Arrogant and bounded rationality

Imagine that you are part of a game with other 50 people. You have to choose a real number between 0 and 100 and a prize of $2,000 will be given to the person whose number was closest to the average divided by two. What number would you choose?

Think a bit more...

If you tried to find a solution, you may have realized that the highest reasonable number is 50, because if everyone else chooses 100, then the average will be close to 100 and you will win the prize. However, if you made an extra effort, you have probably realized that you are not the only one with the same reasoning, so if everyone bids 50, the maximum reasonable bid is now 25. But the game continues, because if you think that the other ones are as intelligent as you are, they will also choose 25, so now it is optimal to bid 12.5. If we are perfectly rational and recognize others as perfectly rational too, then the only reasonable solution for this game - its Nash Equilibrium - is that all players choose 0.

However, several experiments have been done around this game and almost nobody behaves like this. Usually bids are lower than 50 but higher than 0. But, if people can realize that choosing a number higher than 50 is not a best response, why cannot they follow the mental process to its end and arrive to the conclusion that 0 is their best option? The two possible explanations to this phenomenon are that either people are not as rational as economic theory predicts they are, or everyone believes he is the only rational person in a world of irrationals that will not realize that bidding something higher than 0 is suboptimal. In a nutshell, both situations are combined: we are not completely rational and we know that others are not as well.

This last conclusion is crucial and revolutionary, because almost all the economic theory of the last hundred years has been developed from the assumption of perfectly rational individuals that can instantaneously incorporate and process all the available information and take optimal decisions that maximize their utility. Central bankers and policy makers have been using models that assume that we all have limitless reasoning power and memory. Recognizing the real nature of people behavior is necessary to design efficient and effective public policies that may increase people wealth and happiness.

Therefore, a more accurate way of analyzing human behavior is considering that we have a bounded rationality, which means that we try to be rational, but sometimes we cannot, because problems are to complex to be solved in the available time. We simplify the problems with heuristics (rules of thumb) that are described as "judgmental shortcuts that generally get us where we need to go – and quickly". However, the psychologists Daniel Kahneman and Amos Tversky have found that although heuristics are necessary and useful, they sometimes generate systematical and generalized biases (errors). For instance, imagine this situation:

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which is more probable?

1. Linda is a bank teller.
2. Linda is a bank teller and is active in the feminist movement.

Several experiments have demonstrated that under situations like this, most of the people choose the option 2, based on their heuristics and prejudices. However, it is easy to see that this is an error that we make while trying to assign subjective probabilities to unknown events, because option 2 is part of option 1 and, therefore, option 1 must be more probable than option 2. This bias is called the Representativeness Heuristic and the concept that it tries to illustrate is that people fail to distinguish when something is more probable or more representative.

But this is not the only error that people make while assigning probabilities. Imagine the following question:

Are there more words that start with N or words that have N as a third letter (in written english)?

Although there are much more words that have N as a third letter, people usually think the opposite, because it is easier to imagine words that start with N that words that have N as their third letter. This is called the Availability Heuristic and the logic is that if it is easier to imagine a situation, it must be more likely. This bias generates, for example, more demand of "flight insurance" than "any reason insurance" at the same cost.

In conclusion, during the last decades there have been important developments in Behavioral Economics, the new field that studies how people actually behave (with their heuristics and biases) and tries to design efficient public policies based on this behavior. In my opinion the conjunction between Game Theory and Behavioral Economics is a powerful platform to analyze strategic behavior of people and groups under particular circumstances. You may probably not believe me, but you must trust Vilfredo Pareto:

“The foundation of political economy and in general of every social science is evidently psychology.” 

Why do people attend University?

All students have asked themselves - at least once - if it is actually efficient to spend time studying irrelevant concepts at university, while they could be already working for employers that would never require most of those concepts. History dates and facts, integrals, subjects and predicates are just some examples of topics that students need to incorporate during their learning process and demonstrate it in specific examinations.

In my particular experience, I have worked during my bachelor degree and after finishing it. Now, that I am pursuing a MSc in Behavioral Economics and Game Theory, I find myself again studying theories and topics that would never be required in a professional environment and I cannot avoid feeling a bit frustrated by this situation. What is more, although companies do not need their workers to know much of the concepts they learnt at university, they still require that applicants have finished particular programs and with minimum GPA. The apparently irrational situation could be illustrated like the following: Jake goes to undergraduate and graduate school and spends 6 years of his life studying and learning things that he knows will be completely useless in his future professional career. In the meantime, Companies filter applicants that do not have Jake background, although they know that Jake´s knowledge is completely irrelevant and unnecessary.

So, why does Jake still attend University? and why do companies still require unnecessary and irrelevant knowledge?

Fortunately, Game Theory has a brilliant answer to this question. We can think about the job search process as a game with incomplete and asymmetric information. The Applicant goes to a job interview knowing his characteristics, strong and weak points and the Employer has to evaluate and compare this Applicant with other ones, based on the information that the Applicant provides and other references that he may get. Being in a job interview is not an easy task, but all applicants know that they must sellthemselves. Thus, they will try to seem intelligent and the ideal candidate to fulfill the vacancy. If an Employer needs a worker to do an analytical job, then it will be extremely difficult to choose among several candidates that are making their best to seem analytical profiles. Therefore, the Employer will try to find a signal that help him determine the type of the Applicant (for simplicity, assume Good or Bad). All Applicants will try to seem Good candidates, so they will be willing to provide a good signal to the Employer. This means that in order to have a useful signal, it must be costly, because otherwise all the Applicants would be able to deliver the good signal and it will become completely uninformative.

University programs can be interpreted as costly signals that job applicants provide to their potential employers. A Bad Applicant for an analytical position, would need to put much more effort to finish an analytical university program and, if he achieves it, probably will not even try to attend graduate school. On the other hand, a Good Applicant for an analytical position, needs less effort to understand analytical tasks. Considering that effort is a source of disutility (all of us want to maximize our utility, and putting effort in a task lowers our utility), probably just Good Applicants will finish analytical PhD programs and the Employer knows that if he sets a PhD degree as a minimum requirement, he will probably find just Good Applicants. Of course, this does not mean that there are not Good Applicants without graduate studies or that there are not Bad Applicants with PhD degrees, but the signal of studying particular programs is so costly, that the probability of someone not ideal for it actually doing it, is definitely low.

So, why do people attend university? the first answer would be that people attend university to learn but this vision is completely biased. Students decide to go to university in order to get the appropriate signal that will give them the opportunity to be considered for particular jobs. Therefore, the difference between a PhD in Economics from Harvard and an applicant without a university degree is not how much they actually know, but a signal about what type of challenges each applicant can overcome. In other words, if a person was accepted in Harvard and finished his PhD with a Cum Laude diploma, it means that he was able to learn extremely difficult - and probably useless - concepts and that he was able to outperform in complicated and stressful situations. So, if he could learn how to calculate complicated integrals and optimizations, he will probably be ready to learn the particular tasks related with his job position.

Going back to the Game, the Applicant will go to the job interview trying to seem a Good candidate. However, the Employer will use the university degree as a signal about the candidate suitability for the job position. Knowing this in advance, young and ambitious students will decide to go to university and pursue complicated programs, no matter if the content is useful or not. Bad applicants will have to put so much effort that the disutility will be high enough to discourage them to continue and will never get the appropriate signal, while Good applicants will finish the program because the necessary effort will not be so high for them.

In conclusion, this analysis states that students do not go to university to learn, but for signaling. This does not mean that they cannot learn anything in the process; in fact they will probably learn several useful and irrelevant concepts. In addition, there still are job positions for which this theory does not apply, such as medicine, architecture or academia. For all these examples, university is also the first quasi-internship, so the conclusions may be completely different.

References:

Job Market Signaling (Michael Spence). The Quarterly Journal of Economics, Vol. 87, No. 3. (Aug., 1973), pp. 355-374

La dispersión del Real-Time Bidding

Imagine el lector la siguiente situación y piense rápidamente que haría si se lo pusiera en ese escenario:

Se le ofrece elegir entre obtener 5.000 dólares con seguridad o tirar una moneda. Si sale cara, recibirá 10.000 dólares pero si sale seca no obtendrá nada.

Muy probablemente el lector habrá elegido por la opción de recibir 5.000 USD con seguridad a pesar de que el valor esperado de ambas loterías era exactamente el mismo. Esto ocurre porque los humanos somos, la gran mayoría, aversos al riesgo. Es decir, que una situación de riesgo nos quita utilidad y preferimos loterías con menor valor esperado, siempre y cuando disminuyan también el riesgo. Si el lector aún duda acerca de esta afirmación, ¿no aceptaría acaso también recibir 4.500 USD antes de tirar al aire la moneda con la esperanza de ganar 10.000 USD con un 50% de probabilidad?

Para los que aún se resisten a considerarse agentes aversos al riesgo, los invito a pensar si hoy en día son propietarios de algún seguro, ya que estas herramientas son optimamente demandadas por agentes aversos al riesgo, dispuestos a pagar un monto mayor a la probabilidad del suceso con tal de cubrirse del riesgo de su ocurrencia.

La teoría de selección óptima de portafolios de inversión ha incorporado la aversión al riesgo como un aspecto central e incuestionable. Teoría y práctica coinciden en que un inversor únicamente aceptará un mayor riesgo si viene acompañado por un mayor retorno esperado. Quienes adquieren bonos están dispuestos a obtener menores retornos esperados que quienes invierten en acciones, cuyo retorno esperado es superior, pero viene acompañado de un riesgo comparativamente elevado.

Es curioso entonces que los mecanismos de decisión de pujas óptimas en el Real-Time Bidding (RTB) consideren únicamente el valor esperado de la impresión. Actualmente, el RTB evalúa el Response Rate (Click-Through Rate o Conversion Rate según el caso) de impresiones similares en el pasado y puja un valor proporcional al resultado muestral. Es decir, que el RTB en ningún momento incorpora el impacto del riesgo en su análisis y supone, por lo tanto, que los compradores de medios resultamos indiferentes al comprar una impresión que con seguridad nos dará un CTR del 0.5% o una que nos generará un CTR de 1% con un 50% de probabilidad y de 0% con el otro 50% de probabilidad.

La medida ampliamente utlizada para incorporar al riesgo a cualquier proceso de decisión es la varianza. La varianza es la medida estadística que nos permite medir la dispersión y, por lo tanto, ante el mismo valor esperado (Response-Rate), los compradores estarán dispuestos a pagar más por aquella impresión con una menor varianza.

Ahora, para quienes consideran que el problema finalizaría con la inclusión de la varianza al mecanismo de decisión de pujas óptimas aún están equivocados. Si bien implicaría un paso fundamental para volver al RTB aún más eficiente que lo actual, el enfoque de media-varianza solo sería suficiente para distribuciones de probabilidades normales. Si ese no fuera el caso, sería necesario también incorporar el tercer y cuarto momento de la función de distribución de probabilidades. Una distribución con sesgo positivo será más demandada que una con sesgo negativo por un agente averso al riesgo, más allá de que tengan la misma media y varianza. Además, una distribución leptocúrtica (valores concentrados cerca de la media) será preferible, frente a otra platicúrtica (con colas pesadas).

En conclusión, no quedan hoy en día dudas de que el RTB es el mecanismo de compra-venta de medios más eficiente del mercado ya que es el único que arriba a un Equilibrio de Nash, superando económica y estadísticamente a Google Adwords y Facebook Ads. No obstante, en el futuro deberá continuar el mismo camino que han tomado los enfoques de selección óptima de portafolios de inversión, dejando en el pasado el supuesto de neutralidad al riesgo e incorporando la varianza, sesgo y curtósis de la función de distribución de probabilidades logrando, de esa manera, pujar exactamente el valor que cada anunciante desee por cada impresión, sea este último averso, neutral o amante al riesgo.

Header Bidding y el atrapante Teorema de Equivalencia de Ingresos

El Teorema de Equivalencia de Ingresos pertenece a la Teoría de Juegos y plantea que, bajo ciertos supuestos, el precio promedio al que se termina vendiendo cualquier bien es, en el largo plazo, el mismo en Subastas de Primer o Segundo Precio.

Las derivaciones de este -poco- conocido Teorema son monumentales, ya que es la justificación matemática que se utilizó para desestimar la demanda de los Publishers de pasar a esquemas de Subastas de Primer Precio, en lugar de las Subastas de Segundo Precio que hoy en día dominan los mecanismos de compra-venta en el mercado de medios digitales (Real-Time Bidding). El argumento del Teorema, intuitivamente, es que en Subastas de Primer Precio los jugadores ganadores (de mayor valuación) están impulsados a bajar su puja progresivamente en cada repetición, hasta llegar a conocer la valuación del segundo jugador en la lista y ofrecer un centavo más. En cambio, en las Subastas de Segundo Precio, el incentivo de todos los jugadores es ofrecer su propia valuación ya que de esa manera la oferta mayor gana, paga un centavo más que la segunda y obtiene el bien, alcanzando un Equilibrio Bayesiano de Nash.

Sin embargo, la aparición de los Private Marketplaces (PMP) y del Header Bidding son claras demostraciones de que el mecanismo económicamente eficiente para los jugadores (Advertisers) no lo era para los rematadores (Publishers), puesto que estos últimos han encontrado que existe la posibilidad de obtener ingresos mayores a partir de innovaciones tecnológicas y nuevos modelos de negocio.

Ahora, ¿qué es lo que fallaba en el mecanismo anterior que llevó a los Publishers a encontrar mejores resultados con la incorporación del Header Bidding? ¿por qué motivo si los Publishers deberían estar obteniendo con las Subastas de Segundo Precio el mayor ingreso posible, solicitaron durante estos años migraciones a mecanismos con Subastas de Primer Precio y/o la incorporación del Header Bidding?

La respuesta a ambas preguntas parte de un análisis específico del Teorema de Equivalencia de Ingresos. En primer lugar, como dijo Lord Keynes, en el largo plazo estamos todos muertos y esta es una premisa que en la práctica no podemos omitir. En segundo lugar, es fundamental considerar que para poder confiar en las conclusiones de un Teorema primero hay que garantizar el cumplimiento de sus supuestos. Para que el Teorema de Equivalencia de Ingresos se cumpla son necesarias las siguientes condiciones:

1) Cada jugador conoce su propia valuación, que es privada e independiente

2) El pago depende únicamente de las pujas realizadas

3) Los jugadores son neutrales al riesgo

4) Los jugadores son simétricos, es decir, que poseen la misma información

Los primeros dos supuestos no generan grandes debates en este contexto. El de neutralidad al riesgo es interesante por si mismo, ya que podría pensarse en que es una simplificación suponer que los Advertisers evalúan únicamente el valor esperado de la impresión (Response Rate) para determinar la puja y sería más prudente pensar en esquemas que contemplen también la varianza, el sesgo y la curtosis de la función de distribución de probabilidades. No obstante, este supuesto no pone en jaque el mecanismo cuestionado en este artículo ya que todo el sistema está armado sobre el supuesto de neutralidad al riesgo, más allá de que sea adecuado o no.

En contraste, el supuesto de jugadores simétricos es completamente relevante en esta discusión: para que el Publisher pueda recibir en promedio el mismo ingreso por Subastas de Primer o de Segundo Precio, es necesario que los jugadores (Advertisers) tengan todos acceso a la misma información en el mismo momento y que, por lo tanto, puedan realizar las pujas con exactamente la misma prioridad.

El Header Bidding, que consiste en un sistema mediante el cual los Publisherspueden recibir ofertas de todos los Advertisers simultáneamente, eliminando cualquier esquema de prioridades, es la respuesta sistémica a un mecanismo previo matemáticamente ineficiente. En resumen, primero los Publishers propusieron migrar de Subastas de Segundo Precio a Subastas de Primer Precio. Cuando la respuesta fue negativa y fundamentada en el Teorema de Equivalencia de Ingresos, entonces los Publishers optaron por exigir que se cumplan los supuestos necesarios para que el Teorema entre en vigencia y sus conclusiones se observen empíricamente.

En conclusión, la incorporación del Header Bidding al esquema de compra-venta de espacios publicitarios digitales, no hace más que acercar cada vez más a esta industria caracterizada por el uso de Big Data y de las decisiones automatizadas en tiempo real a la eficiencia microeconómica necesaria para que todos los agentes participantes obtengan el máximo beneficio posible por su participación. Abrazar la incorporación del Header Bidding implica entender que es clave en esta industria eliminar cualquier fricción que genere ineficiencias que provoquen luego la necesidad de replantear los mecanismos de compra-venta de medios digitales.