Strategic behaviour often diverges from Nash-equilibrium, in particular in inexperienced play. Studying a class of games in which participants choose their payment and receive it as long as their opponent chooses a different amount, I show that none of the popular models of behavioural game theory predicts the predominant aggregate choice pattern consistently. And yet, noisy introspection (Goeree and Holt, 2004) readily accounts for about half of the individual observations. The reason for the apparent paradox and the mis-match of the aggregate data and the models is a disregarded behavioural type that makes up about 25% of the population. These 25% display a specific form of central-tendency bias, holding beliefs that peak in the centre of the option set and that are roughly symmetric. In addition, the players show a more heuristic process translating their belief into actions, as their choices cannot be explained readily by quantal responding. The behavioural pattern of a ‘centred belief’ in connection with boundedly-rational decision-making is present also in another prominent game from the literature on behavioural game theory, the 11–20 game. Finally, I show that classifying players as ‘heuristic centred-belief types’ by one game’s beliefs has predictive power for behaviour in the other game.