# An Alternative Approach to Compare Three Stationary Concepts: A Dummy Judgment

2 Pages Posted: 18 Dec 2010

Date Written: December 17, 2010

### Abstract

Three stationary concepts, Nash equilibrium, quantal response equilibrium and impulse balance equilibrium, have different predictions for a given completely mixed 2x2-game, which leads experimental economists to conduct relevant experiments with different parameters for testing the performances of theoretical predictions by those concepts.(see Selten et al., 2008; Brunner et al., forthcoming and Goerg et al., forthcoming in AER etc.) Limited by the precision of the data from laboratory experiments, it is, however, usually difficult to evaluate different stationary concepts in terms of the Euclidian distance between the prediction and the data. We suggest that it's important to clarify the internal logic of those theories and determine typical discrepancies between those theories. By mathematical proof, we show that, if a Dummy is added to a constant game and thus we get a non-constant game, only the IBE in those three concepts prefigures a regular change of the predicted equilibrium, namely, if we add a diamond-shaped Dummy on the constant game, the probability of the UP used by the UP-DOWN player, compared with that in the original constant game, will increase in the new non-constant game, and, in contrast, if we add a L-shaped Dummy, the probability of UP will decrease; meanwhile, in both situations that of the LEFT used by LEFT-RIGHT player has no such regular change. However, all other theories do not logically contain such a deduction. Thus, we get another approach for testing those concepts, that is, experiments can be designed to identify whether there are stable discrepancies in those three games - the constant game and two non-constant games transformed from the constant game by adding Dummy. Accordingly, we have three treatments in this paper, the control treatment with payoff matrix of [(5, 0) (0, 5) (0, 5) (5, 0)], treatment 1 with payoff matrix of [(5, 5) (5, 10) (0, 5) (10, 0)] from adding a Diamond-shaped Dummy on the payoff matrix of the control treatment and treatment 2 with payoff matrix of [(10, 0) (0, 5) (5, 10) (5, 5)] form adding a L-shaped Dummy. For QRE, as well as other best response structure concepts, this design ensures that, for any given λ, such additive changes do not have any effect on the quantal response equilibrium. However, IBE tells a stationary state of (0.5, 0.5), (0.667, 0.5) and (0.333, 0.5) for the control treatment, treatment 1 and treatment 2 respectively, which is, in other word, that IBE tells a regular change in the frequency of UP-DOWN but no change of LEFT-RIGHT. The "observed equilibria" are as follows: UP frequencies are 0.511(±0.005), 0.592(±0.029) and 0.384(±0.009) for the control treatment, treatment 1 and treatment 2, respectively; LEFT frequencies are 0.492(±0.005), 0.519(±0.030) and 0.497(±0.014), respectively. Thus, we show that, in terms of predicting the regular change, IBE outperforms in those three stationary concepts. Meanwhile, further analysis of the IBE reveals that the success of IBE is due to "Selten Transformation"( Brunner et al. noted this point in terms of Euclidian distance.), not "Impulse Balance." On this basis, the paper concludes by discussing possible ways to improve those stationary concepts.

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