Poker

Poker is a non-deterministic zero-sum game with imperfect information. A game is said to be zero-sum if the sum of the winnings obtained for each of the players is zero, independent of the strategy followed by each player. In poker, the profit of one player is the loss of other players. The long-term goal of all the players is to leave the table with more money than they had at the beginning. A poker session is played in a series of games with a standard deck of 52 playing cards. Each card is identified by its suit and rank. There are four suits: $\clubsuit$ Clubs, $\diamondsuit$ Diamonds, $\heartsuit$ Hearts and $\spadesuit$ Spades. The thirteen card ranks are (in increasing order of importance): Deuce (2), Three (3), Four (4), Five (5), Six (6), Seven (7), Eight (8), Nine (9), Ten (T), Jack (J), Queen (Q), King (K) and Ace (A).

A poker game is composed of several rounds. A round consists of a number of cards being randomly dealt followed by betting. Every active player is given the chance to act at least once in a round. Every time it is a player's turn to act, there are three alternative actions:

• Fold - drop out of the current game, losing any investment made in the current game,
• call - match the current per-player contribution, or
• raise - increase the amount to call.

If there has not been any previous bet in the round, a call is referred to as a pass, and a raise is said to be a bet. A round ends when each player has either folded or contributed the same amount of money as all the other active players. A poker game has two terminating conditions:

1. all the players have folded except one, who wins all money wagered (the pot), or
2. all the betting rounds have been completed.

In the second case, the game proceeds to a showdown where all the active players reveal their cards and the winner is determined. The winner is the player holding the highest poker hand (see Table 1 for hand ranking). Two hands of the same type are ranked according to the value of the cards (without regard for suits); for example, a pair of Aces beats a pair of Kings. In the case of a tie, the pot is split evenly.

 

Table 1: Poker Hands: weakest to strongest

Hand Type	Example	Probability
Busted	A$\clubsuit$ K $\spadesuit$J $\diamondsuit$ 10 $\diamondsuit$ 4 $\heartsuit$	0.5015629
Pair	2 $\heartsuit$ 2 $\diamondsuit$ J $\spadesuit$ 8$\clubsuit$ 4 $\heartsuit$	0.4225703
Two Pair	5 $\heartsuit$ 5$\clubsuit$ Q $\spadesuit$ Q$\clubsuit$ K$\clubsuit$	0.0475431
Triple	7$\clubsuit$ 7 $\heartsuit$ 7 $\spadesuit$ 3 $\heartsuit$ 4 $\diamondsuit$	0.0211037
Straight (sequence)	3 $\spadesuit$ 4$\clubsuit$ 5 $\heartsuit$ 6 $\diamondsuit$ 7 $\spadesuit$	0.0035492
Flush (same suit)	A$\clubsuit$ K$\clubsuit$ 7$\clubsuit$ 4$\clubsuit$ 2$\clubsuit$	0.0019693
Full House	7 $\spadesuit$ 7 $\diamondsuit$ 7$\clubsuit$ 10 $\diamondsuit$ 10$\clubsuit$	0.0014405
Four of a Kind	3 $\heartsuit$ 3 $\spadesuit$ 3 $\diamondsuit$ 3$\clubsuit$ J $\spadesuit$	0.0002476
Straight Flush	3 $\spadesuit$ 4 $\spadesuit$ 5 $\spadesuit$ 6 $\spadesuit$ 7 $\spadesuit$	0.0000134

 

Games classified as poker are divided into stud games, flop games (some cards of each player are shared), and draw games (no cards are exposed, some are discarded and replaced with cards from the deck). BPP plays the limit version of Five-Card Stud. In the limit version of this game there is a fixed bet size for each round.