  In an earlier experiment, participants played a perfect information game
against a computer, which was programmed to deviate often from its backward
induction strategy right at the beginning of the game. Participants knew that
in each game, the computer was nevertheless optimizing against some belief
about the participant's future strategy. In the aggregate, it appeared that
participants applied forward induction. However, cardinal effects seemed to
play a role as well: a number of participants might have been trying to
maximize expected utility.
  In order to find out how people really reason in such a game, we designed
centipede-like turn-taking games with new payoff structures in order to make
such cardinal effects less likely. We ran a new experiment with 50
participants, based on marble drop visualizations of these revised payoff
structures. After participants played 48 test games, we asked a number of
questions to gauge the participants' reasoning about their own and the
opponent's strategy at all decision nodes of a sample game. We also checked how
the verbalized strategies fit to the actual choices they made at all their
decision points in the 48 test games.
  Even though in the aggregate, participants in the new experiment still tend
to slightly favor the forward induction choice at their first decision node,
their verbalized strategies most often depend on their own attitudes towards
risk and those they assign to the computer opponent, sometimes in addition to
considerations about cooperativeness and competitiveness.
