Questions regarding the calculation ofthe market share without simulating

Insurance pricing game
#1

My first question is: Do you use unordered sampling without replacement to calculate market shares?

Here’s an example assuming that there are 209 models (mine + 208).

Round 1:
Price each policy for 208 models. Let’s say I’m more expensive than 18 and cheaper than 190.

There are 190! / ( 181! * 9!) = 733629525258630 markets where I am the cheapest price, out of a total of 208!/ (199! * 9!) = 1684982353074800 markets, so I will be able to sell the policy in 733629525258630 / 1684982353074800 = 43.54% of markets

Round 2 (assuming I didnt make it into the top 10%
10% of 209 models is 20.9 Let’s round to 21 “top” models. and let’s say i am cheaper than 10 and more expensive than 11. I’ll be the cheapest price in 10!/9!/1! = 10 markets
out of in 21! / (12! * 9!) = 293930 markets , so my market share for round 2 (and the leaderboard) will be 10 /293930 = 0.00340217%

My second question is :What happens if I am in the top 10% (aka 21) competitors?

Are there less markets to compete it in (20!/( 11! * 9!) = 167,960 instead of 293,930 ) or do you bring in a replacement (maybe the 22nd best?)

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#2

Thanks for raising the questions, it would be interesting to understand better how the whole process works.

I might just add that to also allow for players who set the same price for certain policies, the math gets even more complicated. Using your example above, you may be more expensive than 18, cheaper than 180 and same price as the other 10!

And please do not make assumption that same pricing can’t or rarely happen, I can think of 2 scenarios easily: a) setting the price at 0; b) setting the price at the mean of the training data.

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#3

Ah yes that’s almost the correct calculation! :heavy_check_mark:

Let’s dig into it.

Conversion rates and market share :chart:

Just for the sake of terminology, let’s be clear, those % numbers you are computing are conversion rates, not market shares. So you will win that specific policy around 0.003% of the time. In your example this happens because there are only 10 markets where you are the cheapest for that particular policy, so you have no chance of winning that one against the top players!

A simple example: 11 models, 1 policy, 5-player markets :game_die:

Let’s make the example a little simpler (or with easier numbers!). Let’s say the market size is 5, and there are a total of 11 models and our threshold is top 7 models rather than top 10%. Let’s make it even simpler with only 1 policy in the market so that the conversion rate and market share mean the same thing. Otherwise the market share is the mean value of the conversion rates.

Note: In practice you have to perform this calculation for every policy in the dataset.

Round 1 :repeat_one:

Now let’s assume you are cheaper than 5 others. Your conversion rate in round 1 can be computed:

A) Number of possible round 1 markets with you in it: nCr(10, 4) = 210
B) Number of round 1 markets where you are cheapest: nCr(5, 4) = 5
Conversion Rate: B/A = ~2.4%

Round 2 with you not the top of round 1 :3rd_place_medal:

Ok now let’s go into round 2 with the assumption that you’re not in the top 7 round 1 models and you’re cheaper than 4 in the top 7:

C) Number of possible round 2 markets with you in it: nCr(7, 4) = 35
D) Number of round 2 markets where you are the cheapest: nCr(4, 4) = 1
Conversion Rate: D/C = ~3%

Round 2 with you in the top of round 1 :1st_place_medal:

Now what happens if you ARE in the top 7 and you’re cheaper than 4 of the top 7? Well from your perspective it’s exactly the same as before right? Except the total number of markets has to account for this restriction. So it looks something like:

E) Number of possible round 2 markets with you in it: nCr(6, 4) = 15
F) Number of round 2 markets where you are the cheapest: nCr(4, 4) = 1
Conversion Rate: F/E = ~6.5%

Wait, so I get 6% if I’m in the top models and 3% if I’m not? :exploding_head:

Yes because those markets are different “hypothetical worlds”.

Our base case is when your model is cheaper than 4 others in the top 7:

  1. If you are in the top 7 there is 1 less player in the top that you have to play against (yourself)
  2. If you are not in the top 7 then the number of players remains 7 (not 6).

Cases (1) and (2) represent different competitive dynamics and are not really comparable from the perspective of a single model.

Your specific questions :thinking:

Unordered sampling without replacement?

Yes. This is necessary if we are to treat each market as a new simulation.

What happens if you’re in the top 10%?

You still compete only against the others in the top 10% in exactly the same way as round 1. So there are fewer total markets.

What about identical prices for policies?

In that case the calculation does not change very much, remember we are computing conversion rates. So all we have to do is to remember to not count someone as being cheaper than someone else when they are actually equal.

The rest of the mathematics follows normally (see :point_up_2:) . In practice what this amounts to in a simulation is randomly breaking the ties, which ultimately average out over many repeats of the market.

I hope this clears things up, and more importantly, that I have not made an arithmetic mistake! :slightly_smiling_face:

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