Introduction to the Game

"High Show" is a card game where you aim to maximize your profit by selecting high-value cards from a deck. Each card you select affects your game's balance positively, while cards you reject incur a penalty.

Game Setup

• Deck: The game uses a custom deck of cards. You will specify the range of numbers that these cards represent at the start of the game.
• Selection: You will be required to select a specific number of cards during the game.

Objective

The main goal is to finish the game with the highest possible balance, which is the sum of all cards you cash out minus any penalties.

How to Play

1. Start the Game:
   • When you start the game, you will choose how many cards are in the deck, say N. Based on this, a starting value K will be generated at random. It's known that K is at most 3N. Given the starting value is K, the cards are labelled K to K+(N-1), sequentially.
   • You will also choose how many cards you need to select by the end of the game. This must be smaller than N.
   • You will choose a penalty function, which is a method that penalizes rejecting many cards. There are 5 different methods to choose between:
     • No Penalty: The penalty will be set to 0.
     • Linear Penalty: A fixed amount will be removed from your score for each card rejected. For example, if the parameter is 5, a score of 5 is removed per rejected card. Currently, the default value is the number of cards divided by 10.
     • Penalty on Turn: The penalty for rejecting the ith card that appears is i times some multiplier. These are summed up over all turns of rejection. For example, if you reject the 6th card appearing, and the multiplier parameter is set to 3, 18 would be removed from your score on this turn. The default multiplier is the 2 times the quotient of the sum of the number of cards and the lowest card number and the number of cards plus 4.
     • Penalty on Rejects: The penalty for rejecting the card with value i on it is i times some multiplier. These are summed up over all rejected cards. For example, if you reject the card with value 20, and the multiplier is set to 1/2, 10 would be removed from your score on this turn. The default multiplier is 1/4.
     • Turn and Value Penalty: The penalty for rejected the card with value i on it on the jth term is ij times some multiplier. These are summed up over all rejected cards and turns of rejection. For example, if you reject the card with value 10 on the 6th turn, and the multiplier is set to 1/6, 10 would be removed from your score on this turn. The default multiplier is 1/8.
2. Gameplay:
   • Each turn, a card is drawn randomly from the deck.
   • The value of the card is shown to you.
   • You must decide whether to "cash out" (keep) this card or to "see another" (reject and draw another card).
   • Accepting a Card: If you accept the card, it is added to your selected cards. This decreases the number of cards you still need to select.
   • Rejecting a Card: If you reject the card, it's added to a pile of discarded cards. Each rejection is recorded and may incur a penalty depending on the penalty function used.
3. Ending the Game:
   • The game continues until you have selected the required number of cards.
   • After you have selected your cards, no more cards will be drawn.
   • You'll be asked for your estimate of the starting value K and your net profit/loss.
   • After your estimates are entered, the smallest value, the sum of all your cards (gain), the penalty, and the profit/loss is revealed.