2.2 Texas Hold'em

The specific variant under consideration in this thesis is Texas Hold'em, the most popular variant played in casinos. It is used in the main event of the annual World Series of Poker championship to determine the World Champion. It is considered to be one of the most strategically complex poker variants and has ``the smallest ratio of luck to skill" [2]. The script for Texas Hold'em is as follows (each of the four rounds is followed by betting):

• Pre-flop: each player is dealt two face-down cards ( hole cards).
• Flop: 3 cards dealt face-up to the board (community cards).
• Turn: 1 card dealt face-up to the board.
• River: 1 card dealt face-up to the board.

After the betting on the river, the best 5-card hand formed from the two hole cards and five board cards wins the pot.

Specifically, we examine the game of Limit Texas Hold'em with a structured betting system of 2-2-4-4 units, and blinds of 1 and 2 units. This means that the bet size is fixed at 2 (the small bet) for the pre-flop and flop, and 4 (the big bet) for the turn and river. Before the pre-flop, the first player after the button is the small blind and is forced to put 1 in the pot, and the subsequent player is the big blind and forced to bet 2 (meaning the amount to call is 2 for all subsequent callers, 1 for the small blind, and if there has been no raise, the big blind has the option to fold, check or raise to 4 units). Limit Hold'em is typically played with 8 to 10 players, although the minimum is 2 and possible maximum is 23.

In later chapters, we use a special convention for representing Texas Hold'em hands. The designation 8$\diamondsuit$-J$\heartsuit$/4$\clubsuit$-5$\clubsuit$-6$\clubsuit$ represents hole cards of 8$\diamondsuit$-J$\heartsuit$ with a board of 4$\clubsuit$-5$\clubsuit$-6$\clubsuit$.