# Market Making Card Sum Game

I am preparing for an interview with a prop trading firm and wanted to discuss potential strategies for the classic market making games. I have seen similar posts on the forum, but a lot of the answers do not feel complete, so wanted to discuss on here. Here is a proposed game.

1. Game #1 : Make a market on the sum of 3 cards. Each round, a card will be revealed.

I have a statistical value for the expectation, which is 21 (each card has an expected value of 7). Hence, I can make my spread pretty small and make a market of 20 @ 22. Assuming that the interviewer knows the cards and knows that the sum is 25, he will take my offer. Hence, I am now long 1 stock at 22. In the next round, the revealed card is 1, so my hypothetical expected value is now decreased at 15. This is where I would appreciate some input. These are the two choices that I am thinking about:

1. Since I know that the interviewer is an informed trader, I can make the assumption that the next two cards must be very high despite this initial small one, as he had bought at 22. Hence, despite the fact that my theoretical expected value went down, I still make higher prices for the market as I follow the informed trader's direction. Hence, I now quote 22 @ 24.

2. I can treat the interviewer as a noise trader, and hence do not let his trades influence my decisions too much. Hence, as my new expected value is 15, I lower the market to say 14 @ 16 despite the fact that he just took my offer.

In what event do I let the interviewer's trades take me outside of my expected value? I know that I am the market maker, so I ultimately decide on the prices. Is this simply dependent on whether the interviewer is an informed or noise trader? If he is an informed trader, do I follow his trades with no limit, even if they far exceed my personal expected value?

Also, as an additional question, do I try and stay flat and keep a neutral position?

I am a little confused to the available approaches, and would love to hear some input!

Few corrections:

• The expected value is not 21 since all the cards are from the same deck (it's something between 19.5-19.6).
• If he hits your offer at 22, you will be short 1 share.
• Knowing the expected value is not enough to make such a tight spread unless you are asked to make bid/offer within N% confidence interval.

You have to stay flat for this game to make sense, meaning when the true sum is revealed if you are short/long you will have to close that position at the value of the sum.

Since it is an interview question, your reasoning is more important than winning. You need to make assumptions such as bid/ask price is a whole number and size is always one, the other side is playing optimal etc. Only then having a strategy makes sense.

To answer your questions, if you assume the interviewer knows the sum, he is already an informed trader and the only way for you to make money is when he is noise trading. So you have to make market based on his moves to minimize your loss (choice 1). On the other hand if you are both uninformed you can follow expected value and ignore his moves (choice 2).

• I am not sure where the original poster got this question from but I’ve seen this deck MM question asked 2 different ways. Either find the probability that the trader is a noise trader given specific trading patterns, and the other; given a probability $p$ that they’re a noise trader, what is the EV of subsequent card reveals Commented Feb 7 at 23:56