Lecture 13

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!

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)

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

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?

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?

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

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

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

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). Play matlab game

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 A gains -1 +1 -2

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 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.

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?

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).