Is game theory useless?
The ability to quantify and predict phenomenon in the physical world has always been the hallmark of a good theory. While the ancestral fields of inquiry in the natural sciences have enjoyed tremendous success in describing the fundamental interactions by a largely linear approach, the emergence of complex dynamical systems in the applied and social sciences have met with considerable resistance. While there is now a growing consensus that canonical physical concepts such as “equilibrium” and “potential energy” (which we’ve also kind of talked about in class), or even more esoteric principles like “Quantum Game Theory” and “Financial Thermodynamics” [1], are to some extent transferable to the study of strategies and games, there remains a certain hesitation of its direct applicability to the actual complex world [3]. There exists a contention, albeit not from the Game theorist themselves, (but rather the physicist whom they borrowed the ideas from) that Game Theory remains a purely analytical toy model that has little to no descriptive power even in qualitative scenarios.
I chanced upon this article entitled “why game theory is useless?” by physicist and author, Michael Buchanan [3], in which he explained the rationale of the above physicists. In short, his main argument is that the foundational concepts in Game Theory, in particular the canonical notion of “equilibrium” are not an equivalent set of assumptions that hold true in the real world, unlike in certain constrained physical systems. More specifically, he argues that, game theory neglects the problem that irrational players exist in the real world.
I choose to think of this in a probabilistic manner:
While an ideal, rational player have a distinct and significantly reduced parameter space (number of strategies), irrational players have an enormous number of strategies that he might play with small but finite probabilities. This leads to a varying cycle of small adaptations and response, which are actual fluctuations in the true probabilities associated to the “rational” strategies of the irrational player. At first glance, these perturbations can be thought of as reasonably small corrections, but the networks that accompany such real-world games are highly complex (non-linear) and quickly bifurcates towards chaos. Thus, Buchanan concludes that in the dynamical real world, it is almost destined to never find a “neat Nash equilibrium solution”.
Moreover, Buchanan alludes to the fact that these chaotic games do not even need to involve huge parameter spaces. For illustration, he simulated a general two player game with each player having N=50 strategies and payoff matrices whose entries follow a simple Gaussian distribution. While both players “work hard to learn the optimal strategies”, and there were no external “shocks” to the system, no mixed Nash equilibrium was found and the system exhibited highly chaotic behavior as seen below in the payoff difference against time graph.
A study into the chaotic behavior of the simple “rock-paper-scissors” game also seems to suggest the limitations of classical Game Theory [2]
In a rational version of the rock-paper-scissors the payoff matrix can be written simply as follows, and the mixed Nash equilibrium is given by the equal probabilities (1/3) of each strategy.
|
Rock |
Scissors |
Paper |
|
|
Rock |
0 |
1 |
-1 |
|
Scissors |
-1 |
0 |
1 |
|
Paper |
1 |
-1 |
0 |
Through various forms of simulated irrational behavior, including the inertia in changing opinions, and how the memory of recent events affect the probability of the strategy chosen, it was shown by Salvetti et al.[2] that the probabilities deviated chaotically from the Nash equilibrium, and no equilibrium solution was found
Thus, the attempt to understand complex real-world games is probably best done through a detailed, case specific, dynamical analysis as compared to conventional classical game theory.
References
[1] Patel, N., Nature, 445, 144-146 (2007)
[2] Salvetti F., Patelli P., Nicolo S., Applied Soft Computing, v.7-4, 1188 (2006)
[3] Buchanan M., (2011) http://physicsoffinance.blogspot.com/2011/10/why-game-theory-is-often-useless.html