I have been interested in poker theory for a little while now, so I thought I might as well try and create something related to Poker. For now, I am specifically looking into a Texas Hold'em Poker variation, Heads-Up Hold'em. So far, I have set up the simulations and now am working towards creating a GTO poker bot.

To test out JFreeChart and how I want to run simulations, I created a couple charts:

These charts are simply simulating the probability of being dealt a certain kind of hand. For example, if we look at the Pocket Pair Probability Chart, the x-axis represents the number of simulations I ran while the y-axis represents the probability of being dealt a pocket pair. For instance, one simulation means I dealt a hand of two cards and then using some predicate (in this case, card rank equality), I kept track of whether we got a pocket pair. Then, probability is simply $\frac{\text{num pocket pairs}}{\text{num simulations}}$. This exact process was also used for the Suited Hole Cards Probability Chart.

Obviously, these are things we can prove mathematically without the use of simulations. It is, however, a pretty cool example of the central limit theorem and may be helpful to statistics students who like poker¹. Also, JFreeChart and my current structure for simulations has been working decent, so I am planning on following a similar pattern for future additions².

In this simulation, I analyzed how a certain hand plays against an unrestricted range on the flop. In this example, the small blind was dealt pocket deuces (pocket twos), and the big blind was given a random hand (with an unrestricted range). A flop consisting of three cards was randomly dealt, and the small blind and big blind hands were simply the two cards in their hand, plus the three community cards on the board. This simulation was relatively easy to set up and run because there is only one combination of possible hands, namely because $\binom{N}{k} = \binom{\text{num hole cards } + \text{ num cards on board}}{5} = \binom{2 + 3}{5} = 1$. On the turn each player has $\binom{N}{k} = \binom{2 + 4}{5} = 6$ combinations of possible hands. On the river each player has $\binom{N}{k} = \binom{2 + 5}{5} = 21$ combinations of possible hands. This makes simulating entire games where there are 21 combinations of possible hands on the river for each player quite tedious. I am still trying to figure out an efficient way to do this. However, because determining which player is winning on the flop is the simple calculation of who has the better five-player hand, this simulation ran fast and without problem. It seems like pocket deuces are winning around ~66.5% of the time.

It seems like pocket aces are winning around ~95.5% of the time on the flop against an unrestricted range (random hand). However, it is important to note that this probabiity vastly overestimates the equity³ pocket aces really have against a random hand preflop. For this we will need to simulate the entire game, not just to the flop. Conversely, draw heavy hands (e.g., suited hands, connectors) will have their preflop equities underestimated using this flop analysis method.

It will be useful to use the hand 56 suited to see how you can easily overvalue pocket pairs and undervalue draw heavy hands on the flop if you are not thinking in terms of equity. On the flop it seems like 56 suited is winning ~45% of the time against an unrestricted range.

I was able to figure out how to simulate games to the river. As mentioned before, simulating games to the river is a decent bit harder than just to the flop because we need to determine what the best 5-card hand is for each player (because there are 7 to choose from). Also, to make this nice graph and visually show the convergence, we are running this simulation $\sum_{n=0}^{995} 50 + 10n = 5004900$ times. This is because I start with $50$ simulations, then I simulate in increments of $10$ all the way to $10000$, which totals to $5004900$ simulations. On my computer this simulation takes around six minutes to fully run, which I think I can improve but for now I am just happy that it actually ran. As we can see from the graph, it seems like pocket deuces have ~51.5% equity preflop against an unrestricted range. This is significantly less than the probability that deuces are winning on the flop.

We can see a similar thing with pocket aces. From the graph it looks like pocket aces have ~85% equity preflop against an unrestricted range⁴. As we saw from the flop analysis, the probability of pocket aces winning on the flop is significantly higher.

We see the opposite with 56 suited. It seems like 56 suited has ~47% equity against an unrestricted range. Albeit small, there is a slight improvement in overall equity of 56 suited compared to its probability winning on the flop. This is because 56 suited is a draw heavy hand, which means that it has a higher chance of improving with the addition of cards (e.g., to a flush or a straight).

The code needed to create and run these equity convergence calculationns and create the graph is given below. You can run this in Main.kt and it should take ~5 minutes though you can modify this by running less or more simulations.

    val cardStrings = CardStrings()
    val simulator = PokerEquitySim()

    simulator.runProgressiveSimulation(
        startTrials = 50,
        endTrials = 10000,
        step = 10,
        card1 = cardStrings.queenDiamond,
        card2 = cardStrings.tenDiamond,
        chartTitle = "Queen-Ten Suited Equity"
    )

I set up a simulation where two poker bots play 10 million hands against each other. I set up both bots to use the check or call strategy, so all the bot does is check or call depending on what action is available (i.e., these hands always go to the river no matter what). I kept a record of the winning hand ranks for each poker game.

This data is somewhat useful if you are playing heads-up and want to know what the winning hand rank looks like on average. So, for example, the winning hand rank for heads-up is less strong than multiway poker, and roughly ~75% of hands are won with a two pair or worse. If you are playing against someone who seems to never fold, then this is a useful statistic to keep in mind.

I used a relatively simple approach to try and tackle Monte Carlo CFR.