# Math behind SoS approach

Dear Sawtooth team,

I conducted a CBC analysis and used the Sampling of Scenario (SoS) approach in the Lighthouse market simulator to generate WTP values. I used the share of preference simulation method. It worked well.

I was wondering if you could provide me with the math (or function) behind the SoS approach? Further, could you please explain to me the effect of competition in the approach? I see that a monopoly is different from a market with competitors. I can also see from my results that with more competitors (and holding all else equal), WTP values decrease. It that because the choices are split between them?

Thanks so much for your help and time in advance and best regards
Ulrike

Sampling of Scenarios (SOS) is just drawing random sets of competitors and random specifications of the firm's product, and summarizing the WTP across those 100s or 1000s of scenarios.

The fundamental WTP calculation is based on simulating the firm's product vs. a set of competitors.  Then, writing down that simulated share of preference for the firm's product.  Then, enhancing the firm's product and finding the increase in price that drives the share for the firm back to the original share amount prior to enhancing the product.

Details about the approach may be found in this white paper:  https://sawtoothsoftware.com/resources/technical-papers/estimating-willingness-to-pay-in-conjoint-analysis

As shown in the white paper, with the assumption of more competitors, WTP decreases somewhat.  But, depending on the CBC dataset and attribute list dimensions, the WTP tends to stabilize after about 5, 10, or 20 competitors are assumed.  Again, it just depends on the attribute list and the data set.  I think with more levels within an attribute, it will take more competitors under the Sampling of Scenarios until WTP stops decreasing.  We show two examples in the appendix of the paper I cite above related to this issue.
answered Nov 24, 2021 by Platinum (195,590 points)
Thank you for your detailed response. Could you please provide me with the equations or guide me to a source where I might find them?
The methodology does not involve a closed-form equation.  It involves simulating market share for a firm's product vs. competition.  Simulating market share from conjoint analysis utilities can involve your choice of multiple market simulation approaches.  The WTP is found by enhancing the firm's offering and finding the increase in price that drives the share back to the original amount prior to enhancement.

Sampling of Scenarios (SOS) simply involves drawing random staring points for the firm's offering and the competitors' offerings...then repeating the same approach to find the price difference that equalizes the share of preference.

You can find the details in our white paper:  https://sawtoothsoftware.com/resources/technical-papers/estimating-willingness-to-pay-in-conjoint-analysis
Ok, I understand. Thank you Bryan.