an over-estimation of the number of unprotected sexual acts. Second, while HIV status is not perfectly observable, it might be partly so, and if individuals choose not to have unprotected sex with individuals they know with certainty are HIV-positive then our assumption that individuals randomly choose their partners will lead us to under-estimate the number of unprotected sexual acts. Finally, the assumption that individuals choose partners only from their own village may not be a good approximation. The direction of the bias this will create is, however, unclear.

_Estimated sexual behavior change_ Table 6 reports the results with _π_ set to _π_ = 0.0128.<sup>14</sup> The effect of the intervention on the estimated number of risky sexual acts is presented in columns (1)-(2). On average we observed a reduction of two risky sexual acts in the pooled intervention group over the 2 year study period, or a 20% reduction relative to the control group, and 2.7 fewer risky sexual acts in the high lottery arm. The mean number of risky sexual acts in the control group was 9.75 over the trial period.

Columns (4)-(5) report the results for the number of unprotected sexual acts. For the pooled intervention group we estimate a reduction of approximately 11 unprotected sexual acts during the two year study period, with a mean of 59 unprotected sexual acts in the control group. The treatment effect was larger for the high-lottery arm (15.5 fewer unprotected sexual acts as compared to the control group) than for the low-lottery arm (5.4 fewer unprotected sexual acts as compared to the control group), although we cannot reject the equality of these effects ( _χ_<sup>2</sup> -stat=1.39, p-value=0.24).

# Columns (3) and (6) in table 6 exploit the random variation in prizes; i.e. we estimate

> 14Our preferred estimate of _π_ ( _π_ = 0.0128) is taken from Baeten et al. (2005). Their sample population (from Mombasa, Kenya) looks similar on a number of observable characteristics to our sample from Lesotho (similar age ranges of the participants, low condom use, most study participants are uncircumcised, and similar rate of baseline HIV prevalence rates among prospective partners). Our estimate of the per-act transmission probability _π_ is consistent with the review and meta-analysis in Powers et al. (2008) and Boily et al. (2009). Powers et al. (2008) conclude that in a low-income context where the partner has STI or is uncircumcised, heterosexual infectivity probably exceeds 1%. Boily et al. (2009) estimate lower pooled female-to-male and male-to-female per-act transmission probabilities in a low income setting (0.38% and 0.30%, respectively). These estimates, however, are drawn from studies primarily implemented within the context of interventions involving an important counseling component (use of condoms) and thus do not provide an accurate estimate of the per-risky act transmission probability. Inconsistent use of condoms is positively correlated with infectivity. Hira et al (1997), for example, estimate a 4.7 times higher infection rate among couples who report using condoms less consistently compared to couples who reported using condoms at every intercourse. Adjusting Boily’s et al. (2009) pooled number for sex without condoms, would imply an upper-bound per-risky act transmission rates of _π_ = 0.0179 and _π_ = 0.0141 (it is an upperbound because the estimates in Boily et al. (2009) are weighted averages of condom and non-condom based per-act transmission probabilities). Moreover, risk factors that increase the per-act infectivity, and that likely play an important role in Lesotho, such as the presence or history of genital ulcer, usually caused by a sexually transmitted infection, and the low rate of male circumcision, are not properly accounted for in the pooled per-act estimates in Boily et al. (2009).

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