Experimental data for two types of bargaining games are used to study the role of strategic risk in the decision making process that takes place when subjects play a game only once. The bargaining games are the Ultimatum Game (UG) and the Yes-or-No Game (YNG). Strategic risk in a game stems from the effect on one player’s payoff of the behavior of other players. In the UG this risk is high, while it is nearly absent in the YNG. In studying the decision making process of subjects we use the time elapsed before a choice is made (response time) as a proxy for amount of thought or introspection. We find that response times are on average larger in the UG than in the YNG, indicating a positive correlation between strategic risk and introspection. In both games the behavior of subjects with large response times is more dispersed than that of subjects with small response times. In the UG larger response time is associated with less generous and thus riskier behavior, while it is associated to more generous behavior in the YNG.