# Influencer Campaigns

*Onur Solmaz, Kemal Apaydin, Sylvia Janoskova*
*Theory for 2Gather, our submission to Disrupt Berlin 2019*
*Dec 12, 2019*

**tl;dr our idea:** *Influencer campaigns* are a feature to our gamified analytics platform, specialized for influencer marketing. Influencers launch campaigns together to collectively increase their conversion rate, owing to what we call the "snowball effect". Sales revenue is then redistributed to reward the influencers that have performed the best.

## Snowball Effect

In organic influencer marketing, conversion rate increases with each exposure from an additional influencer. This is called the "snowball effect".

Let $F$ be the set of followers, and $F_i\subset F$ be the set of followers of each influencer $i\in I$. Influencers in $I$ start a campaign to market a product too all of their followers.

The exposure degree $\epsilon:F\to \mathbb{Z}_\geq 0$ maps each follower to the number of influencers that have exposed them to the product:

$$
\epsilon(f) = \lvert\{i\mid f \in F_i \}\rvert
$$

Let the probability of conversion of a single follower be a monotonically increasing function of $\epsilon$, defined as $p: \mathbb{Z}_{\geq 0} \to [0,1]$. Below is an example of such a function:


Note that the effect of exposure on conversion decreases after some point, and the probability should either stay constant or decrease. Such behavior depends on the product and should be captured by the corresponding phenomenological model. We take the existence of such functions at face value and build our formulation based on that.

Conversion rate for the case when each follower is exposed by only a single influencer is simply calculated as

$$
c_1 = p(1)
$$

Given the follower and influencer sets, the overall conversion rate is then calculated as

$$
c = \frac{\sum_{f\in F} p(\epsilon(f))}{|F|}
$$

The snowball surplus is then defined as the increase in conversion rate compared to the case when each follower is exposed by only a single influencer:

$$
\Delta c = c - c_1
$$

An example to understand the snowball surplus: If conversion rate without influencer collaboration is 5%, and $\Delta c$ is calculated to be 3% from the social connectivity graph, the conversion rate is expected to be 5+3=8% as a result of collaboration.

If a group of non-collaborating influencers are able to sell a total of $s$ products with conversion rate $c_1$, their total sale for a collective campaign is expected to be

$$
s' = s \frac{c_1+\Delta c}{c_1}
$$

Continuing the example above, if non-collaborating influencers make a total sale of 1000 units, then the campaign would help them sell 1000*(5%+3%)/5%=1600 units with a 600 unit surplus.

## Performance Based Sale Revenue Redistribution

Let $s_i$ be the sale amount of influencer $i$, $c_i$ be their conversion rate and $p$ be the revenue coming from the sale of a single unit.

Without pooling, each influencer $i$ would receive $r_i = s_ip$.

If influencers pool, their conversion rate is factored in their revenue. Letting $\mu$ be the mean conversion rate, the normalized conversion rate is calculated as

$$
\bar{c}_i = \frac{c_i}{\mu}-1
$$

Then the pseudo T-score of each influencer $i$ is calculated as

$$
t_i = 5 + 10 \frac{\bar{c}_i}{\sum_j|\bar{c}_i|}
$$

This score isn't used in revenue redistribution, but as a metric for influencer performance.

Influencers pool their revenue, and the total revenue is calculated as $\sum_j r_j$. Then the original share of revenue of an influencer $i$ is calculated by $r_i/\sum_j r_j$.

If an influencer overperforms (has a higher conversion rate) with respect to the mean, they will be rewarded by getting a higher share of the total revenue. If they underperform with respect to the mean, they will be punished by getting a lower share.

The performance based score of $i$ is then defined as

$$
\sigma_i = \frac{s_i}{\sum_j s_j}(1+\alpha\bar{c}_i)
$$

where $\alpha \geq 0$ is a parameter controlling the degree of change in revenue due to performance.

The scores are used to calculate the new shares of the influencers. Each influencer $i$ receives

$$
\boxed{
r'_i = \frac{\sigma_i}{\sum_k \sigma_k} \sum_j r_j
}
$$

The performance based reward, or surplus, of $i$ is calculated as

$$
\Delta r_i = r'_i - r_i
$$

The goal is to have a positive surplus for all influencers, regardless of the competition between campaign participants. The parameter $\alpha$ defined above needs to be set in a way that achieves this goal. If $\alpha$ is set too high, participating can be less profitable than working alone. In that case, $\alpha$ should be lowered enough to allow a positive surplus for all participants. This way, collaborating becomes a positive sum game for every influencer.

## Conclusion

We have presented a formulation to estimate the surplus that can be generated by collective influencer marketing using the social connectivity graph. Additionally, we derived a redistribution formula that rewards better performing influencers, thereby incentivizing competition among themselves. With these incentives, we think that influencers will produce better content and increase sales considerably.