We develop the theory of behavioural types in social dilemma games by experimentally studying four public goods voluntary contributions games in which the incentive structures are varied. We replicate previous studies by identifying free-riders, as well as two distinct groups of conditional cooperators (strong and weak), as comprising a supermajority of participants when payoffs are linear. Strong conditional cooperators, who match contributions one-forone in the linear game, display sophisticated behaviour in games with nonlinear payoffs: they match one-for-one only when doing so improves social welfare. This mode of conditional cooperation appears to be underpinned by a sophisticated understanding and assessment of the financial incentives presented by the game.