Computability of Nash Equilibria...

An excellent post on the computability of Nash equilibria of normal form games. I quote:

"Bottom line: Rational players can reason about and make use of Nash’s rational prediction, but they cannot deduce it. The prediction should somehow magically pop up in their minds."

I particularly like the author's reference to a paragraph of John Nash's doctoral thesis that hints at the issue.  Apparently this paragraph never made it into the published version of the thesis.  Here is a Google Scholar link to a bunch of other papers on the subject...

Why should we care about the computability of Nash equilibria? If agents are not able to deduce the Nash equilibria from the payoff matrix, then this cripples game theory as a predictive model of human behaviour (except in the subset of normal form games that are simplistic enough that the Nash equilibrium can be deduced from the payoff matrix by the agents).  The question then becomes: How well do simplistic normal form games map to real world decisions?

1st Year of my PhD (A Rough Outline)...

As my PhD studies approach I have been thinking about how best to structure my first year to make sure that it is in line with my overall PhD agenda.  Here is my first attempt:

1st Year Topic of Focus: Formation of Credit Networks

Need to build at least two small-scale, analytically tractable models (i.e., solvable via pen & paper)of credit network formation.  Goal is to build intution and gain a deeper understanding of how credit effects network incentives.  There are two main approaches to explore:

• Traditional utility-based approach to credit network formation.  Analysis will rely heavily on game threory.  Equilibrum concept will be some version refinement of Nash.  Work will rely heavily on previous research by Jackson, Goyal, Vega-Redondo, et al.
• Random network approach to credit network formation.  Relies heavily on research done by Jackson, Barabasi and Albert, et al.  Analytical techniques borrow heavily from statistical physics.

Need to begin building simple agent-based computational models of credit network formation.  Starting point will be Axtell and Epstein's model of emerging credit networks from Growing Artificial Societies.  Other more advanced models will be considered for year's two and three.

• I will need to establish relationships with the School of Informatics in order to develop my programming skills to the point where I can write and compile my own code.  Programming is especially important as I will be applying to the Sante Fe Institute's Summer School in 2011.
• Also I need to continue to develop my EVT skills.  Much of this can actually be done in conjunction with my responsibilities as a QM tutor and TA.

These 1st-year objectives fit nicely into my overall PhD agenda which starts with network formation, then moves to diffusion of credit/liquidity over the networks, and then finishes (ideally) with a detailed dynamic computational model that ties everything together.  In addition to increasing my understanding of economic theory, I want to finish my PhD with significant computer programming skills.  These skills should be sufficient to build complex computational models of network formation that could be used to address policy issues.

Teaching the Prisoner's Dilemma...

The Prisoner's Dilemma features prominently in every undergraduate and graduate course in game theory, microeconomics, or international relations. Having had the good fortune to sit through multiple presentations of the Prisoner's Dilemma to a number of different audiences (academic audiences in both undergrad and grad school, private industry, government, etc) I have noticed that the game just doesn't make much sense to people when it is first explained. The audience typically has a hard time understanding why defecting is clearly the dominant strategy...

This is a very rough first draft of notes on the Prisoner's Dilemma that I am hoping to expand into a self contained teaching module at some point...my plan is to develop the module using Joshua Epstein's computational model and ASCAPE software as the major teaching tools.  Robert Axtell's The Evolution of Cooperation: Revised Edition is also a major influence.

I think it would help if the initial presentation did not focus so much on the one-shot version of the game.  Prisoner Dilemma games can be divided into four distinct classes:

1. The One-Shot Version: This is the version emphasized in introductory level courses.  In this version the dominant strategy is for both players to defect.
2. Repeated Game Version #1: In this version of the Prisoner's Dilemma the players will play the game for an a priori known, finite number of rounds.  The result...since the last round is in essence a one-shot game, mutual defection is the dominant strategy. Since both players know this, they will find it optimal to defect in the next to last round, this dynamic devolves all the way back to the beginning.  The strategy of mutually defection still rules in this version.
3. Repeated Game Version #2: In this version of the Prisoner's Dilemma the players will play the game for an infinite number of rounds! In this version mutual cooperation can be sustained as an equilibrium as long as the players care a sufficient amount about future rounds of the game (i.e., their discount factors are not too low).
4. Repeated Game Version #3: In this version of the Prisoner's Dilemma the players will play the game for an unknown but finite number of rounds.  In this world, which to me is the one that most closely resembles reality, the backwards induction argument used in repeated game version #1 doesn't work.  Computational simulations can be explored to understand what ingredients are necessary/sufficient for cooperation to be sustained in the population and in what situations where defectors are able to invade the population.    

As mentioned above...this is just an idea I am playing around with...comments/suggestions/constructive criticism would be appreciated...