Analysis of optimal strategies and player behavior in card games
Advisor: Ms. Julia Wendel
Internship Location: Baruch College
Onsite Mentor: Dr. Richard Gottesman, Adjunct Assistant Professor of Mathematics at Baruch College
I will first do analysis on the strategies of the card games using combinatorics and probabilities. This part will be entirely remote and done theoretically. I will then do analysis comparing these strategies to actual player behavior in game. No change has been made since the proposal presentation.
Project Blog
Week 0: Introduction
This is an introduction to my senior project, "Analysis of optimal strategies and player behavior in card games". Not much research has been done, so I will explain the context of my project as well as the plan for the next few months.
Firstly, the motivation. Over summer camps, I learned a lot of different card games, and I especially like "Fish" and "Cambio". I introduced it to my grade and the class of 2024, and everyone is getting addicted to it. However, as we play more of it, I felt frustration over how much the game seems to depend on luck, instead of memory or skill. Therefore, I will be doing an in-depth analysis of whether "optimal strategies" exist in these 2 card games, and whether players are following them.
Then, the games themselves. You can hopefully find the specific rules of these games on Yourui's blog, but I will post the link anyways: http://cambiocardgame.com/ for Cambio and http://www.pagat.com/quartet/literature.html for Fish. (Yes, Fish has an alternate name called "Literature").
Prior to this project, not much research has been done on these 2 card games specifically due to its low popularity. Therefore, I will be relying on their methodology and my own understanding of game theory, combinatorics, and these card games.
For Cambio, the plan for now is diving into deciding when a player should call cambio, whether or not a player should draw a card, and whether the player should keep or play the card. This will be a lot of casework, and it will depend on the number of players, the additional rules added, and others. I will start simple, from the 4-card golf version of this game, and expand on the number of players, sticking cards, as well as card powers.
For Fish, I will split it into when "trolling", or asking for a card one already owns, is allowed or not in the rules, and do analysis on how much information each question communicates and how much utility it gains. I will use game theory and combinatorics to determine the optimal strategies given perfect information.
Week 1: Cambio
Overall, this week has just been attempting to progress through my analysis of Cambio.
I first tried to continue my old approach of analyzing optimal decisions at each point of the game given all available information, but I realized that this is dependent on everything that has happened in the game, and is impossible to generalize or be viewed as a "strategy". Chess technically has an "optimal strategy" where you simply play like Stockfish, but we don't really view it as an optimal strategy in the sense that it is too long and dependent on memorization and even exhaustive search. On the other hand, if I were to continue my previous approach, the end result would not only depend on the cards you have, you know, but also depend on cards other players have, other players played or discarded, as well as kept. It would be as complicated as "Solving" 5D chess.
Therefore, I attempted to work on the game directly, and simplify the decisions I need to make or analyze.
To summarize, here are the list of observations I made that turned this project a lot more manageable.
Instead of analyzing what happens when you draw a card, it is better to simply analyze decisions you should make after knowing the card you draw. Specifically, when you draw a 7 or 8 (cards that allow you to look at one of your own cards), when you draw a 9 or 10 (cards that allow you to look at someone else's cards), and when you draw a J or Q (cards that allow you to make blind swaps). The rest of the cards don't need to be discussed because either they have no special power or there is no practical incentive to keep them.
The only strategic decision to make in this game is when you call Cambio. If you never call unless you have a guaranteed win, the game becomes entirely luck based, and whoever draws the red kings win. On the other hand, though, if you make the right calls to call Cambio, especially early in the game, even though you don't know your entire hand, you might be able to improve your winning chances.
There is no correlation between players' actions and their cards. For example, simply because a player discards a 2 they already know doesn't mean the card they got has a value lower than a 2. They might decide that it is more likely for them to get rid of a certain card in a shorter period of time than the 2. Players might also just do random actions (cough cough Rohan, Antony, and a few others I won't mention) instead of going with conventions. This includes swapping 2 unknown cards with a J or Q, intentionally sticking his own cards that he doesn't know, etc.
Who gets to stick a card in the case where multiple players know the same card is completely based on reaction, and after doing some analysis I realized that if we assume perfect reaction and memory, it is almost impossible for someone who is slower in general to win the game, when playing against a faster player. The reason is that the faster player can simply use Rohan's strategy of missticking every card they have, and repeating. Therefore, I decided to use the following order of priority when sticking a card:
The player who played or discarded the card gets priority over everyone else
If that is not the case, then the player the card belongs to gets priority
The only remaining case would be multiple players playing a 9, 10, or black K and look at the same card the owner doesn't know. For the sake of simplicity, I will ignore this case and assume that players play optimally and always look at the player who went right before them's cards.
I also started my analysis, computations, and writing for the case of drawing a 7 or 8.
Also HUGE thanks to Jonathan Du's Cambio website: https://cambio-online.vercel.app/. This really helped me proceed on the analysis by letting me play as multiple players in Cambio and simulating a random deck.
Week 2: Cambio
Overall, this is another week of just trying to analyze the game. Some progress is made, and here is the summary:
-I finished the analysis for drawing a 7 or 8, a 9 or 10, and I am almost done with the analysis for drawing a J or Q. To summarize my findings, assuming the game goes long enough and the goal is to stick the card you keep/look at, it is almost always optimal to keep a 9 or 10, it is sometimes optimal to play a 7 or 8, depending on the number of cards you know and the specific cards that are gone. I am approaching the conclusion that it is always optimal to keep a J or a Q as long as there are cards you don't know, and playing the J or Q to swap 2 other players' cards if otherwise currently.
-There isn't much detail I can go into, since a lot of the findings are reached through repeatedly playing against myself in multiple-player Cambio. I will describe the process used:
I am essentially using Monte Carlo simulations, except I can't go through every possible orientation of the deck. Therefore, I will hold the discard pile constant, shuffle the rest of the deck, and play with cards that I know face up, the rest face down. I play a round, assigning different strategies to different players, and see who gets rid of their cards more often or in a shorter amount of time.
This process is extremely time consuming, and feel free to ask me for any other details (I don't think there is any). This is where most of my time went.
Week 3: Cambio
Overall, this week is just another week of trying to analyze the game. I finished up the analysis for when to call Cambio, and did A LOT of playing against myself.
To summarize my findings, when to call Cambio heavily depends on your hand, as well as the total number of players and the number of cards each player has. We will split into cases where there are 2-4 players in total or more. The reason for setting the bar at 4 players is that beyond 4 players, it is very likely for at least 1 player to get rid of at least 1 card after you call cambio, and potentially more.
The first case would be when there are 5 or more players.
If at least 1 player already knows all their cards, in other words, if it is the 2nd round or later (or 3rd if you go first), never call Cambio unless you have a guaranteed win. If you are at 0 cards, and both red kings are unknown, do not call cambio. Instead, misstick someone intentionally and draw a card, hoping it is a red king. The reason for this is that in over 70% of the games I played against myself, players who call with 0 points will lose to someone with -1 points from a red king. There are simply too many opportunities for players to stick their cards, and it is hard to still be at 3 or 4 cards by turn 3.
On the other hand, if it is the first turn, only call Cambio if the cards you know have a combined score of 0 or less. This limits the combination to joker joker, joker red king, red king red king, and red king ace. This is because in over 60% of the games I played against myself, even if the player who goes first gets red king joker and calls Cambio turn 1, at least 1 other player is able to get rid of both cards they know, and then it kind of becomes a coinflip as to who wins, except the first player has an disadvantage since the other player knows 1 extra card.
The second case is for 4 players or less.
A relatively simple and strong rule for calling Cambio would be assuming every unknown card on your hand to have a point value of 6, every card your opponents have but don't know to have a point value of 6, 5.5, and 5, and every card your opponents know to have a point value of 5, 4.5, and 4, in 2, 3, and 4 player games respectively. If the point value of your hand is currently less than (or equal to, if you feel like gambling) all other players' hands, call Cambio. Otherwise, keep drawing and playing cards.
The reason for this rule is that you should account for your opponents' cards getting stuck or replaced the turn after you call Cambio. The larger the number of players, the more likely the card gets stuck, and thus, the lower the point value of the card. This is fairly accurately estimates the total point value of your opponent's hand, assuming none of the players have insane luck (start with both red kings or something, I don't know). But even in the case of insane luck, there is not much you can do against it.
This should conclude analysis to Cambio strategies, and I will finish writing up the strategy guide on Overleaf this week.
I also discussed the potential for more on-site experiments in Ms. Wendel's (my faculty advisor) classes in Trimester 3 for both Cambio and Fish, after I am close to finishing the analysis for both. The idea would be splitting students into 2 groups, teaching them the game, and giving only one group my strategies. Then, when the groups compete against each other in Cambio or Fish, we can observe which group performs better, to see if my strategies are understandable enough, and whether it works or not.
Week 4: Fish
I am still writing up the guide to Cambio. Trying to format the guide in the way comprehensible to someone who is new to the game is a long process, and it is even harder to put in the reasoning behind my arguments in the middle. Regardless, I will work on this every week in the future, and I started my analysis on Fish.
Fish (Literature), unlike Cambio, is in some way, simpler to analyze because the number of players and cards is fixed. Every game utilizes 6 players and the entire deck of 54 cards, and there aren't that much variability in terms of the rules. However, it is also a lot harder to analyze due to the large number of cases for the number of cards and the specific cards an opponent might have, as well as the existence of almost infinitely many possible strategies.
There are a lot of versions of Fish I did not include in my analysis, either because they change the game entirely and add too much work, or because they are trivial to analyze and don't affect the game at all. Examples include changing the game to 8 players, calling stalemates, challenges, different point values for high and low sets, and a certain number of trolls allowed.
Overall, my project will be focused on analyzing 6 player fish with trolling, and 6 player fish without trolling. These are completely different games, and the one with trolling will be more oriented towards the exchange of cards and information between 2 players only, mostly because the interpretation of what cards your teammates have and whether they trolled or not heavily depends on the observation of details in person, or just conventions.
I started the analysis to Cambio with trolling by making a few key assumptions. They are not 100% accurate or realisitc, but the game is impossible to approach without them.
Firstly, instead of strictly following that every player starts with 9 cards, I assume that every card is distributed independently. This doesn't change the probability distribution of the initial configuration of cards much, because of the central limit theorem, but simplifies down calculations dramatically. Now I can write a simple c++ program to count the total number of possible initial configurations of cards in a set given your hand, making the calculations a lot easier.
Secondly, I assume that players troll with a fixed probability p_i, which only depends on the number of cards in the set they have. This is not accurate, and in real life, this depends on their opponents, their cards, and other factors, but I can't factor that into my analysis. Although this adds 4 variables into my analysis, I have to cope with it. Moreover, I assume that players ask back for your card with a fixed probability P, assuming you asked for a card they previously did not ask for.
Finally, to simplify the payoffs, I modeled every possible configuration with a function E(a,b,c), but I am still trying to decide if I can somehow reduce the number of variables. E(a,b,c)=1 means you successfully gets all cards from the set from your opponent, whereas E(a,b,c)=1 means you lose all cards from the set to your opponent. Any value in the middle represents the probability for which you get the entire set, assuming neither of you divert from the set. This simiplifies the game by making the game symmetric, and making the recursion easier to compute.
I also used a lot of conditional probability in my calculations, which I will hopefully not have to show. If you don't understand any of it, just ask :)
Week 5: Fish
Overall, this is another week of just doing analysis. After doing a lot more calculations using conditional probability and expected values, I reached a few conclusions that are important but not really useful. All of them are statements of your optimal strategies at very specific situations given the frequencies at which the opponent trolls. However, in real games, it is not useful for a few reasons. Firstly, it is almost impossible to determine how often the opponent trolls, unless you collect data of the entire game. Secondly, it is very unlikely that the opponent has a fixed probability of trolling given the number of cards they have. Thirdly, if the opponent also has full access to this strategy and know that you strictly follow it, then they can choose a certain combination of frequencies to troll and to not troll to ensure that each action you take have a 50% chance of succeeding.
A major portion of my time has been spent on trying to make the strategy more understandable and applicable to real games, for example by making reasonable estimations or assumptions. So far, the one that simplifies the strategy the most is assuming that players troll with a certain percentage that is not related to the number of cards they currently have, then the optimal strategy simply becomes asking for a random card if the rate at which they troll is below a certain threshold, asking for the card they just asked for if the troll rate is above the threshold, and a mixed strategy if their troll rate is at the threshold or indeterminant.
I also started with the calculations on what you should do when you are starting the game or starting a new set by computing the expected utility of asking a question given the number of cards in the set you have. It is slightly more complicated and computationally intensive.
Week 6: Fish
After a whole bunch of calculations, I realized that my recursion formula had been completely wrong. This is mostly due to the fact that changing perspectives in a game that allows for trolling is hard. For example, after you lose a "50-50", you may think that the opponent will take the entire set from you guaranteed, but the opponent does not know that. They don't know how many cards you have, as well as whether you trolled or not.
Therefore, I added a 4th variable in the formula for Expected payoff. Currently, E(a,b,c,d) means you have asked your opponent a questions from this set, your opponent has asked you b questions from this set, c is the number of cards you know your opponent does not have for sure, either because you asked them or because you own those cards, and d is the number of cards your opponent actually has in the set.
The advantage to using a 4th variable that represents the number of cards your opponent actually has is it makes the recursion a relatively simple equation, where you sum all the expected values of what can happen next based on the currrent 4 variables.
Moreover, this fixes the issue of the model potentially having the opponent repeatedly thinking you trolled, and thus asking you for the same card 2 rounds in a row (this happens with probability P^2, if you ask them for a card they previously asked for). This also differentiates the expected values of you before asking a question and you right after trolling on a question.
Week 7: Fish
I finished revising the calculations for when you start with 6 cards, 5 cards, or 4 cards from a set. This has been so much work and pain. Now I think about it, looking at the results, there are a few things I could've simplified: using recursion to calculate the expected values, especially when 5 or more cards from the set is split between your opponent and you; doing more computations and shortening expressions using the fact that the game is symmetric and has a constant sum, etc.
One problem I encountered is that using our current model, the opponent does not adjust their strategies based on your trolling frequencies, and does not change their own strategies based on how often you think they troll. This sometimes leads to very obvious dominant strategies of to troll versus not to troll, trivializing the entire project. So, I thought, instead of maybe computing the amount of information given in Fish without trolling, I should do another analysis on Fish, but both players play "optimally".
In fact, there is a clear dominant strategy in Fish without trolling, which is already summarized by experienced players: always work on the same set as your teammates, if a card from the set has been asked and you don't have the card, ask a player on the other team for that card. If otherwise, start a new set. I will skip the proof in the blog, but it isn't long. In fact, I already have examples of why it works: my friend won the Fish tournament with this precise strategy and good memory.
Week 8: Fish
This week I had an epiphany. Instead of wondering what a logical opponent would do, or what variables I should add to my recursion, I should just assume that my strategy is optimal, and both sides would follow it perfectly. In other words, I will hold the assumptions constant, and try to minimize or maximize expected payoffs from both sides. The difference is that no matter what you do, the opponent will follow the exact same strategy, or in other words, troll, ask randomly, and ask for a card already asked for, with the same probability as you for every situation.
The reason this works is the symmetric properties of the game. If both players play optimally, then they should have the exact same optimal strategy, assuming it is unique. Combining this with the fact that it is sort of a zero-sum game, a mixed strategy nash equilibrium must exist.
From here, I continued working on the computations of the version where the opponent follows a fixed strategy, and I am almost done with the case where you start with 3 cards of the set.
Week 9: Fish
After a series of more algebra bash, I have realized that it is practically impossible to minimize or maximize a fifth degree expression that contains 4 variables using lagrange multipliers.
Though, I have concluded that at least 2 Nash Equilibria must exist in this game of one set fish with trolling. One where neither players troll at all, and ignore the other player's question, and another one where both players follow the exact same mixed strategy.
On top of that, I tried computing the actual expected values using a C++ program, plugging in actual values for the different constants we assumed (using actual frequencies in games), and obtained expected values that are overall lower with a less number of initial cards, and higher when the opponent has less cards. This fits with my expectations, as you are more likely to get the entire set when you have more cards from the set.
One exception to the trend would be when the 2 players have the entire set, in which case it is almost guaranteed for one player to take it, but other than that, the trend seems fairly consistent. Moreover, after running the recursion with different initial trolling probabilities, the strategies don't change that much, except for when a player trolls 100% of the time, but that does not make sense anyways.
With this, I concluded my research, and started formulating my thoughts and findings into a final paper.
Week 10: Conclusions
To conclude my project, I will reflect back on my past progress and see what I could've improved on.
Overall, the Cambio part of the project went very smoothly. I did simple probabilities and card counting that would help decision making, and concluded the project with an easily understood strategy guide. Although the results of the project weren't that helpful in terms of actual gameplay, the process was certainly insightful. One thing I could've done is better approximate the lowest possible points from an opponent. For example, when rolling a die, the expected minimum value is 3.5. When rolling 2 dice, the expected minimum value drops to 2.527, and with 3 dice, it further drops to 2.04. Right now my model predicts linearly for this decrease, which causes problems when it predicts that an opponent has a hand value lower than -2.
Then, the part of Fish without trolling. This part was briefly touched on in the project, due to the existence of an obvious strategy that dominates others. Plus, this strategy is more or less already prevalent in more experienced players, leaving the outcome of the game to memory and luck. I guess something I could've done is not asking my friends for suggestions or existing strategies, and try to come up with them on my own, whether through simple analysis or logical reasoning, to see how intuitive they are and how hard it is to come up with them independently, but it would not change my results.
Finally, the part of Fish with trolling. This part was the hardest, and also took the most time and effort. If I started with the parts of the calculation that were harder to perform, or if I estimated how much work my original plan would've required, a lot of the tragedy that happened near the end could have been avoided. Moreover, I could've saved a lot more time for in-person data collection as well as testing out my strategies. So far, the only Fish games I've played after my project started were to test out Yourui's bot, but it did not go as expected.
There were also a lot more minor details that could be improved, whether it is submitting blogs on time, being more productive during the meetings with advisors, etc.