Uncovering Seeds

Abstract

We provide the theoretical foundations for a new estimation algorithm that non-parametrically infers higher-order beliefs in generalized guessing games with heterogeneous interactions within an extended level-k framework.

The algorithm takes the strategic dependencies of the game and subjects' choices as input and returns a detailed histogram (a 'pseudo-spectrogram') of seeds that represent population beliefs about the behavior of level-0 players. As a by-product, the algorithm also returns the estimated population composition of reasoning levels. Estimating individual seeds in a highly parametrized model without requiring strong distributional assumptions is made possible by incorporating the game-theoretical structure into a mixture model.

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 seeds are alike. These results are obtained without any distributional assumptions on the seeds. 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 non-parametric maximum likelihood estimation algorithm that fully identifies the seed pattern. Fourth, we apply the algorithm to existing experimental data. It is found that seeds 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.

Keywords: beliefs, expectation-maximization algorithm, level-k, maximum-likelihood estimation, mixture model, networks

JEL Classification: C91,D90,C14,C87