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Generic statements ('Tigers have stripes') are pervasive and early-emerging modes of generalization with a distinctive linguistic profile. Previous experimental work found that generics display a unique asymmetry between their introduction conditions and the implications that are typically drawn from them. This unique asymmetry effect of generics was also thought to have serious social consequences: if speakers tend to accept and communicate negative generics about social groups based on prevalence levels that are systematically lower than what is assumed by their recipients, then using generics is likely to exacerbate negative social stereotypes and biases. This paper presents evidence against the hypothesis that only generics display an asymmetry effect. Correcting for limitations of previous designs, we found a generalized asymmetry effect across generics, various kinds of explicitly quantified statements ('most', 'some', 'typically', 'usually'), and variations of predicated properties (striking vs. neutral). To determine if generalized asymmetry may exacerbate biases, we examine whether speakers systematically choose, from a range of options, generalizing sentences based simply on their acceptance conditions, without regard for the implications which their recipients are likely to draw. We found that, in neutral or cooperative scenarios, speakers reliably choose generalizing sentences whose implied prevalence levels tend to closely match actual levels. In non-cooperative scenarios, speakers can exploit asymmetry effects to further their own goals by choosing generalizing sentences that are strictly true but likely to mislead their recipients about actual prevalences. These results refine our understanding of the source of asymmetry effects and the conditions under which they may introduce biased beliefs into social networks.