Uncovering seeds

We provide the theoretical foundations for a new estimation algorithm that non-parametrically infers level-k beliefs from laboratory choices in generalized guessing games with heterogeneous interactions. The algorithm takes the strategic dependencies of the game and subjects' choices as an input and returns a detailed histogram (a ``pseudo-spectrogram'' of seeds) that represents population beliefs about the behavior of level-0 players. As a by-product, the algorithm also returns the estimated population composition of reasoning levels. The main contributions are as follows. First, we study the equilibrium properties of generalized guessing games and provide an ordinal (visual) characterization for uniqueness. Second, within the level-k model, our key theoretical results establish conditions on the subjective beliefs or the game structure so that the population distributions of level-k choices and the population distribution of beliefs are alike. These results are obtained without any distributional assumptions. We also present a central limit result that supports the use of parametric gaussian approaches often used in the literature. Third, on the basis of the theoretical results, we construct a new a non-parametric maximum likelihood estimation algorithm that fully identifies the belief pattern. Fourth, we apply the algorithm to experimental data. It is found that beliefs cluster around a few focal points and that a few seeds are able to explain a high percentage of observed behavior. Finally, our theoretical results can also be useful in the design of laboratory guessing games with good estimation properties.