Lecture 13

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 J).

• Sandbagging: Passing on a winning hand (in this case a K).

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

cards (J, Q) (J, K) (Q, J) (Q, K) (K, J) (K, Q)

probability 1/6 1/6 1/6 1/6 1/6 1/6

A gains -1 -1 +1 -2 +1 +1

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 K and pass-fold with a J or Q).