Essays on Game Theory

Abstract

This dissertation studies mixed strategy equilibria with uncertainty aversion in strategic games. Experimental evidence has found a substantial discrepancy between the prediction of mixed-strategy Nash equilibrium and observed choice frequency. The quantal response equilibrium model of Mckelvey and Palfrey (1995) has been successful to explain observed aggregate choice frequencies at least qualitatively. However, its fundamental aspect, inherited from the purification theorem of Harsanyi (1973a), that the mixing is not made intentionally is not supported by experimental studies which confirmed the wide use of the mixing device when it is given to subjects. This paper answers this question by proposing a model which explains players’ intentional mixing in strategic games. In the first chapter, we illustrate the main idea of this paper with examples. Then, we compare and contrast the main conclusion of our study with the relevant literature. In the second chapter, we develop the game theoretic model in which strategic uncertainty is introduced and studies the effect of it to equilibria, especially mixed strategy equilibria, when players are uncertainty averse. The players’ perception of uncertainty in opponent players’ strategies is modeled with random perturbation. Uncertainty averse players may strictly prefer the properly mixed strategy in order to hedge against this uncertainty. We define an equilibrium under perturbation as a profile of best response strategies to perturbed belief. We show that for any regular mixed-strategy Nash equilibrium in finite games and under the mild assumptions on perturbation structure, there exists a sequence of strict equilibria under perturbations which converges to the given mixed-strategy Nash equilibrium as uncertainty in perturbation vanishes. This result has two implications. First, in almost all finite games, we can find the corresponding equilibrium under perturbation of any given mixed-strategy Nash equilibrium. A discrepancy between them could be substantial if the degree of uncertainty in perturbation is significant. Second, the theory predicts that in equilibrium under perturbation, players intentionally mix their pure strategies due to hedging motive. This implication is clearly distinguished from Harsanyi’s purification theorem (and therefore the quantal response equilibrium model’s interpretation for mixed equilibrium strategy), and consistent with experimental findings. In the third chapter, we test our model’s predictive power by re-examining the experimental studies; Ochs (1995) and Selten and Chmura (2008). We then compare the performance of our model with the QRE model and show that our model outperforms the QRE model in both re-examinations. In the re-examination of Ochs (1995), our model’s predictive power is significantly better than the QRE model in the reasonable range of risk aversion which might not be perfectly controlled in the original experiment of Ochs (1995). Re-examining Selten and Chmura (2008) also shows that the change in equilibrium prediction caused by the payoff transformation is actually observed in the data, which only our model can explain.