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Chapter 5 SUMMARY, CONCLUSIONS, AND FUTURE RESEARCH DIRECTIONS 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. 114 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. 115 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 116 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. 117