Card Poker
Poker, card game, played in various forms throughout the world, in which a player must call (i.e., match) the bet, raise (i.e., increase) the bet, or concede (i.e., fold).Its popularity is greatest in North America, where it originated.It is played in private homes, in poker clubs, in casinos, and over the Internet.Poker has been called the national card. Basic Poker game where you are given 5 cards, you select which ones to discard and you are given new cards. A payout table determines your winnings. I hope you enjoy the game.
Welcome to my poker page! Here you can play one-card pokeragainst the computer. Of course, the bets are not for realmoney.Here are the rules of the game: you and the computer each get one cardand ante $1. You bet first, either $0 or $1. Then the computergets a chance to match you (if you bet $1) or raise you (if you bet$0). If you bet $0 and the computer raised you, you get a chanceto call. Betting $0 when your opponent has already bet $1 meansyou fold and lose your ante. If no one folds before the end ofbetting, the higher card wins the pot; that results in a net gain ofeither $1 or $2, equal to the other player's ante plus the bet of $0or $1.
One-card poker is a simple game; nonetheless it has many of theelements of more complicated games, including incomplete information,chance events, and multiple stages. And, optimal play requiresbehaviors like randomization and bluffing. The biggest strategicdifference between one-card poker and larger variants such as draw,stud, or hold-em is the idea of hand potential: while 45679 and 24679are almost equally strong hands in a showdown (they are both 9-high),holding 45679 early in the game is much more valuable becausereplacing the 9 with either a 3 or an 8 turns it into astraight. In other words, hands need to be evaulated accordingto both their current value and the possibility that they might becomemore valuable.
Even the simple game of one-card poker is difficult to solveoptimally: the first efficient solution that I know of is presented in[Kollerand Pfeffer, 1995]. That paper applies the sequencerepresentation of extensive form games to one-card poker. Beforethe invention ofthe sequence form, the standard algorithm was to convert the game toits normal form, which is exponentially large in the size of thedeck.
For this page, we are using a single-suit (13-card) deck. Withthis deck size, the normal form has 2^26 (about 67 million) purestrategies per player. By contrast, the sequence form has only26 information states and 52 sequences per player. Real-worldpoker variants are much larger, with many more information states thancould possibly fit in memory. Even so, modern techniques(which include trickslike grouping together sets of similar hands and ignoring some of thecoupling between very early and very late rounds of betting) can nowfind approximately-optimal solutions for games like two-player Texashold-em.
If you play optimally, you should be able to keep your losses down toabout 6.4 cents per deal on average. To limit your performanceto this level, the computerized second player plays according to thebetting tables below. By contrast, if the computer decidedwhether to bet by flipping a coin without looking at its card, youcould win up to 50 cents per deal on average. These tables werecomputed in Matlab by solving a small linear program generated fromthe sequence form of the game tree. (Getthe source.) The tables are not unique; theanswer we compute depends on the details of the linear programmingsolver we use. (SamGanzfried pointsout, interestingly, that there are other equilibrium strategies thatweaklydominate this one (perform better against some opponents withoutperforming worse against any): e.g., in the tables we calculated,player 1 never bets first holding a 5-8. So when we are decidingwhether to call a bet, our hands 5-8 are completelyequivalent—the opponent must have something lower than a 5 orhigher than an 8. Because of this equivalence, the LP solverchose arbitrarily how to distribute the bets we make on thesehands. But we can take any bets we'd make on a low hand like a 5and move them to happen on a higher hand like an 8; doing so doesn'tchange our performance against this particular opponent, but strictlyimproves our performance against any opponent who happens to bet firstholding 5-8. In particular, since we can only improve ourpayoff, the new strategy is still an equilibrium; and since there aresome opponents where we actually do improve our payoff, the newstrategy weakly dominates the current one.)
Here is the betting table for player 2. To use it, look upplayer 2's card in the column headers. Then choose a rowaccording to whether player 1 bet or passed. The correspondingentry says how often to bet: for example, 0.632 means bet 63.2% of thetime.
| Holding: | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | T | J | Q | K | A |
| On pass: | 1.000 | 1.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
| On bet: | 0.000 | 0.000 | 0.000 | 0.251 | 0.408 | 0.583 | 0.759 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
In case you're interested, here is the analogous betting table forplayer 1. You can play optimally using just this table and a1000-sided die. The first row says how often to bet in thefirst round; the second row says how often to bet if you passed in thefirst round and the computer raised to $1.
| Holding: | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | T | J | Q | K | A |
| 1st round: | 0.454 | 0.443 | 0.254 | 0.000 | 0.000 | 0.000 | 0.000 | 0.422 | 0.549 | 0.598 | 0.615 | 0.628 | 0.641 |
| 2nd round: | 0.000 | 0.000 | 0.169 | 0.269 | 0.429 | 0.610 | 0.760 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
Here are the betting tables plotted as graphs:
Notice that these strategies contain interesting behaviors such asbluffing (for example, player 2 always bets holding a 2 if player 1passes) and slow-playing (for example, player 1 doesn't always bet onthe first round even if he holds an ace).
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