Market Making Card Game Strategy

Question

Recently i played a market making card game during a job round of a company. Following is the description of the same -

In a game there are 5 players and 13 cards, used in total. Each card has a value. At the start of the game, each player is given a card from the set of 13 cards. There will be 4 cards face down on the table. The rest of the card will be discarded and will never come back in the game.

Trading - The sum S of all the values of the cards will be traded in the game,i.e, the sum of the values of 9 cards. This will be done in several steps.

Round 1 - Each player will know nothing but his own card, but in the subsequent rounds, cards faced down will be revealed one at a time. One of the player, say Player 1, starts by market making S. The initial market cannot be more than 20 points wide, i.e, the sell price cannot be more than higher than the buy price. All the players are free to trade. Once a trade happens, the market maker(i.e, player 1 will make a new market. This continues untill there are no trades.

Round 2 - Before the start of the round, one card on the table will be turned up. Player 2 then makes the market at most 15 points wide. As before, others can trade or improve.

Round 3-5 - In the further rounds, one card at a time will be revealed. Remaining players will start making markets. The market can be at most 10 points wide in round 3, and 5 wide in subsequent rounds.

Settlement - At the end of the game, we shall calculate the sum of all the values of the cards on the table. The sum will be settlement value.

What strategy i should apply for making most profit?

Answer 1

There are no clear cut strategies for games like this, but I think a basic framework to utilise is the following:

At all times we should track two values, the expected total of the values of the cards using public information E and the expected total values of the cards using private information as well as public information P.

I'm assuming the cards are numbered 1,2,...,13.

Before any of the table cards are turned over E = 58.5 (6.5 x 9). If, say, you hold a 10, you can adjust to obtain P = 62. If a 4 is turned over after round 1, we have E = 56, P = 59.5.

When you are making a market you should begin by centring it on E and taking a spread of your choice. When somebody trades with you, you should adjust with respect to your position i.e. if you are long, that means you are paying too much to buy, so adjust down, if you are short, you are selling for too cheap, so adjust up. How you do this is up to you.

When you are a market taker, you should buy and sell based on your value of P i.e. buy for less, sell for more. This means you are adequately using all available information. It is possible and desirable to trade off of a modified value of P, however the strategies surrounding how such modifications should be made are qualitative and similar to reading in poker. For example, if you see that another player is playing with a strategy (i.e. not just cluelessly executing trades at random), you may be able to place him on a card e.g. you think he has a 12, so you can adjust your value of P by adding 5.5 (excess of 12 above the average of 6.5)