Turn-in Checklist:
  1. Each team should bring to class a stapled turn-in containing your written and printed answers to all questions (including the code you also submitted electronically) by 10:01am Monday 14 September. Your writeup must include both partners UVA IDs (e.g., mst3k) in big block letters at the top.
  2. Use our automatic adjudication service to submit a single Scheme file (a modification of ps2.ss) containing your answers to Questions 3 - 7. You must define an author list. Each partner must separately submit a copy of your joint file. Do this before 9:50am on Monday 14 September.

Collaboration Policy - Read Carefully

For this problem set, you are required to work with an assigned partner. Parter assignments are posted here.

Before starting to work with your partner, you should go through Question 1 yourself on paper. When you meet with your partner, check if you made the same predictions. If there are any discrepancies, try to decide which is correct before using DrScheme to evaluate the expressions.

You and your partner should work together on the rest of the assignment. However, you will each turn in separate stapled documents containing your answers to the written questions. You should read the whole problem set yourself and think about the questions before beginning to work on them with your partner.

Remember to follow the pledge you read and signed at the beginning of the semester. For this assignment, you may consult any outside resources, including books, papers, web sites and people, you wish except for materials from previous cs1120, cs150, and cs200 courses. You may consult an outside person (e.g., another friend who is a CS major but is not in this class) who is not a member of the course staff, but that person cannot type anything in for you and all work must remain your own. That is, you can ask general questions such as "can you explain recursion to me?" or "how do lists work in Scheme?", but outside sources should never give you specific answers to problem set questions. If you use resources other than the class materials, lectures and course staff, explain what you used in your turn-in.

You are strongly encouraged to take advantage of the scheduled help hours and office hours for this course.


These robots are much better than the average player...
It would for sure make money online.

Phil Laak, Los Angeles Times, July 18, 2005.

In this problem set, you will develop procedures that can calculate odds for poker. Actually creating a poker bot involves making decisions based on incomplete information about what the other players have and how they behave. This is much harder than just calculating odds, but knowing the odds is important for any poker strategy.

We will consider the game Texas Hold 'Em, a popular version of poker that is used in most major tournaments. Each player is dealt two hole cards. These are kept hidden from the other players and can only be used by the player who was dealt them. Then, five community cards are dealt. Every player may use these cards. There are betting rounds after the hole cards are dealt, and after the third, fourth, and final community cards.

At the end of the hand, each remaining player makes their best five-card hand using as many of their own hole cards as they want and the community cards. So, a player may make a hand using just the five community cards (and none of their hole cards), either of their hole cards and any four of the community cards, or both of their hole cards and any three of the community cards.

To calculate the odds a player will win a hand, we need to know all possible hands that player could get and how many of them beat the other player's hand. For example, when there is one community card left to be dealt, that means we need to consider how many of the remaining cards in the deck will allow the player to make a hand that is better than the other players hand.

Reading: Before going further, you should finish reading at least through Section 5.4 from the course book.

Becoming Pros at Cons

It never hurts for potential opponents to think you're more than a little stupid
and can hardly count all the money in your hip pocket, much less hold on to it.

Amarillo Slim
Question 1: For this question you should not use the Scheme interpreter. Do not work with your partner (yet). For each fragment below, either:
  1. Explain why the fragment is not a valid Scheme expression; or,
  2. Predict what value the expression will evaluate to. If the expression evaluates to a compound data structure, draw a picture showing the structure. ("Draw the structure" means "draw a box-and-pointer diagram showing all cons pairs," like in the book.)
Your answers do not have to be correct to receive full credit, but you must clearly explain your reasoning. Assume that all the procedures from Chapter 5 are defined. Your answers do not have to be correct to receive full credit, but you must clearly explain your reasoning.
  1. (cons (cons 1 2) 3)
  2. (cons 1 (cons 2 3))
  3. (car (cons (cons 1 2) null))
  4. (car (cdr (cons (cons 1 2) null)))
  5. (list? (car (list 1 2 3)))
  6. (list? (cdr (list 1 2)))
  7. (list-length (cdr (cdr (cdr (list 1 2 3)))))
  8. (list-length (list-append (list 1 1) (list 2 0)))
  9. (list-length (list-append (list 1 2 3) 4))
  10. (list-length (list-append (list 1 2 3) null))
  11. (list-map (lambda (x) x) (list 1 2 3))
  12. (car (list-map (lambda (x) (> x 2)) (list-map (lambda (x) (+ x 3)) (list 1 2 3))))

For the remainder of this problem set, you should work with your partner. First, compare and discuss your answers on Question 1. Then, download the PS2 code so you can check your answers in the interpreter.

Download: Download ps2.zip to your machine and unzip it into your home directory J:\cs1120\ps2.

This file contains:

  • ps2.ss — A template for your answers. You should do the problem set by editing this file.
  • ch5-code.ss — Code for the list procedures from Chapter 5. You should already understand all this code from reading Chapter 5. (Yes, you should definitely have finished reading through Section 5.4 before this point!)
  • poker.ss — Provided code for this problem set. You should examine, but not modify, this file.
Question 2: After you have compared and discussed your Question 1 and 2 answers with your partner, evaluate each one in DrScheme to check your predictions. Remember to load ch5-code.ss first for the list procedure definitions.

For each fragment that you mispredicted indicate the correct answer and what you misunderstood. If the expression evaluates to a compound data structure, draw a picture showing that structure.

If you address all fragments in Question 1, and for each fragment you either predicted the correct answer (in Question 1) or explained the correct answer (in Question 2) you will receive full credit for Questions 1-2.

Representing Cards

To represent a card we need to keep track of both its rank (2 through Ace) and its suit (Hearts, Diamonds, Clubs, or Spades). We can do this using a cons cell (these definitions are in poker.ss):
   (define (make-card rank suit) (cons rank suit))
   (define card-rank car)
   (define card-suit cdr)
We use the numbers 2 through 10 to represent the normal card ranks, and 11 (Jack), 12 (Queen), 13 (King), and 14 (Ace) to represent the special cards. The definitions in poker.ss allow us to use the names Ace, King, Queen, and Jack for these ranks.

Question 3: Define a procedure higher-card? that takes two cards as operands and evaluates to #t if and only if the first card has a higher rank than the second card. Note that the card suit does not matter in this comparison.

If your higher-card? procedure is correct, you should get the following interactions:

> (higher-card? (make-card Ace Diamonds) (make-card King Spades))


> (higher-card? (make-card 3 Clubs) (make-card 2 Clubs))


> (higher-card? (make-card Ace Diamonds) (make-card Ace Spades))


Sorting Hands

In order to evaluate a hand, it will be useful to sort the hand according to the card values. In addition to the list procedures introducted in Chapter 5, the built-in sort procedure will be useful for sorting hands:
sort: List x Procedure → List
sort takes as input a List and a comparison Procedure, and evaluates to a List that contains the elements in the input List reordered according to the comparison Procedure.

For sort to make sense, the input Procedure must evaluate to a transitive comparison procedure. This means if (Procedure x y) evaluates to true and (Procedure y z) evaluates to true, then (Procedure x z) must also evaluate to true. Then, the sorted list will have the least element (according to Procedure) at the front.

Read poker.ss and look at the functions we have already defined before starting Question 4.

Question 4: Define a procedure sort-hand that takes one operand that is a list of cards, and evaluates to the cards sorted by decreasing card value (the highest card should be first in the list).

If your sort-hand procedure is correct, you should get the following interactions (the display-cards procedure provided in poker.ss makes it easier to see the cards; the sample hands are also defined in poker.ss):

> (display-cards (sort-hand royal-flush))

"Ah Kh Qh Jh 10h"

> (display-cards (sort-hand ace-higher))

"Ad 10s 8c 7c 3s"

Hint: you should not need more than one line for your definition.

Since the rankings of poker hands don't just depend on the card values, but on having pairs, triples, and quads of a given card, it will be more useful to sort the cards according to first the number of duplicates of each rank, and then by rank. For example, if the hand is Ac Js Jc 7d 4c the procedure sort-by-ranks should produce ((Js Jc) (Ac) (7d) (4c)) since the pair of Jacks are more important than the single A. Note that instead of being just a list of cards, the result is now a list of lists of cards.

The procedure sort-by-ranks is defined:

(define (sort-by-ranks cards)
  ;;; sorts cards into lists of cards of each rank, ordered by most
  ;;; cards and highest cards within group
  (sort (combine-adjacent-matches  ;; combine them into lists of matching rank
	 (sort-hand cards))        ;; cards sorted by rank
        (lambda	(r1 r2) 
          (if (= (length r1) (length r2))
              (higher-card? (car r1) (car r2))
	    (> (length r1) (length r2)))))

Ranking Hands

There are few things that are so unpardonably neglected in our country as poker. The upper class knows very little about it. Now and then you find ambassadors who have sort of a general knowledge of the game, but the ignorance of the people is fearful. Why, I have known clergymen, good men, kind-hearted, liberal, sincere, and all that, who did not know the meaning of a flush. It is enough to make one ashamed of the species.
Mark Twain
There are two ways that a hand could beat other hands:
  1. It is in a better category of hands (e.g., a flush beats a straight)
  2. It is in the same category of hands, but with higher cards
The ranking of the hand categories is shown next.

Straight FlushFive cards in sequence all of the same suitKs Qs Js 10s 9s
Four-of-a-Kind ("Quads")Four cards of the same rank3h 3d 3c 3s Jd
Full HouseThree cards of matching rank and two different cards of matching rankAc As Ah 7d 7h
FlushFive cards of the same suitQh 10h 8h 3h 2h
StraightFive cards in sequence9d 8h 7c 6h 5s
Wheel StraightAce-2-3-4-5 straight5d 4h 3c 2h As
Three-of-a-Kind ("Trips")Three cards of matching rank5d 5h 5c Kh Js
Two PairTwo different pairs of matching rankJd Jh 5c 5h As
One PairTwo cards of matching rank8d 8h Ac 7h 3s
High CardAnything elseKd 9h 7c 5h 2s

(Note: we've listed wheel straight as a separate hand category, even though it is usually listed just as a straight. It is the only straight that can use A as a low card, and the lowest possible straight.)

We have provided the higher-hand? procedure that defines the poker rules:

(define (higher-hand? hand1 hand2)
    ((straight-flush? hand1)     (or (not (straight-flush? hand2)) 
                                     (and (straight-flush? hand2) 
					  (higher-similar-hand? hand1 hand2))))
    ((four-of-a-kind? hand1)     (or (and (not (straight-flush? hand2)) 
					  (not (four-of-a-kind? hand2)))
                                     (and (four-of-a-kind? hand2)
					  (higher-similar-hand? hand1 hand2))))
    ((full-house? hand1)         (or (and (not (straight-flush? hand2)) 
					  (not (four-of-a-kind? hand2)) 
					  (not (full-house? hand2)))
                                     (and (full-house? hand2) 
					  (higher-similar-hand? hand1 hand2))))
    ((flush? hand1)              (and (not (beats-flush? hand2))
                                      (or (and (flush? hand2) 
					       (higher-similar-hand? hand1 hand2))
                                          (not (flush? hand2)))))
    ((wheel-straight? hand1)     (not (or (any-straight? hand2) (beats-straight? hand2))))
    ((three-of-a-kind? hand1)    (and (not (beats-trips? hand2))
                                      (or (and (three-of-a-kind? hand2)
					       (higher-similar-hand? hand1 hand2))
                                          (not (three-of-a-kind? hand2)))))
    ((three-of-a-kind? hand1)    (and (not (beats-trips? hand2))
                                      (or (and (three-of-a-kind? hand2) 
					       (higher-similar-hand? hand1 hand2))
                                          (not (three-of-a-kind? hand2)))))
    ((two-pair? hand1)           (and (not (beats-two-pair? hand2))
                                      (or (and (two-pair? hand2) 
					       (higher-similar-hand? hand1 hand2))
                                          (not (two-pair? hand2)))))
    ((one-pair? hand1)           (and (not (beats-pair? hand2))
                                      (or (and (one-pair? hand2) 
					       (higher-similar-hand? hand1 hand2))
                                          (not (one-pair? hand2)))))
    (#t                          (and (not (beats-high-card? hand2))
                                      (higher-similar-hand? hand1 hand2)))))
Your job is to define the higher-similar-hand? procedure it uses to compare two hands in the same category.

The rules for comparing poker hands of the same category specify that the most important part of the hand should be compared first. The most important part is the highest card with the highest number of duplicates. If the most important parts are equal, than the next most important part of the hand determines the higher hand. Note that the sort-by-ranks procedure we defined sorts the cards in a hand according to importance, so you can determine the higher hand by considering each element of the list produced by sort-by-ranks in order until you find one that is unequal.

Here are a few examples:

Question 5: Define the higher-similar-hand? procedure, completing the template we have provided in ps2.ss. Notably, you should change word REPLACE to some valid scheme code that does the job. It should take two poker hands as operands, and can assume the hands are of the same category. It should evaluate to #t if and only if the first hand beats the second hand.

If your procedure is correct, you should get the following interactions:

> (higher-hand? pair-jacks pair-kings)


> (higher-hand? pair-kings pair-jacks)


> (higher-hand? queens-up queens-up)


> (higher-hand? ace-high ace-higher)


> (higher-hand? ace-higher ace-high)


> (higher-hand? kings-full-of-aces kings-full-of-jacks)


Finding the Monster

To find the best possible hand, we will need to consider all possibilities and use the higher-hand? procedure to identify the best hand among them. There are five community cards and two hole cards, so a player can make a hand by using both hole cards and choosing any three of the community cards, by choosing one of the hole cards and any four of the community cards, or by just using the five community cards.

To find possible hands, you will find this procedure (defined in poker.ss) useful:

(define (choose-n n lst)
  ;; parameters: a number n and a list (of at least n elements)
  ;; result: evaluates to a list of all possible was of choosing n elements from lst
  (if (= n 0) 
      (list null) 
      (if (= (length lst) n) 
          (list lst) ; must use all elements
	   (choose-n n (cdr lst)) ;; all possibilities not using the first element
	   (map (lambda (clst) (cons (car lst) clst)) 
		(choose-n (- n 1) ;;; all possibilities using the first element
			  (cdr lst))))))) 
Question 6: Define the possible-hands procedure, completing the template we have provided in ps2.ss. It should take two operands: the first is a list of two cards representing a player's hole cards; the second is a list of five cards representing the community cards. It should evaluate to a list of all possible hands that could be made using 0, 1, or 2 of the player's hole cards and enough of the community cards to make a five-card hand.

Note that your possible-hands procedure doesn't depend on the elements of the operands being cards. You may find it easier to test by using scalar values instead. For example, (possible-hands (list 1 2) (list 'a 'b 'c 'd 'e)) should evaluate to a list containing these elements (in any order):

((a b c d e) 
 (1 b c d e) (1 a c d e) (1 a b d e) (1 a b c e) (1 a b c d)
 (2 b c d e) (2 a c d e) (2 a b d e) (2 a b c e) (2 a b c d)
 (1 2 c d e) (1 2 b d e) (1 2 b c e) (1 2 b c d) (1 2 a d e)
 (1 2 a c e) (1 2 a c d) (1 2 a b e) (1 2 a b d) (1 2 a b c))

Now that we have a list of all possible hands, we can find the best hand using the higher-hand? procedure.

Question 7: Define the find-best-hand procedure, completing the template we have provided in ps2.ss. It should take two operands: the first is a list of two cards representing a player's hole cards; the second is a list of 5 cards representing the community cards. It should evaluate to the best possible hand that could be made by the player using 0, 1, or 2 of the player's hole cards and enough of the community cards to make a five-card hand. (If there is more than one equally good best possible hand, your procedure can output any one of those hands.)

If your procedure is correct, you should get the following interactions:

> (display-cards (find-best-hand aces-in-hole trip-nines))

"9d 9c 9s Ah Ad"

> (display-cards (find-best-hand big-slick community-clubs5))

"Ac Qc Jc 7c 5c"

> (display-cards (find-best-hand big-slick royal-flush))

"Ah Kh Qh Jh 10h"

For the first hand, both hole cards are used with the three nines to make a full house. For the second hand, the Ace of clubs hole card is used with the four club community cards to make a flush. For the third hand, no hole cards are used and the five community cards make a straight flush.

Hint: as in Question 4, you should be able to define this using only one line. Don't worry about efficiency in your definition (for now).

After the Turn

After the fourth community card (known as the turn or fourth street) has been dealt, there is one more community card to come (known as the river or fifth street). To make good decisions, a player needs to know the likelihood that she will have the winning hand after the final card is dealt, but doesn't know what the final card will be.

For simplicity, we will assume the player is playing against only one opponent, and has good enough card-reading skills to know the other player's exact hand. (Of course, no real poker player is that good, but in many cases players do have a reasonable guess what cards the other players are holding.)

To analyze a hand, we determine how many of the possible final cards would allow the player to win or draw. The analyze-turn-situation procedure is defined below (and in poker.ss):

(define (accumulate-outs lst)
  ; lst is a list of triples representing  cards
  (if (null? lst) (list null null null)
      (let ((rest-outs (accumulate-outs (cdr lst))))
        (list (append (car (car lst)) 
		      (car rest-outs))
              (append (car (cdr (car lst))) 
		      (car (cdr rest-outs)))
              (append (car (cdr (cdr (car lst))))
		      (car (cdr (cdr rest-outs))))))))
(define (analyze-turn-situation hole1 hole2 community)
  ;; remove all known cards from the deck
  (let ((current-deck (remove-cards (append hole1 hole2 community) 
    ;; we want to find out how many of the remaining cards produce each
     (map (lambda (final-card) 
            (let ((outcome 
		    (find-best-hand hole1 
				    (cons final-card community))
		    (find-best-hand hole2 
				    (cons final-card community)))))
              (if (eq? outcome 'higher)
                  (list (list final-card) null null)
                  (if (eq? outcome 'equal)
                      (list null (list final-card) null) ; chop
                      (list null null (list final-card))))))
The analyze-turn-situation procedure determines the cards left in the deck by removing the known hole cards and community cards as current-deck. Then, it uses map to try each possible final card from the current deck. If the final card would allow player 1 to produce a better hand than player 2, it puts that card as the first element in the list; if the hands would be equal, it puts that card as the second element; if player 2 would win, it puts that card as the third element. The accumulate-outs procedure combines all the sublists to form a list of all the winning and chopping outs.

Here's an example:

> (show-analysis (analyze-turn-situation connect67 aces-in-hole straight-draw4))

Winning outs (15): 2c 3c 5h 5d 5c 5s 9c 10h 10d 10c 10s Jc Qc Kc Ac
Chopping outs (0):
Losers (29): 2h 2d 2s 3h 3d 3s 4h 4d 4s 6h 6d 6s 7h 7d 7s 8h 8d 8s 9h 9d
Jd Js Qh Qd Qs Kh Kd Ks As three-clubs))

To consider the situation after the flop (the first three community cards have been dealt), we need to look at all possibilities for both the fourth and fifth card. The analyze-flop-situation does this:
(define (analyze-flop-situation hole1 hole2 community)
  ;; operands: hole cards for player 1 and play 2 and community cards
  ;;     there must be 2 cards in each players hole cards and 3 community cards
  ;; result: a list of three elements (winning-outs, chopping-outs, loser) showing
  ;;     the turn and river cards that will lead for the each outcome for player 1.
  (let ((current-deck (remove-cards (append hole1 hole2 community) full-deck)))
    ;; we want to find out how many of the remaining cards produce each
    (map (lambda (turn-card) 
           (analyze-turn-situation hole1 hole2 (cons turn-card
The show-flop-analysis procedure displays a flop analysis in a (barely) human-readable way.

Question 8: Predict how long it will take to evaluate an application of analyze-flop-situation (for example, (analyze-flop-situation connect67 aces-in-hole straight-draw3)). You should start by timing analyze-turn-situation. You can do this using the time special form:
   (time (show-analysis (analyze-turn-situation connect67 aces-in-hole straight-draw4)))
This will print out something like
   cpu time: 3523 real time: 4823 gc time: 203
(the actual numbers here are made up for the example). The number after real time gives the number of milliseconds it took to do the evaluation (in this case 4.823 seconds). If you want to try evaluation and timing analyze-flop-situation you can, but you should predict how long it will take first.

Hint 1: Call and example display-analysis and display-flop-analysis to get a better idea of what is going on.

Hint 2: Phrase your answer in generic terms such as "If analyze-turn-situation takes X seconds, analyze-flop-sitatuion will take ...".

Question 9: This implementation is obviously too slow for many practical uses, including building a poker bot. Suggest some approaches you would use if you wanted to make a faster implementation. (It is not necessary to actually implement your improvements, although especially impressive implementations will receive extra credit.)

Automatic Adjudication: Submit a single Scheme Definition file containing all your code until you are satisfied with your test results. Your scheme file should be a modification of the ps2.ss file. Each partner must submit the file separately.

For More Information

The Loki poker-playing algorithm is described in Using Probabilistic Knowledge and Simulation to Player Poker by Darse Billings, Lourdes Pena, Jonathan Schaeffer, Duane Szafron in AAAI 1999.

Credits: This problem set was originally developed for UVA CS200 Fall 2005 by David Evans
and tested and improved by Dan Upton and Westley Weimer.