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Brain Teasers

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Quantitative Finance Society

Gambling Strategies & Statistics

- Basic Rules
- Expected Value (EV)
- Hand Odds
- Pot Odds

- Betting
- Position
- Implied Odds, Bet-Sizing, C-betting/Floating/Triple-barreling
- Fold Equity, Fold vs Call vs Re-raise
- Board Texture, Other Plays with Bets

- Poker Statistics, Adjusted EV
- Putting your opponent on a hand range
- VPIP, PFR, other metrics for reading your opponent’s play style

- Analogies to Markets & Trading

- The concepts covered today will only be touched upon lightly, so do not follow them blindly and assume they will work. Just like investing, there are lots of intricacies about when to use certain strategies at the right time, but we’re just trying to give a broad overview of things to know, and if you’re interested you can read more (recommended sites: twoplustwo.com [COTW Concept of the Week <-GOOD READ], deucescracked.com)
- Variance. In poker, just like finance, past results do not indicate future returns, and there will be up and downswings even if you are correct in theory. There is no foolproof strategy. The market can stay irrational longer than you can stay solvent.

- Standard 52-card deck, 2-10 people
- Dealer button, small/big blind
- 2 hole cards each, community board, your hand=top 5 cards (chops)
- Standard Poker Hand Strength
- Royal/straight flush
- Four of a kind (quads), Full house (boat)
- Flush, Straight, Trips, Two-pair, Pair, High Card…

- Betting: Pre-Flop, Post-Flop, Turn, River

- Chance of having a hand:
- AA (or any pocket pair): .45%
- Any pocket pair: 6%
- AK (any suit): 1.2% [same for any two card combos]
- AK (suited): 0.3%
- 2 suited cards: 23.5%
- 2 connected cards: 15.7%
- Suited connectors: 4%

- Probabilities of hitting the flop:
- With any two cards
- Flopping a pair: 27%
- Flopping two-pair: 4%
- Trips: 1.3%

- With a pocket pair:
- Hitting a set: 11%

- With suited cards:
- Flopping four-to-a-flush: 11%

- With connected cards:
- Flopping four-to-a-straight: 11%

- With any two cards

- Matchups:
- Over-pair: 80-85% win/15-20% lose
- E.g. AA vs 66, AA vs J7

- Domination: sharing a card, one higher, ~75% win/25% lose
- E.g. AJ vs A7 (one card same, one bigger)

- Pair vs one over-card: 70%/30%
- E.g. QQ vs A9

- Slight favorite: 60-65%/35-40%
- E.g. A8 vs 79

- “Coin-flips”: ~55% win/45% lose
- E.g. A3 vs 79 (high card+low card vs two middle cards), 66 vs AK (pair vs two over-cards)

- Adjustments: +3% or so for straight/flush potential (suited/connected/gap cards)

- Over-pair: 80-85% win/15-20% lose

- Post-Flop Drawing Hands:
- “Rule of 4”: If you have 9 “outs”, your chance of hitting by the river is 4x the # outs, so ~36%
- “Rule of 2”: If you have 4 outs on the turn, your chance of hitting on the river is 2x #outs, so ~8%
- So if you have two clubs, and the flop has two clubs, you have 13-4 = 9 more clubs to hit so your chance of having a flush by the river ~36%

- Sum all the money that people have bet: this is the money in the “pot”
- Say the pot is $6. It’s you vs another person, post-flop. He bets $3 (pot is now $9). What are your pot odds?
- You have to pay $3 to win $9, so that’s 3/9=33%. You have to win 33% of the time to break-even.

- Combining pot odds and hand odds
Same scenario, $6 pot, $3 bet. You have four-to-a-flush. Is it profitable to call?

Pot-odds: 33% (3/9)

Hand-odds: 9 outs ~=36%

36%>33%, so you’ll win more often than is needed to break-even. YES! (positive EV)

- Problems/assumptions/missing factors?
- You may have more/less outs than you think, if he has the same draw, or if a pair is strong enough, or if he already has the nuts
- If you hit the draw, you can win more money later (implied odds)
- If you miss on the turn, it may not be profitable to call more bets
- Raising can be more profitable than just calling (fold equity)

- Adjustments to pot odds, based on more money that you can earn from future betting rounds.
- Same example, but he bets $4 instead. So your pot odds are now 4/10 = 40%, but your hand-odds are still only about 36% (assuming you must hit the flush to win, otherwise lose). Do you call?
- If you hit, you can possibly win his whole stack if he keeps betting his pair
- If you miss, there’s little downside because you can just fold

- What if there are 3 other people in the pot?
- Even higher implied odds, because the odds of 1 of the 3 people calling you is higher than just 1 person

- Problem: don’t get overly optimistic
- People will see the flush on the board and won’t call with a pair
- If you miss on the turn, you will likely fold to another bet

- Position is everything: it gives you information
- If you’re first to act/early, you don’t know the strength of anyone’s hand. If you bet, you have 8 people behind that might raise.
- If you’re on the button, people have folded/limped/bet, so you have an idea what people have, and can make plays accordingly.

- Example: You have A5, you raise, 1 caller. Flop comes A98 suited. You have top pair but a weak kicker. He has position. What do you do?
- If you bet and he re-raises, he probably has a better ace, and you will have to fold.
- If you check and he raises, he still probably has a better ace, so you have to fold. If he checks too, he might have a flush draw.

- Now instead, you have position. He checks.
- Sign of weakness! You should bet because he probably doesn’t have a better ace.
- What if he bets? You can safely fold.

- Maximizing profit: Remember in the pot/implied odds example. The pot was $6. If you only bet $3, then the flush draw is going to call you. If you bet $4, then a loose player might call while a tight player might fold.
- Bigger bets increase chance of folding (obviously), but bet appropriately so the odds are slightly in your favor, but that people might still call. You want the flush draw to call the $4 bet, because it’s positive EV for you.
- If you bet $5+, nobody will call you, and you lose potential profit.
- At the same time, if you bet $3 or even less with your top pair, then you’re losing potential profit because drawing hands would have called a higher bet, and now more weaker hands will call you and probably suck out on you.

- Bet consistently: If you bet $4 one time and $3 the next, the opponent will think you have a weak hand when you bet $3 and will re-raise you.

- C-betting
- If you raised pre-flop (and everyone just called), you should continuation bet (c-bet) on the flop in most scenarios, because you are showing the most strength
- E.g. you bet, 2 callers. You have A9. Flop comes J95. Nothing dangerous and you have two overs. Both people check. You should bet, and they will probably fold.

- Double/Triple-Barreling
- Say one person calls that raise. Turn comes something useless, like a 2. He checks again. You should bet AGAIN to get him to fold, since he keeps checking and showing weakness. Maybe even a 3rd time on the river, if nothing dangerous comes out (and definitely if you hit).

- Floating (bluffing technique, representing strength)
- Say you’re the person that check-called flop in the above example. You have nothing, like K7. On the turn you just bet. This is good because the PFR likely c-bet with anything, so you can get him to fold.

- As you can see from that last example, your hand gets more value if you bet because you can get people to fold.
- This is why re-raising is better than calling most of the time too. (E.g. Pre-flop, you should 3-bet/4-bet AKs/KK/QQ to get more value, and to fold out weaker hands)
- This also boosts your implied odds. If the board looks really scary (3 to a flush, 3 to a straight) you can bet big with anything and a pair might fold because he is afraid of the straight/flush (playing board texture)
- Taking into consideration present/implied odds and all possible scenarios, KNOW WHEN TO FOLD and don’t be too optimistic

- Given all the above betting strategies, if a person bets strange/unconventional bet sizes, what does it mean? What level is his gameplay/mind at? E.g. he goes all-in pre-flop and you have JJ:
- Level 1: he is just betting his hand (he has AA/KK/QQ/AK)
- Level 2: he is considering your reaction, so he is trying to price you in/out (he is bluffing)
- Level 3-infinity: he knows that you might think it’s a bluff, so he has something good but is trying to induce a call
- Etc.

- Is it better to assume that people know nothing, or people know everything/some things? (weak/strong/semi-strong efficient markets)

- VPIP: essentially, % of hands played
- PFR: % of hands raised pre-flop
- Using VPIP/table-image/bet-sizing to put the opponent on a range of hands that they might play in that certain way. These are SUBJECTIVE PROBABILITIES that depend on PSYCHOLOGY.
- With this knowledge, calculating a new EV
- You have AJ, flop is A89. Pre-flop you bet and 3 callers. On the flop you bet, two fold, one caller. The caller has a VPIP of around 25%. Given that he called twice and he’s loose, he probably has a straight draw or a weak ace.
- Say he has an equal chance of either hand. Your new hand odds are .5*(90% chance of your AJ beating a weaker ace)+.5(68% chance of you winning if he doesn’t hit the striaght) = 79%

- Reading the skill of opponents ~ Efficient Markets Hypothesis
- Hand ranges and subjective probabilities ~ stochastic interest rate/price modeling
- Knowing when to fold and raise ~ closing/increasing a trade position. Discipline, sticking to your thesis
- Being able to handle variance, huge upswings/ downswings without getting emotional and still playing with positive EV. Not being attached to money, but being attached to winning.