Playing Poker Optimally

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Playing Poker Optimally

• Gijs Schoenmakers
• Robbert de Zeeuw

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Poker

• Cards are dealt
• Players make moves in turn
• Fold Leave the game
• Check / Call Match the highest bet
• Bet / Raise put more money in the pot than other
  players
• Best 5-card hand wins and gets the pot

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Ranking of hands

• High Card

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Ranking of hands

• One Pair
• Two Pair
• Three of a kind
• Straight 5 in a row

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Ranking of Hands

• Flush 5 cards of the same suit
• Full House Three of a kind a pair
• Four of a kind
• Straight Flush 5 in a row of the same suit
• Royal Flush Straight Flush with an ace high

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Types of poker

• Draw Poker
• Draw 5 cards, bet, change some, bet
• Texas hold em
• 2 Hole cards, 5 community cards
• Limit (one can raise by 1 unit)
• No limit (one can raise by any amount)

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• Texas Hold em
• The Game after the
• hole cards are dealt

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"Call"
"Raise"
"Check"
2
"Call"
"Check"
"Fold"
Let's Play!
Blinds
Flop
3
1
Hole Cards
Community Cards
Turn
River
"Fold"
"Call"
"Raise"
"Reraise"
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Optimal Play

• Backwards Induction 4th round of betting
• Player 1 check or raise?
• Player 2 fold, check/call or (re-)raise?
• Depends on strength of hand
• A At least 90 chance of winning (very strong)
• B 70 - 90 (strong)
• C 40 - 70 (mediocre)
• D at most 40 (weak)

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Optimal play

• Goal Maximize expected payoff
• Player 1 A-hand
• a PrRaise
• 1 a PrCheck
• Player 2 B-hand
• after check raise (theory)
• after raise
• q PrReraise
• 1 q PrCall
• Player 1 Reraise (if player 2 raised)
• Player 2
• After CRR p PrCall 1 p PrFold
• After RRR s PrCall 1 s PrFold

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Game Tree (A,B)-hand
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Expected Payoffs to player 1 (A,B)-hand

• (1 a )p2 (1 a)(1 p)1 a(1 q)1
  aqs3 aq(1 - s)2
• If players already put 4 in the pot
• (1 a )p6 (1 a)(1 p)5 a(1 q)5
  aqs7 aq(1 - s)6

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(B,C)-hand

• Player 1 (B-hand)
• Raise with probability 1 (theory)
• Player 2 (C-hand)
• v PrReraise
• 1 - v PrFold (theory Do not call)
• Player 1 After raise by player 2
• b PrCall
• 1 b PrFold

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Game Tree (B,C)-hand
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Expected Payoffs to player 1 (B,C)-hand

• (1 v)0 vb2 v(1 b)-1
• If players already put 4 in the pot
• (1 v)4 vb6 v(1 b)-5

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Table

• Payoff-table for all combinations (X,Y)
• (1 v)4 vb6
• v(1 b)-5
• (1 a )p6
• (1 a)(1 p)5
• a(1 q)5 aqs7
• aq(1 - s)6

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Results

• If there is 4 in the pot already
• Player 1
• A-hand always (re)-raise (up to 3 times)
• B-hand raise (theory), always call after a
  reraise
• C-hand always check, always call after raise
• D-hand
• PrRaise 0.98 !!
• After Check Raise Fold
• After Raise Reraise PrReraise 0.68,
  PrFold 0.32
• Payoff against Best Reply player 2 0.1221

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Results

• Player 2
• A-hand Always reraise (up to 3 times)
• B-hand
• After a check, always raise and then after a
  reraise, always call.
• After a raise Prraise 0.384, Prcall
  0.616,
• and after another reraise, always call.
• C-hand
• After a check, either check or raise and then
  after a reraise, always call
• After a raise, always reraise and then after a
  reraise, always call
• D-hand
• After a check, raise with probability between
  0.15 and 0.30, and after a reraise fold
• After a raise, reraise with probability 0.217,
  otherwise fold, and after a reraise fold

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Results

• Player 2 does not have to pay more than 0.2014
• 2 Strategies against each other 0.1850

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Questions?