Sa[BB]ermetrics: The Hidden Value of Motivation in Backdoor Success Rate – 08/20/13

With 13 seasons of Big Brother (USA), 1 season of Big Brother (CAN) and one oft-forgotten season of the Glass House mislabeled as a Big Brother in the books, the sample size is just large enough to do what no one has tried to do before: analyze Big Brother game play and strategy by the numbers.

Much of the strategy discussion this week in the Big Brother house centered on whether the presumed eviction target, Helen, should be nominated straight away or backdoored. The argument against option B is that even though nominating her straight away, ensures she plays in the veto competition, it doesn’t significantly change the probability of winning POV since only 8 people remain in the house. If we assume that Helen is an average competitor (a reasonable assumption as per our BB15 competition metrics) she holds a 1/6 chance of winning the veto when nominated straightaway. This probability drops to 1/9 (2/3 chance she is picked to play in the veto * 1/6 that she wins) if she is left off of the block. If we assume that whomever wins the veto will use it (either Helen when on the block, or another competitor in order to backdoor Helen), this means the chances that Helen remains on the block come eviction night are 83% if frontdoored and 89% if backdoored. This marginal difference in success rate is probably not large enough to offset the small chance that Helen wins the veto while off the block and uses it on Elissa, causing both the primary and secondary targets to be safe from eviction. Thus, from this rudimentary statistical calculation, nominating Helen straight up appears to be the better move.

This analysis requires the assumption that the vantage from which you play veto competitions — that is, whether you are HOH, a nominee or a randomly chosen competitor — has no effect. However, it seems likely that motivation would play a role in how players perform in the competition. Intuitively, it makes sense that nominees will fight harder to win the veto since they need to save themselves from the block. On the other hand, non-nominees will often throw these competitions in order to avoid making waves and/or being forced to decide between using and not using the veto. If this postulate is correct, it should be visible in the stats.

The ability of the houseguests playing in veto competitions as either HOH, a nominee or a non-nominee should even out towards average ability over a large enough sample size. Thus, if motivation has no effect, we would expect that competitors of each type (HOH, nominee, other) would win POV at a similar rate over our 140-veto competition sample. The expected probability of a win in this case would simply be one divided by the total number of competitors in that particular competition. Represented below is the expected and observed Veto competition win rate by type of participant for big brother (USA) seasons 4 (the first season where all veto’s competition were played for the golden power of veto) through 14 and big brother (CAN) season one:

¹Probabilities are calculated for 106 6-person veto comps, 12, 5-person veto competions and 12, 4-person veto competions included in the sample

The statistics show that observed win frequencies for nominees and other houseguests deviate significantly from the expected frequency in the direction we would predict if motivation plays a roll. Specifically, nominees win POV at a greater frequency than random chance whereas non-nominees win POV at a frequency lesser than expected. The win probability of the HOH is similar but still lower than the expected rate. This suggests that the vantage of the player significantly affects their likelihood of winning the veto competition, most likely due to nominees having increased motivation to win the competitions.

In turn, this effect of motivation on veto performance must be factored into the calculations of the efficacy of backdooring. The advantage of backdooring (assuming the victim doesn’t see it coming) is seen in both the chance they aren’t picked to play in the veto competition, and in their decreased motivation to win that competition when they are picked. If we substitute the observed win frequencies into our calculations from above, when Helen is frontdoored, she will remain on the block 77% of the time. However, if she were backdoored she would win the veto at a lower rate (12% versus 23% when a nominee) during the 66% of the time she is randomly selected to play. Thus, a backdoor will result in Helen being on the block on eviction night 91% of the time, a much higher frequency than when she is frontdoored. When the hidden value of motivation is taken into account, the correct move by 3AM would have been to backdoor Helen; the numbers were in their favor.

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