegos no cooperativos”. a ıa La Academia justifica este premio en econom´ a tres de los “grandes” ıa en teor´ de juegos con el argumento de que esta ha probado ser muy util ıa ´ en el an´lisis econ´mico. 60 a˜ os despu´s de la publicaci´n de la obra a o n e o pionera de John von Neumann y Oskar Morgenstern (Theory of Games and Economic Behavior (1944)), la teor´ de juegos hab´ reıa ıa cibido el merecido reconocimiento como herramienta fundamental del an´lisis econ´mico moderno y el aporte de John Nash fue fundamena o tal.

2. ¿Qu´ es la teor´ de juegos? e ıa
La teor´ de juegos (o teor´ de las decisiones interactivas es el estudio ıa ıa del comportamiento estrat´gico cuando dos o m´s individuos interact´an e a u y cada decisi´n individual resulta de lo que ´l (o ella) espera que los o e otros hagan. Es decir, qu´ debemos esperar que suceda a partir de las e interacciones entre individuos.

3. ¿Con qu´ estructuras estudiamos la teor´ de juegos? e ıa
Existen, fundamentalmente, dos formas distintas de aproximarnos al an´lisis de una situaci´n de interacciones entre individuos a o I) La primera (que es quiz´s la dominante dentro del ambiente de a los economistas) es la teor´ de juegos no cooperativos, en la que, b´siıa a camente, tenemos un conjunto de jugadores, cada uno con estrategias a su disposici´n, y unas asignaciones de pagos que reciben por llevar o a cabo tales estrategias. La caracter´ ıstica “no cooperativa” est´ en la a manera de c´mo eligen y en lo que saben de los otros jugadores cuando o

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eligen: en general, se supone que los individuos toman sus decisiones independientemente unos de otros aunque conociendo sus oponentes y las posibles estrategias que estos tienen a su disposici´n. Es decir, son o individuos ego´ ıstas pero que tratan de predecir lo que los otros agentes har´n para obrar entonces en conveniencia propia. En esta estructura a de an´lisis los agentes no alcanzan ning´ n nivel de cooperaci´n. a u o Nada mejor que un ejemplo bien ilustrativo del modus operandi de este tipo de modelos. Y quiz´s el m´s elocuente de los juegos no-cooperativos a a elementales es el dilema del prisionero. La historia de este juego va como sigue: dos individuos son detenidos debido a que cometieron cierto delito. Ambos son separados en celdas diferentes y son interrogados individualmente. Ambos tienen dos alternativas: cooperar uno con otro (no-confesar) o no cooperar (confesar el delito). Ellos saben que si ninguno confiesa, cada uno ir´ a prisi´n por dos a˜os. Pero si uno de los a o n dos confiesa y el otro no, entonces al que confiesa lo dejar´n libre y al a que no confiesa lo condenar´n a 10 a˜os. Si ambos confiesan, los dos a n a o n o ir´n a prisi´n por 6 a˜os. La situaci´n se resume en la siguiente bimatriz (es decir, una matriz cuyos elementos son parejas n´ meros): u Prisionero 1 C C Prisionero 2 NC (−2, −2) (0, −10) NC (−10, 0) (−6, −6)

C = cooperar (no confesar), N C = cooperar(confesar)

La pregunta natural es: ¿qu´ har´n los detenidos? ¿cooperar´n entre e a a s´ (no confesar´n) o se traicionar´n el uno al otro (confesar´n)? Alı a a a guien desprevenido que est´ observando este juego podr´ pensar que e ıa los dos jugadores cooperar´ (no confesar´n) puesto que en ese caso ıan a ambos obtendr´ el menor castigo posible. Sin embargo, la estructura ıan no cooperativa del problema hace que este arreglo no sea cre´ ıble: si se pactara la no-confesi´n por parte de los dos, ambos tendr´ incentivos o ıan particulares para romperlo, pues dejando al otro en cumplimiento del

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pacto de no confesar y ´ste confesando, el que rompe el pacto obtiene e la libertad mientras al otro lo condenar´n a 10 a˜os. Y, similarmena n te, estudiando las otras tres posibilidades del juego (es decir, (C, N C), (N C, N C), (N C, C)) observamos que el unico acuerdo cre´ (que sig´ ıble nifica que ninguno de los dos querr´ romper el pacto unilateralmente ıa porque perder´ es (N C, N C). En definitiva, la predicci´n de lo que ıa) o ocurrir´ en el juego es que ambos confesar´n y permanecer´n en la carcel a a a 6 a˜os. n La conclusi´n en situaciones similares a ´sta (que son comunes en la o e vida diaria) es que la competencia ego´ puede conducir a estados que ısta son inferiores (en t´rminos de beneficio personal y social) a los estados e cooperativos, pero que estos ultimos no podr´n implementarse a menos ´ a que existan reforzmientos externos (contratos firmados por ley, con verificaci´n, etc.) que obliguen a las partes a cumplir con el acuerdo de o cooperaci´n. o Esta es la idea esencial de Nash al definir el concepto de equilibrio de Nash en su tesis doctoral en Matem´ticas en la Universidad de Princeton a (Non-cooperative Games (1950)): un equilibrio de Nash de un juego es un acuerdo que ninguna de las partes puede romper a discreci´n sin perder. o Es decir, si alguien quiere romper el pacto y lo hace unilateralmente, se arriesga a ganar por debajo de lo que hubiese ganado dentro del pacto. Sin embargo, como queda claro en el juego del dilema del prisionero, esto puede no ser lo mejor socialmente para los jugadores. Uno de los resultados que hacen del equilibrio de Nash un punto a de referencia para casi todo an´lisis en el que las interacciones entre individuos est´n involucradas es que e todo juego finito (es decir, finitos jugadores y finitas estrategias de cada jugador) tiene al menos un equilibrio de Nash, aunque involucre ciertas probabilidades objetivas de juego de las estrategias por parte de los jugadores. Este resultado es del mismo Nash. En un art´ ıculo previo a su tesis de doctorado, y publicado tambi´n en 1950 (Equilibrium points in e n−person games) ´l prueba, utilizando un teorema de punto fijo (el coe nocido teorema de punto fijo de Brouwer), que un equilibrio de Nash es

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un “punto fijo”: las expectativas de los gentes con respecto a lo que los dem´s har´n, coinciden, todas, en el equilibrio de Nash. a a II) La segunda estructura fundamental para el estudio de la teor´ ıa de juegos para desde all´ predecir resultados de la interacci´n, es la ı o teor´ de juegos cooperativos o coalicionales. Aqu´ todav´ tenemos los ıa ı ıa mismos agentes ego´ ıstas, pero ahora se asume que, si pueden obtener alg´n beneficio de la cooperaci´n, no dudar´n en formar coaliciones que u o a son cre´ ıbles. Por supuesto, bajo una estructura como la de juegos no cooperativos, un acuerdo de cooperaci´n puede no ser la “soluci´n”, o o de manera que los agentes deben tener una estructura de informaci´n o diferente si queremos un comportamiento acorde. En una estructura cooperativa tenemos el mismo conjunto de jugadores ego´ ıstas, solo que ahora tienen informaci´n sobre cierta valoraci´n o o a priori de las coaliciones. Es decir, se reconoce cu´les coaliciones son a las m´s “valiosas” y cu´les las “menos valiosas”. Para concretar ideas, a a vamos a presentar un modelo que muestra bien el poder predictivo que puede tener este tipo de estructura y las soluciones asociadas. El modelo del peque˜o mercado muestra una estructura de mercado en la que n hay un vendedor (jugador 1) de cierto bien que no podemos dividir en partes (pi´nsese, por ejemplo, en un autom´vil como bien de uso), y dos e o compradores (jugador 2 y jugador 3) que desean comprar ese bien. Las a valoraciones que a priori se le asignan a las coaliciones ser´n, en este caso, un reflejo del ´xito o fracaso de la negociaci´n entre el vendedor e o y el comprador dependiendo de c´mo se emparejen, y no una predico ci´n sobre qui´n de los dos obtendr´ el bien (este problema tiene otra o e a valoraci´n). En este caso, asignamos la valoraci´n a todas las posibles o o coaliciones de la siguiente forma: V ({1, 2, 3}) = V ({1, 2}) = V ({1, 3}) = 1
(“si hay vendedor y comprador, el negocio se lleva a cabo”)

V ({1}) = V ({2}) = V ({2, 3}) = 0
(“si solo hay compradores o vendedor, no se realiza el negocio”)

Existen varias “soluciones” a ese tipo de juegos en forma cooperativa. Por supuesto, una “soluci´n” debe significar una repartici´n de la riqueza o o

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o valoraci´n total de todo el grupo de jugadores como gran coalici´n, de o o tal manera que a cada jugador le corresponda su “aporte” a ella. C´mo o determinar el nivel de este aporte es algo que analizamos con las dos m´s importantes soluciones a juegos cooperativos: el n´cleo y el valor a u de Shapley. a) El n´cleo del juego cooperativo (Gillies (1953)) u La idea de repartici´n detr´s de la soluci´n de n´cleo es que si esta le o a o u asigna x1 al vendedor, x2 al comprador-jugador 2, y x3 al compradorjugador 3, entonces debemos, por lo menos, tener que: x1 + x2 + x3 = 1 (ef iciencia) (i)

(los tres agentes se reparten el “poder” del mercado que est´, a priori, a en la uni´n de los tres). o V ({1, 2}) ≤ x1 + x2 , V ({1, 3}) ≤ x1 + x3 , V ({2, 3}) ≤ x2 + x3 , V ({1}) ≤ x1 , V ({2}) ≤ x2 , V ({3}) ≤ x3 (“lo que los agentes reciben a trav´s de la asignaci´n de n´cleo es mejor e o u que lo que recibir´ antes de la asignaci´n a priori V , y esto sucede en ıan o cualquier coalici´n que formen”). o Obs´rvese que la asignaci´n de n´cleo es un acuerdo muy b´sico. Es e o u a una “invitaci´n” a formar coaliciones: si usted forma alguna coalici´n o o ganar´ m´s de lo que gana en cualquier coalici´n del status quo detera a o o minado por la asignaci´n a priori V . Un poco de aritm´tica nos muestra que la unica asignaci´n de n´cleo e ´ o u en nuestro peque˜o mercado es x1 = 1, x2 = x3 = 0. En otras palabras, el n n´cleo est´ pronosticando que la importancia total del mercado la tiene u a el vendedor y que los compradores no tienen niguna. Pobre predicci´n: o no existe casi ning´n mercado en el que los compradores no tengan u ninguna importancia estrat´gica. e (ii)

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b) El valor de Shapley del juego cooperativo (Shapley (1953) Otra forma de distribuci´n con consideraciones m´s sut´ que las de o a ıles n´cleo, es el valor de Shapley. Podr´ decirse que el valor de Shapley es u ıa a los juegos cooperativos, lo que el equilibrio de Nash es a los juegos no cooperativos. Como ya hab´ ıamos descrito con el ejemplo anterior, un juego cooperativo consiste en un conjunto de jugadores N (= N = n) y una asignaci´n o monetaria V (S) para cada subcoalici´n S ⊆ N. El problema que se ino tenta resolver es: ¿C´mo distribuir la riqueza total V (N ) entre todos o los participantes? El valor de Shapley busca solucionarlo imponiendo ciertas condiciones a la distribuci´n: o o ıa o Si xi , i ∈ N , es la asignaci´n que recibir´ en la distribuci´n de Shapley el jugador i, entonces a) (Eficiencia) x1 + x2 + ...xn = V (N ) b) (Jugador “dummy” o fantasma) Si para alg´n i ∈ N , V (S ∪ {I}) = u V (S) para toda coalici´n S, entonces xi = 0 o c) (Simetr´ Si las valoraciones de las coaliciones no cambian cuando ıa) se reemplaza un jugador por cualquier otro, entonces, todos reciben lo mismo. Es decir, x1 = ... = xn . d) (Aditividad) Si V y W son dos valoraciones distintas sobre el mismo conjunto N de jugadores, entonces la asignaci´n de cualquier o jugador para la valoraci´n V y para la valoraci´n W es aditiva. Es o o decir, para todo i ∈ N , xi (V + W ) = xi (V ) + xi (W ) Solo el valor de Shapley satisface estas cuatro condicions. Y m´s a a´n, existe una f´rmula expl´ u o ıcita sobre c´mo calcularlo, basada en las o “contribuciones marginales” (es decir, una medida de cu´nto un jugador a le aporta a las coaliciones en donde est´): a (∆i (Si (r))) (∗) n! donde es el conjunto de los n! ordenes de N , Si (r) = conjunto de ´ jugadores que preceden a i en el orden r, ∆i (S) = V (S ∪ {i}) V (S) = contribuci´n marginal del agente i a la coalici´n S. o o xi = r∈ 144

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Como un ejemplo de la “mejor” distribuci´n total de la riqueza del o juego entre los participantes que el valor de Shapley realiza, es f´cil a calcular (con la f´rmula (*) de arriba) que en el juego del Peque˜o o n Mercado que analiz´bamos antes, el valor de Shapley es a x1 = 2/3, x2 = 1/6, x3 = 1/6. A diferencia del n´cleo, cuya asignaci´n es (1, 0, 0), el valor de Shapley u o s´ est´ asign´ndole importancia a los compradores, aunque reconoce un ı a a poder mayor al vendedor. IIa) Una muy importante categor´ en el estudio de la teor´ de juegos ıa ıa cl´sica es, realmente, una subdivisi´n de la teor´ de juegos cooperatia o ıa vos: los modelos de negociaci´n. En estos juegos, dos o m´s jugadores o a buscan ganar a trav´s de la cooperaci´n, pero deben negociar el procedie o miento y la forma en que se dividir´n las ganancias de esta cooperaci´n. a o Un modelo de negociaci´n espec´ o ıfica: c´mo y cu´ndo se alcanzan los o a acuerdos y c´mo se dividir´n las ganancias, dependiendo de las reglas o a de negociaci´n y de las caracter´ o ısticas de los negociadores. John Nash realiz´ contribuciones fundacionales a la teor´ de jueo ıa gos de negociaci´n. En su art´ o ıculo de 1950, The bargaining problem, se aparta radicalmente de la teor´ econ´mica ortodoxa que consideraba inıa o determinados los problemas de negociaci´n. En contraste, Nash asume o que la negociaci´n entre agentes racionales conduce a un unico reultado, o ´ y lo determina imponi´ndole al modelo ciertas “propiedades deseables”. e La formulaci´n de Nash del problema de negociaci´n y su soluci´n (la o o o o o ıa soluci´n de negociaci´n de Nash) constituyen el fundamento de la teor´ moderna de la negociaci´n. o Para aclarar un tanto lo dicho arriba con una versi´n muy elemental o del problema de negociaci´n de Nash, supongamos que tenemos dos juo ıstas que buscan dividirse una cantidad de dinero M y que gadores ego´ no est´n de acuerdo en que los dos no obtengan nada de la negociaci´n, a o pero que si no llegan aun acuerdo, entonces s´ les corresponder´ “irse ı a con las manos vac´ ıas”. John Nash mostr´ que bajo ciertas condicioo nes plausibles, los dos jugadores acordar´ la repartici´n (x1 , x2 ) que ıan o maximiza, en este caso, el producto de las dos. Es decir, resuelven M´x x1 x2 , a sujeto a la condici´n x1 + x2 = M. o

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La soluci´n a esto es, como se ve f´cilmente, (x1 )∗ = (x2 )∗ = 1 M . Esta o a 2 es la soluci´n de negociaci´n de Nash a este simple problema: distrio o buci´n equitativa de la cantidad de dinero disponible. Esta soluci´n se o o ilustra en el diagrama de abajo. Por supuesto, problemas de negociaci´n o m´s complicados dan or´ a ıgen a soluciones de Nash menos obvias. grafico!!!! Aquellas “propiedades deseables” mencionadas arriba, que satisface la soluci´n de negociaci´n de Nash son las siguientes: o o a) Eficiencia: las soluciones de negociaci´n de Nash agotan todas las o oportunidades de mejorar las ganancias de ambos jugadores. b) Simetr´ si el cambio de un jugador por otro no cambia el probleıa: ma de negociaci´n, en la soluci´n ellos deben obtener lo mismo. o o c) Invarianza: la soluci´n no puede depender de las unidades arbitrao rias en las que se midan los beneficios de la negociaci´n. o d) Independencia de alternativas irrelevantes: es la hip´tesis m´s suo a til. Dice que el resultado de una negociaci´n no puede depender de o alternativas de negociaci´n que los negociadores no escogen aunque o pudieran hacerlo. Estas cuatro hip´tesis conducen al conoocido teorema de negociaci´n o o de Nash: La soluci´n de negociaci´n de Nash es la unica soluci´n o o ´ o que satisface las propiedades a), b), c), d) en un modelo de negociaci´n. o

3. El programa Nash: ¿Es mejor cooperaci´n que o competencia?
Nash aseguraba que la teor´ de juegos cooperativa y no cooperativa ıa eran complementarias, que cada una ayudaba a justificar y clarificar la otra. Si una soluci´n cooperativa pod´ ser obtenida a partir de un o ıa conjunto convincente de hip´tesis, esto indicaba que tambi´n pod´ ser o e ıa apropiada en una variedad de situaciones m´s amplia que las encontradas a en un modelo no cooperativo simple. Y de otro lado, si reformulamos

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juegos cooperativos como no cooperativos y buscamos los equilibrios de Nash de estos ultimos, la discusi´n abstracta sobre lo razonables ´ o que puedan ser los principios o los resultados se reemplazan por una discusi´n m´s mundana acerca de lo apropiadas que son las reglas del o a juego. El programa Nash buscaba la posibilidad de la unificaci´n te´rica, y o o algunos logros ya se tienen en este sentido. Uno de ellos es el famoso teorema de Aumann (1975) que afirma, en t´rminos elementales, y a la e vez un tanto vagos, lo siguiente: En un grupo conformado por muchos agentes que tratan de disputarse cierta cantidad de dinero, las distribuciones de equilibrio de Nash, de n´cleo, de valor de Shapley y u o de negociaci´n de Nash coinciden. Aqu´ la palabra “muchos” intenta capturar la idea de que cada jugador, ı, por s´ mismo, tiene un poder estrat´gico pr´cticamente nulo. Solo tienen ı e a poder, para la asignaci´n, grupos verdaderamente “grandes”, no indivio duos aislados. Una implicaci´n directa de este teorema es que cuando o las interacciones individuales tienden a anularse, ¡¡la competencia y la cooperaci´n conducen a los mismos resultados!! o Otro modelo b´sico muy importante en la direcci´n del programa a o Nash es el de Rubinstein (1982) que muestra que cualquier resultado de negociaci´n Nash entre pocos agentes, puede aproximarse mediante o equilibrios de Nash de juegos no cooperativos secuenciales. Pero, por encima de esto, el gran impulso del programa Nash se recibi´ en 1994 o en el seminario Nobel sobre el trabajo de John Nash en la Universidad de Princeton (publicado en Les Prix Nobel 1994). Despu´s de estos, e se han obtenido resultados que involucran otros valores diferentes al de a Shapley en estructuras de juegos m´s sofisticadas que muestran que la afirmaci´n de Shapley en el sentido de que ambas teor´ (coopeo ıas rativa y no cooperativa) son, no solo complemento una de otra, sino que realmente son una sola teor´ vista con dos lentes diferentes, estaba ıa en el camino correcto (v´ase Hart & Mas Colell (1996), Hart & e Monsalve (2001)).

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5. Otras contribuciones de Nash a la teor´ de juegos ıa
Se reconocen muy poco las contribuciones de John Nash a la primera literatura en econom´ experimental (v´ase Roth (1993)). Entre ıa e otras, Nash hace algunos comentarios importantes respecto a la cooperaci´n observada cuando repetimos infinitamente el dilema del prisioneo ro. Adem´s, el trabajo pionero de Nash, conjuntamente con Kalish, a Milnor y Nering sobre experimentos en juegos cooperativos (Kalish et al. (1954)) fue un importante est´ ımulo para Reinhard Selten en sus experimentos (Selten (1993)) que han conducido al florecimiento de la escuela alemana en econom´ experimental. ıa

6. Final
El par´grafo final de los peri´dicos anunciando el premio Nobel en a o Econom´ de 1994 dice: “a trav´s de sus contribuciones al an´lisis de ıa e a equilibrios en teor´ de juegos no cooperativos, los tres laureados consıa tituyen una combinaci´n natural: Nash dio los fundamentos para el o an´lisis, Selten los desarroll´ con respecto a la din´mica, y Harsanyi a o a con respecto a la informaci´n incompleta”. Esta es la unidad “natural” o de este premio. Sin embargo, n´tese que la teor´ de juegos cooperativos o ıa fue completamente ignorada e, indirectamente, tambi´n los aportes de e John Nash a esta teor´ incluyendo los resultados del Programa Nash. ıa, El razonamiento de la teor´ de juegos es ahora el fundamento de ıa importantes areas de la teor´ econ´mica, y est´ r´pidamente entrando ´ ıa o a a en disciplinas aparentemente dis´ ımiles como finanzas, ciencia pol´ ıtica, sociolog´ derecho y biolog´ Las contribuciones de John Nash (junıa, ıa. to con Harsanyi, Selten, Aumann & Shapley) constituyen importantes piedras angulares en el desarrollo de la teor´ de juegos y en el ıa establecimiento de una metodolog´ com´n para analizar la interacci´n ıa u o e estrat´gica dentro de todas las ciencias sociales, e incluso (este es el reto) en otras ciencias.

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Referencias
[1] Aumann, R. J. (1975) Values of markets with a continuum of traders, Econometrica 43:611–646 [2] Aumann, R. J. (1977) Game Theory. In The New Palgrave: A Dictionary of Economics Vol. 2 ed J. Eatwell, M. Milgate, and P. Newman, 460–482. London: Macmillan. [3] Aumann, R. J. (1988) On the state of the art in game theory: and interviewwith Robert Aumann, Games and Economic Behavior 24:181–210. [4] Gillies, D. B. (1953) Some theorems on n−person games. Ph. D. Dissertation, Department of Mathematics, Princeton University. [5] Hart, S. & Mas Colell, A. (1996) Bargaining and Value, Econometrica 64: 357–380. [6] Hart, S. & Monsalve, S. (2001) The Asymptotic approach of the MaschlerOwen values Journal of Game Theory (forthcoming). [7] Kalish, G. K., Milnor, J. W., Nash, J. F., Nering, E. D. (1954) Some experimental n person games, Cap´ ıtulo 21: 301–327 En: Thrall R. M., Coombs, C. H., Davids, R. L. (eds) Decision Processes. Wiley, New York. [8] Mayberry, J. P., Nash, J. F. y Shubik, M. (1953) A comparison of treatments of a duopoly situation, Econometrica 21:141–155. [9] Monsalve, S. (1999) Introducci´n a los conceptos de equilibrio en econom´a. o ı Editorial Unibiblos, Santaf´ de Bogot´. e a [10] Monsalve, S. (2002) Teor´a de Juegos: ¿Hacia d´nde vamos? (60 a˜os ı o n despu´s de von Neumann y Morgenstern) Revista de Econom´ Institucional e ıa, No. 7 Universidad Externado de Colombia, Bogot´ (Pr´xima Publicaci´n). a o o [11] Myerson, R. (1991). Game Theory: Analysis of Conflict. Cambridge, MA: Harvard University Press. [12] Nash, J. F. (1950) Equilibrium points in n person games, Procedings from the National Academy of Sciences, USA 36:48–49. [13] Nash, J. F. (1950) The Bargaining Problem, Econometrica 18:155–162. [14] Nash, J. F. (1950) Non cooperative Games, Ph. D. Dissertation Princeton University. [15] Nash, J. F. (1953) Two person cooperative games, Econometrica 21:128– 140. [16] Roth, A. (1993) The Early history of experimental economics, Journal of History of Economic Thought 15: 184–209. [17] Rubinstein, A. (1982) Perfect equilibrium in a bargaining model, Econometrica 50:97–109 [18] Selten, R. (1993) In search of a a better understanding of economic behavior. En : Heertje A. (ed.) The Makers of modern economics I:115–139. Harvester Wheatsheaf Hertfordshire.

JOHN NASH Y LA TEOR´ DE JUEGOS IA

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[19] Shapley, L. S. (1953) A value for n-person games. En: Contributions to the Theory of Games, vol 2, H. W. Kuhn y A. W. Tucker, eds. [20] Von Neumann, J. & Morgenstern, O. (1944) Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.

(Recibido en junio de 2002; la versi´n revisada agosto de 2003) o

Sergio Monsalve e-mail: email: monsalvesergio@yahoo.com ´ Departamento de Matematicas, Universidad Nacional de Colombia ´ Bogota, Colombia

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