تجربیات بازخورد انتظارات مثبت و تعداد حدس زدن بازی ها به عنوان مدل های بازارهای مالی

In repeated number guessing games choices typically converge quickly to the Nash equilibrium. In positive expectations feedback experiments, however, convergence to the equilibrium price tends to be very slow, if it occurs at all. Both types of experimental designs have been suggested as modeling essential aspects of financial markets. In order to isolate the source of the differences in outcomes we present several new experiments in this paper. We conclude that the feedback strength (i.e. the ‘p-value’ in standard number guessing games) is essential for the results.

In our earlier papers on (positive) expectations feedback experiments we found very slow or no convergence to the equilibrium price. Number guessing games are very much related to expectations feedback experiments but typically show fast convergence to the Nash equilibrium. This striking discrepancy was the reason for designing five additional experiments where we searched for the driving force behind this difference. We found that expectation feedback games are robust to changes in the incentive structure and changes in the information provided to the participants. We consider this robustness to be good news. On the other hand, the low feedback strength experiments TL67, TC67 and QC67 show that presenting the number guessing game in the context of a financial market, with an interior Nash equilibrium results in very fast convergence like in the traditional number guessing game. Since both prediction accuracy and coordination of expectations appear to be independent of the feedback strength, prediction behavior of participants in the low feedback strength experiments is not substantially different from prediction behavior of participants in the high feedback strength experiments. Instead, the convergence properties seem to be mainly due to the structure of the price generating mechanism itself. To see this, consider Eqs. (7) and (8) again. Both price generating mechanisms give the realized price as a weighted average between the mean predicted price and the fundamental value of 60. However, the weight on the fundamental value in Eq. (7) is only 1/21, whereas in Eq. (8) it is 1/3. As an illustration, if the mean prediction equals 50, a high feedback strength experiment would give an expected realized price of 50.48, whereas the low feedback strength experiment would give a price of 53.33. Clearly, prices in the low feedback strength experiments are therefore more strongly pushed towards the fundamental price and this explains the stronger convergence in those experiments. This is confirmed by simulations with the so-called heuristic switching model that was developed in Anufriev and Hommes (in press). In their model they assume that participants switch between four typical prediction heuristics on the basis of past prediction accuracy of these heuristics. This model is quite successful in explaining the results from the QL95MC experiment. Simulations of this model with the same heuristics and parameter specification but with a feedback strength of 2/3 leads to quick convergence of prices and predictions. Let us now consider again the original beauty contest game as described by Keynes (see Section 1) and compare this with the number guessing game. In the beauty contest game the task is to choose the pictures that are most often chosen by others; this is comparable with the number guessing game with α = 0 and β = 1. In the beauty contest game there are many equilibria where all participants choose the same pictures and therefore the game in essence corresponds to a coordination problem. When β = 1 players who have higher order beliefs on different levels can still make the same decision. A number guessing game with β < 1 (and not too close to 1 in order to be able to differentiate between different levels) is a good tool to study higher order beliefs in experiments but it is not necessarily a good behavioral model of an asset market. A β that is much smaller than one corresponds to an enormous interest rate in a financial context (e.g. 50% in the experiment with low feedback strength) and a price that is mainly driven by dividends. A β close to 1 corresponds to a more realistic interest rate and investors/speculators who focus on capital gains rather than on dividends. This seems to be more in line with modern financial markets. The stylized facts about excess volatility in modern markets also point in that direction.24 Another possible objection to an interpretation of the number guessing game as a model of financial markets is that an asset market is clearly not a tournament where the winner takes all. However, the incentive scheme appears not to be crucial for the number guessing game: in the low feedback strength experiments we find about the same results as in standard number guessing games with a tournament structure. Concluding, we find that the β in the number guessing game is the essential design parameter: a β much smaller than 1 makes it possible to study higher order beliefs but the game is in that case not a realistic model of a modern asset market. A β closer to 1 makes a more realistic behavioral asset market model, but at the same time makes it harder or impossible to distinguish different levels of higher order beliefs. The next question is whether Keynes was right in his proposition that higher order beliefs are an important element of asset markets. Agent based models in quantitative finance typically assume that heterogeneous agents learn by reinforcement or by simple belief learning25 (LeBaron, 2006), that is, beliefs about market behavior and not higher order beliefs. Maybe some investors act on higher order beliefs, but anecdotal evidence (e.g. internet forums) suggests that many investors/speculators view the market like a living organism whose movements you try to predict and not as a game in which you try to form beliefs about the beliefs of others. This interesting question cannot be answered here but is a topic for future research.