complex that most of the players deviate from them
in any way. A good player should notice and exploit
such deviations to increase his own gain. In this
case, the applicable concept is the best response.
Let
be the profile of strategies of player’s
opponents - all strategies from profile except
.
Then the best response for player to opponents’
profile
will be
∗
∈
(
,
).
In the case of Nash equilibrium strategies of the
players are the best responses to each other.
Definition 7. A player's strategy
is weakly
dominated if there is a different strategy for this
player
′
, such that
1)
(
,
)
≤
′
,
∀
∈ Σ
,
2) ∃
∶
(
,
)
<
′
,
If the second condition is satisfied for all the profiles
of opponents’ strategies
∈ Σ
, a strategy is
called strictly dominated.
For each type of domination iteratively dominated
strategy can be defined recursively as any strategy
that is dominated at present, or becomes dominated
after excluding iteratively dominated strategies from
the game.
2.2 Poker
Poker is a class of card games with two or more
players. There are more than 100 poker variants with
different rules. General elements of all types of
poker are completely or partially closed opponents’
cards, card combinations, and the presence of trade
during the game. Poker has interesting properties
that cannot be analyzed by conventional approaches
used in games with the full information:
• Incomplete information about the current
state of the game. In most variants of poker
players cannot see opponent's cards.
• Incomplete information about opponents’
strategy. The hidden information is not always
revealed to players at the end of the game,
which is an obstacle to the definition of
strategies of opponents. Therefore, opponents
have to use simulation approaches based on the
frequency of their actions.
• Stochastic events. The random distribution of
cards makes the players to take risks.
• Various numbers of players (2 to 10). Games
with more than one opponent are strategically
different from the games of two players, as the
Nash equilibrium is not giving a break-even
guarantees. Opponents may accidentally or
deliberately choose a strategy so that a player’s
strategy from equilibrium profile will lose.
• Repeated interaction. Poker is a series of short
games or hands, where after each hand the
player receives partial information about his
opponent, which allows him to make
adjustments in their strategy.
• Importance of opponents’ errors
exploitation.
2.2.1 Kuhn Poker
Even the smallest variant of competitively played
poker has a large number of game states (Johanson,
2013), and the game requires a lot of computing
power to solve. So much smaller variations of poker,
retaining its basic properties, are often applied for
testing algorithms.
Kuhn poker was proposed by Harold Kuhn in
1950. It is a very simple version of poker, which is
attended by two players. The deck consists of three
cards: queen, king and ace, the game takes place
only one round of bidding, and only 4 actions are
available to players: bet, check, call and fold. The
game is played as follows:
• Both players initially make a bet of 1 chip into
the pot, called the ante.
• Each player is dealt one card
• The first player can choose to bet or check.
The amount of betting is fixed to 1 chip.
• If the first player selects a bet, the second player
can play call, or fold
o In the case of call the showdown
occurs. The player with the highest
card wins the pot.
o In the case of fold the pot goes to the
first player.
• If the first player chooses a check, the second
player can play a bet or a check
o If the second player selects the check
the showdown occurs. The player with
the highest card wins the pot.
o If the second player selects the bet, the
first player can play call, or fold
In the case of call the showdown
occurs. The player with the
highest card wins the pot.
In the case of folding pot goes to
the second player.
Kuhn poker can be generalized to multiple
players version. In this case, a deck for players
will consist of +1 cards. Kuhn poker with three
players is used as one of the games at the Annual
Computer Poker Championship.