1. Lecture 13

2. Project 2 Select and analyze a game of chance DUE: on November 10.
Use calculation, R or other mean to make QUANTITATIVE observations about the game
Write a 5 page paper. Include:
Description of the game
Your analysis
Conclusions
References
DUE: on November 10.
Come to my office to discuss what makes a suitable project!

3. Game theory So far we discussed: roulette and blackjack Roulette:
Outcomes completely independent and random
Very little strategy (even that is curbed by minimums and maximum bet rules)
Blackjack
Dealers outcome completely random (no strategy)
There is some dependence between draws
Strategy can be useful (counting cards and adjusting play accordingly)

4. Game theory Poker What is a rational (best?) strategy? MAXIMIN
All the participant are following a strategy that uses the information based on the cards and behaviour other players
What is a rational (best?) strategy?
MAXIMIN
Select a strategy S
If the opponent new my strategy, what is the worst they can do to me?
Select the strategy that maximizes this worst case scenario

5. Understanding role of strategy
Coin matching game:
You and your friend each have a coin. You each secretly choose to show “heads” or “tails”. You simultaneously show your choice. The payoff:
If the coins are both “heads”, your friend pays you $1.
If the coins are both “tails”, your friend pays you $9.
If the coins do not match, you pay your friend $5.
Is this a fair game?

6. Coin matching game you\friend H T 1 -5 9 If both of us flip a coin:
Expected gain = 1*0.5*0.5+(-5)*0.5*0.5+(-5)*0.5*0.5+9*0.5*0.5=0
(Looks fair but is it?)
Play matlab game
We do not need to flip a fair coin – we can choose any proportion of H/T we want!
Consider our proportion of H p and friends proportion of H q. For now assume independence between us and friend
Expected gain = 1*p*q+(-5)*p*(1-q)+(-5)*(1-p)*q+9*(1-p)*(1-q)=9-14p-14q+20pq
How to select p?

7. Maximin Main idea: We want select p to maximize expected win
if positive we might also want to minimize variance
What is best depends on the friends strategy: (Depends on friends choices unknown to me)
If I select p – if my friends selects all T gain =9-14p all H gain=-5+6p

8. Maximin Notice that smaller of the two lines is less than -0.8
The best option is p=0.7 – gain =-0.8 

9. Minimax (maximin for friend)
Friends choices
What is best for the friend depends on my strategy (unknown to him)
My friends select q – if I selects all H gain =9-14q all T gain=-5+6q
Notice that larger (my gain his loss!) of the two less than -0.8
The best option is q=0.7 – expected gain =-0.8 (friend gains 0.8 on average)
Nash equilibrium
if the other player is playing optimally there is nothing to gain from changing strategies

10. 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) 

11. 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)

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

13. 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.

14. 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.

15. Ploys Bluffing: Opening on a losing hand (in this case a J).
Sandbagging: Passing on a winning hand (in this case a K).
Play matlab game

16. 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

17. 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

18. 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.

19. “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

20. 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)

21. 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!

22. 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

23. 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
A gains -1 +1 -2

24. Abbreviated Game matrix
We removed obviously bad strategies to fit on the page.

25. 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

26. Optimal strategies Player A: Play In layman’s terms :
(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.

27. Optimal strategies Player B: Play In layman’s terms
(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?

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

29. ``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).