Chapter 5
The Nash equilibrium concept, a generalization of Von Neumann's
minimax theorem, is a crucial component and a basic building block of post-war
research of Game Theory. Though its mathematical properties have been studied,
published, and agonized over by thousands of brilliant minds, its intuition is still
far from clear. In fact, there are two different intuitions, both entertained in
Nash's dissertation [see Nash, 1996, pp. 32-33]. The first is deductive notion of
equilibrium that the Nash equilibrium is the result of rational players solving a
mathematical problem in their heads: "What would be a rational prediction of the
behavior to be expected of rational playing of the game in question? By using the
principles that a rational prediction should be unique, that the players should be
able to deduce and make use of it, and that such knowledge on the part of each
player of what to expect the others to do should not lead him to act out of
conformity with the prediction, one is led to the concept of the solution defined
before." [Nash, 1996, pp. 32-33]. The second intuition on the Nash equilibrium
concept is that the Nash equilibrium is the resting point, the steady state, of a
dynamic process of players learning each other’s choices.
While these two
intuitions both lead to a Nash equilibrium outcome, they strongly clash when
multiple equilibria exist. In other words, multiple equilibria force one to choose
between intuitions.
We have demonstrated in this dissertation the lack of empirical support for
deductive equilibrium principles as the main driving force in human players’
strategic choices. Evidence against deductive selection cannot be derived without
a competing theory. A statistical rejection of a principle (a null hypothesis) can
only be made in the presence of an alternative hypothesis. In our case, the
alternatives presented were in the form of simple boundedly rational rules. The
boundedly rational rules found to contribute significantly to explaining realistic
human reasoning processes were (1) random behavior, also known as level-0, (2)
level-1 bounded rationality, best-responding to a uniform distribution over actions
by other players, (3) level-2 bounded rationality, best-responding to a level-1 type
population, (4) maximax behavior, best-responding to an optimistic assessment of
others' actions, and (5) worldly behavior, best-responding to a convex
combination of evidences.
Behavioral rules 1, 2, 3, and 5 are based on a
hierarchical notion of rationality, where an iterative self-referential process is
truncated after a few rounds of iteration. Nash equilibrium beliefs were found to
contribute significantly to likelihood but could not be attributed to a large portion
of the experimental subject population.
Furthermore, the Nash equilibrium
behavior that was found could not be characterized as following any of the major
deductive selection principles of payoff dominance, risk dominance and security,
all modeled in a variety of manners. These findings would seem to support the
second intuition of Nash equilibrium—pertaining to an equilibrium being the end
result of a dynamic process.
Though evidence is strongly in favor of the above boundedly rational rules
in a parametric likelihood-based estimation framework, one may nonetheless
question their validity since the types were postulated in advance of the estimation
(along with the number of sub-populations). The argument may be made that
good fit does not equal correct specification. Bluntly, the assumptions regarding
the types may be ad hoc. For that reason, data on beliefs would be most useful
for determining both what types exist and how many types exist. Analysis of
hypothesis data in a non-parametric kernel-density framework allows us to test
some of the basic hypotheses without the need for the usual assumptions on types.
Conducting a non-parametric analysis we find that: (1) strong evidence exists to
support the hypothesis that the population is heterogeneous. Depending on the
methodology used, in terms of both the kernel function selected and the boundary
correction used, it appears that the population, in terms of the nature of the
hypotheses entered, is composed of four to six sub-populations. (2) These subpopulations remarkably correspond to the hierarchical notion of bounded
rationality, specifically to level-1, level-2, and level-3. Evidence of Nash thinking
is relatively weak and inconclusive. (3) Level-1 bounded rationality is the most
prominent in the population, accounting for the largest portion of behavior, as
seen by the portion of the density mass it accounts for.
Having proved the heterogeneity of the population (chapter 2 and 3),
characterized that heterogeneity non-parametrically (chapter 2) and parametrically
(chapter 3), in particular for games with multiple equilibria (chapter 3), we set to
deal with the dynamic portion of our inductive approach. Chapter 4 discussed the
alternatives in choosing a dynamic theory, estimating its behavioral parameters,
and assessing its performance. In chapter 4, models of adaptive dynamics were
shown to have greater predictive powers than deductive equilibrium selection
principles. However, there are experimental settings in which adaptive dynamics
fail to explain the data. Particularly hard to explain are cases where a population
of players begin in one equilibrium's best-response basin of attraction, crosses to
another's, and ends up in the latter's equilibrium. In chapter 4, we carefully
demonstrated the "failure" of standard theories of dynamics in a particular game,
namely, game 16. The standard theories apparently failed because they did not
correctly model players' abilities to deduce dominated strategies (action C) and, to
a large extent, avoid such strategies. The ability of players to identify dominated
strategies resulted in the reduction in frequency of action C much faster than the
standard theories would have predicted, partially accounting for the crossing of
basins. On the other hand, there seemed to be a role to "frustration" in the game
as well, leading us to believe that there might be an aspiration-experimentation
connection that could explain the dislodging of the dynamic to another basin.
Preliminary results on one version of such an aspiration-experimentation model
show promise. This apparent success is gauged in both a significant improvement
in the likelihood and better prediction as measured by simulated densities of final
choice distributions.