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Poker

Two basic concepts:

- Poker is a game of skill, not luck.
- If you want to win at poker,
- make sure you are very skilled at the game, and
- always play with somebody worse than you (and who doesn’t cheat)

Baby poker

There are 2 players, Player A and Player B.

- The deck consists of only 3 cards, (J, Q, and K).
- The players begin by putting $1 into the pot. Each player is then dealt 1 card(one card is left in the deck)

Baby poker

- Player A goes first.
- He either opens by putting $1 into the pot,
- or passes by not betting.

Baby poker

- Player B then plays.
- If Player A passed, Player B may
- also open by putting $1 into the pot,
- or pass
- If Player A opened, then Player B
- must either call by also putting $1 into the pot,
- or fold.
- If player B opened, then Player A (who passed the first time) must either call Player B or fold.

Baby poker

This ends the play.

- If either player folds, then the other player gets everything that was put into the pot to that point.
- If neither player folds, then the players com- pare cards. The player with the highest card gets everything in the pot.

Ploys

- Bluffing: Opening on a losing hand (in this case a 1).
- Sandbagging: Passing on a winning hand (in this case a 3).

Game play

- First need to describe strategy for the entire game
- This is a list of what the player would do in all possible situations
- Player actions for Player A
- O: open initially (no further decisions for A)
- PC: pass initially, then call if B opened
- PF: pass initially, then fold if B opened

strategies

- Player actions for Player B
- O/C: open if Player A passes and call if Player A opens
- O/F: open if Player A passes and fold if Player A opens
- P/C: pass if Player A passes and call if Player A opens
- P/F: pass if Player A passes and fold if Player A opens

Using Strategies

- Each of these decisions must be made based only on the value of the card seen by the respective player.
- Thus the above strategies must occur in triples, indicating what specific plays must be made upon being dealt J, Q, or K.

“Pure” strategies

- Possible Player A strategy (P-F,P-C,O)
- Pass then fold if J
- Pass then call if Q
- Open if K
- Possible Player B strategy (P/F, O/F,O/C)
- Pass or fold if J
- Open or fold if Q
- Open or call if K

Ploys

- The bluffing strategies are the ones where an O appears in the 1 slot.
- The sandbagging strategies are the ones where a P appears in the 3 slot.
- Example:
- Strategy (O,P − C,P − C) for player A is both bluffing and sandbagging
- Strategy (O/F, P/C, O/C) for B is bluffing (sandbagging is ineffectual for B in this simple game)

Probabilistic strategies

- Probabilistic mixture of pure strategies.
- Example Player A:
- (P-F,P-F,P-C) with probability 1/3
- (P-F,P-C,O) with probability 1/3
- (O,P-F,P-C) with probability 1/3
- Might be useful to bluff/sandbag sometimes but not all the time!

Assessing Strategies

- The outcome of a particular round of poker depends upon
- the strategies chosen by each of the players,
- and the cards dealt to each of the players (this component is random)
- Example:
- Strategies (P-F,P-C,O) vs(P/F, O/F,O/C)
- Cards dealt Q vs K
- Game will develop as Pass, Open, Call
- The pot will be $4 at the end, noone folded
- B wins $2, equivalently A wins -$2

Expected gain

- Cards are random; given both strategies we can compute expected gain for player A
- Example:
- Strategies (P-F,P-C,O) vs (P/F, O/F,O/C)
- Expected gain for A = -$1/6

Abbreviated Game matrix

We removed obviously bad strategies to fit on the page.

Recall maximin

- Select a strategy S
- Here a probabilistic mixture of pure strategies
- If the opponent knew my strategy, what is the worst they can do to me?
- This will be one of the pure strategies (Why?)
- Select the probabilistic strategy that maximizes this worst case scenario

Optimal strategies

Player A:

- Play
- (P-F,P-F,P-C) with probability 1/3
- (P-F,P-C,O) with probability 1/2
- (O,P-F,P-C) with probability 1/6
- In layman’s terms :
- holding a 1: open 1/6 of the time, and pass and fold 5/6 of the time
- holding a 2: always pass initially, and then call 1/2 of the time and fold the other 1/2
- holding a 3: open 1/2 of the time and pass and call the other 1/2.

Optimal strategies

Player B:

- Play
- (P/F,P/F,O/C) with probability 2/3
- (O/F,P/C,O/C) with probability 1/3
- In layman’s terms
- holding a 1: always fold if B opens, but open 1/3 of the time, and pass 2/3 of the time if A passes.
- holding a 2: always pass if A passes, but call 1/3 of the time and fold 2/3 of the time if A opens
- holding a 3: always open or call.
- Optimal strategy is not unique! Can you find another one?

Value of game

- In Nash Equilibrium
- Value of game: -1/18, i.e. Player A loses $1/18 per game to player B.

``Reward’’ for being honest

- If Player A cannot bluff or sandbag:
- Value of game is -1/9: Player A now loses $1/9 to Player B
- If Player B cannot bluff:
- Value of game is 1/18: Player B now loses $1/18 to Player A
- If neither player can bluff or sandbag:
- Value of game is 0: it is an even game (and a pretty boring one; both players always open with a 3 and pass-fold with a 1 or 2).

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