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I have a final round with a market making firm coming up and will be asked to play several market making games. I wanted to ask for advice on how to approach these games, especially with an information imbalance. Here is my general strategy:

  • quote evenly around the EV
  • toward the beginning when uncertainty is highest, give the smallest quantity and largest spread allowed (e.g. for a market on the sum of 5 dice rolls and a min quantity of 1 on each side and a max spread of 5, before observing any of the rolls I would go 15 @ 20, 1 up). this is to minimize the maximum potential loss (e.g. imagine you had instead gone 15 @ 20, 10 up. say you get lifted on your 20 offer and the contract settles to all 6's, or 30. You would lose $100 from that).
  • can tighten spread and increase quantity as more information is observed (e.g. in the case of sum of 5 dice, once you've observed 4 rolls, you know the range of the contract is 6 so you can tighten your spread and increase quantity, as maximum potential loss per quantity is now much lower than before)

I think this strategy is pretty good in general, but I wanted to ask in the case of adversarial input (e.g. if the interviewer comes up with some input which he knows ahead of time and tries to throw you off). Consider the game of making markets on the sum of the digits of a phone number (i.e. 10 digits, each are 0-9). The interviewer could manipulate the sequence ahead of time to be something like 1119999999, or something of that flavor.

Following my strategy, I would make quotes around the EV and keep getting lifted. For concreteness, I would start by quoting 42.5 @ 47.5, 1 up (as the EV is initially 45). I would get lifted. But then I see a 1, and the new EV decreases to 41.5. Now I have a dilemma. What do I do?

a) Stick with the EV and go 39 @ 44, 1 up
b) See that my 47.5 offer got lifted, so go something like 48 @ 53?

I think b) probably makes more sense (e.g. this is what you would probably do in the real world when trading with a toxic counterparty), but also this is just throwing EV in the trash which doesn't seem good for the purposes of this game (e.g. perhaps interviewers want to see you are taking EV into account when making markets).

Any advice?

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  • $\begingroup$ You must factor in existing position when making markets. If you got lifted at 47.5 and therefore have a sold position and the EV is adjusted to 41.5, and then you bought it back at that price, you cannot falter that strategy. 6 units of profit and a neutral position for making the next market. $\endgroup$
    – Attack68
    Sep 15, 2022 at 17:36
  • $\begingroup$ Yes I know position is important. But let's say I just keep getting lifted on my offer. And for the purposes of this game, I'm always forced to make a market with max spread 5 and quantity at least 1 on each side after each card reveal. Also let's say I have a fixed number of money, e.g. $100, to start with and can never put myself in a position where at settlement, I could lose all my money. What should I do if I simply keep getting lifted after card reveals? $\endgroup$
    – Max
    Sep 15, 2022 at 20:24

2 Answers 2

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If the interviewer is the only person you can trade with, and he has full information about the result (e.g. all the numbers of the phone number), while you have none, why would you ever make a trade with him at all?

The only way it makes sense to ever trade with him is as a "fee" to get information in order to make money off of other people who do not have that information. Put a value on what it's worth to know it. For example, say the answer is 60, and you know there are 1000 contracts worth of people who think it's random, and will sell at 49, then you know there's $11,000 worth of money to be made off of them by knowing the right answer. Your goal at that point is simply to try to buy this information as cheaply as possible. If the guy is willing to part with it cheaply, good. If he's sophisticated, then you de facto become a broker for him.

There are all sorts of other considerations. Time value of money (when do they add the numbers up? A year from now?) and other uncertainty. Maybe the guy with the asymmetrical information advantage is playing you as well? If you try to buy the info with 1 lots, then maybe he just makes you think it's 60 when it's actually 46, and he puts in an order through a friend of his to sell you the 1000 at 49? Then you might want to look at just doing an arbitrage that leaves you flat.

If instead you're doing the sum of dice rolls with no information advantage, then you're right to be smaller when the result is less known, and larger when it is, but it's not "as small as possible." There's plenty of money to be made when it's more unknown. They probably want to make sure you're just offering equivalent amounts of risk at each stage, which you can adjust by statistical variance quite easily.

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  • $\begingroup$ Thank you for the response. Let's say hypothetically though that I HAD to quote a market with a max spread and min quantity of 1 on each side. We know the interviewer has more information, so let's say I'm just trying to minimize my losses. What should I do in this case in my example above, choice (a), choice (b), or something else? $\endgroup$
    – Max
    Sep 15, 2022 at 20:21
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    $\begingroup$ If that really is the game, yeah. Your "maximize gains" is actually just minimizing losses. But I wouldn't just take that for granted. Quote War Games or say the system is temporarily offline or something. If you do have to, then work out the math. Neither of your plans relies on a mathematically rigorous conditional expectation. Make a wide quote around EV. Then after they buy, what's the conditional expectation of the result if they buy your offer? Then make a wide quote around that. Rinse. Repeat. $\endgroup$
    – kolbe
    Sep 15, 2022 at 20:41
  • $\begingroup$ What mathematically rigorous approach are you suggesting? For example, for the sum of 10 digits, if I quote 42.5 @ 47.5 and get lifted, am I quoting around $\mathbb{E}[S|S\geq 47.5]$, where $S$ is the sum of the 10 digits? This quantity does not seem easy to compute quickly, right? $\endgroup$
    – Max
    Sep 15, 2022 at 21:08
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    $\begingroup$ Certainly hard. People smarter than I am might find it easier, but approximation is your friend. The distribution is the sum of 10 iid discrete uniform distributions. Use the central limit theorem to approximate as normal. (I'd have to look up the variance of that distribution, which is apparently ~8). 8*10=80. sqrt(80)~9. So, your distribution is N(45,9). You should have a vague mental model of the normal. 1sd covers about a 2/3 range, So, p>47.5 is probably about 40-42. I picture trying to balance that tail section. Seems like it would balance at 1sd. So, 45+9 is the EV. Market is 51.5-56.5 $\endgroup$
    – kolbe
    Sep 16, 2022 at 1:22
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    $\begingroup$ I certainly know people whose math would be far more on point than mine in the industry, though. And it's a whole other level of game theory when the other person chooses numbers to be intentionally deceptive. Then you're into the game theoretic nash equilibrium (which is a distribution he'd draw from). I think that might be uniform. Bleh. $\endgroup$
    – kolbe
    Sep 16, 2022 at 1:23
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I'm also in the process of learning and will just give you my thoughts here. Might not be fully right so please correct me or enter a discussion with me.

I think it depends on the information you are given. If you know that the other party knows the true value and you have only an estimate, there is information in their trading behaviour. If he buys from you despite a decrease in current EV, you can deduce that the other values are probably higher and raise your quotes. If, on the other hand, you are both unaware of the other numbers, don't let their trading take you too far away from the EV. To me, it does not make sense to quote bids far above the EV, even not to lose inventory right? At one point you will have to think: no further than this. At his point you can set a high ask and place a bid with an unlimited amount of lots on a value equal to or below your average position. If you have to get rid of inventory, you might have to go above this at one point.

Furthermore, they might be bluffing and trying to lure you out. If they keep lifting you and you have to keep a maximum spread of 5 and you are not paying attention to the amount of lots you offer, the following could happen:

  1. 43@48 Your ask is lifted for 1 lot
  2. 48@53 Your ask is lifted for 1 lot
  3. 53@58 Your ask is lifted for 1 lot You are now short 3 lots with average value 53
  4. 58@66 Your bid is lifted for 3 lots You now no longer have a position and made a loss of 15.
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