This paper reports an experiment that elicits subjects' initial responses to 16 dominance-solvable two-person guessing games. The structure is publicly announced except for varying payoff parameters, to which subjects are given free access. Varying the parameters allows very strong separation of the behavior implied by leading decision rules. Subjects' decisions and searches show that most subjects understood the games and sought to maximize payoffs, but many had simplified models of others' decisions that led to systematic deviations from equilibrium. The predictable component of their deviations is well explained by a structural nonequilibrium model of initial responses based on level-k thinking.