Expectation and Hourly RateAprecierea matematica si Rata orara de castig

luni, 28 februarie 2011

David Sklansky show us in this chapter how important is for a poker player to know to use mathematical statistics and correct assessment of the chances of winning.

Mathematical expectation is the amount a bet will average winning or losing. It is an extremely important concept for the gambler because it shows him how to evaluate most gambling problems. Using mathematical expectation is also the best way to analyze most poker plays.

Mathematical Expectation in Poker

Poker plays can also be analyzed in terms of expectation. You may think that a particular play is profitable, but sometimes it may not be the best play because an alternative play is more profitable. Let’s say you have a full house in five-card draw. A player ahead of you bets. You know that if you raise, that player will call. So raising appears to be the best play. However, when you raise, the two players behind you will surely fold. On the other hand, if you call the First bettor, you feel fairly confident that the two players behind you will also call. By raising, you gain one unit, but by only calling you gain two. Therefore, calling has the higher positive expectation and is the better play.

Here is a similar but slightly more complicated situation. On the last card in a seven-card stud hand, you make a flush. The player ahead of you, whom you read to have two pair, bets, and there is a player behind you still in the hand, whom you know you have beat. If you raise, the player behind you will fold. Furthermore, the initial bettor will probably also fold if he in fact does have only two pair; but if he made a full house, he will reraise. In this instance, then, raising not only gives you no positive expectation, but it’s actually a play with negative expectation. For

if the initial bettor has a full house and reraises, the play costs you two units if you call his reraise and one unit if you fold.

Taking this example a step further: If you do not make the flush on the last card and the player ahead of you bets, you might raise against certain opponents! Following the logic of the situation when you did make the flush, the player behind you will Told, and if the initial bettor has only two pair, he too may fold Whether the play has positive expectation (or less negative expectation than folding) depends upon the odds you are getting for your money – that is, the size of the pot – and your estimate of the chances that the initial bettor does not have a full house and will throw away two pair. Making the latter estimate requires, of course, the ability to read hands and to read players, which I discuss in later chapters. At this level, expectation becomes much more complicated than it was when you were just flipping a coin.

Mathematical expectation can also show that one poker play is less unprofitable than another. If, for instance, you think you will average losing 75 cents, including the ante, by playing a hand, you should play on because that is better than folding if the ante is a dollar.

Another important reason to understand expectation is that it gives you a sense of equanimity toward winning or losing a bet; When you make a good bet or a good fold, you will know that you have earned or saved a specific amount which a lesser player would not have earned or saved. It is much harder to make that fold if you are upset because your hand was outdrawn. However, the money you save by folding instead of calling adds to your winnings for the night or for the month. I actually derive pleasure from making a good fold even though I have lost the pot.

Just remember that if the hands were reversed, your opponent would call you, and as we shall see when we discuss the Fundamental Theorem of Poker in the next chapter, this is one of your edges. You should be happy when it occurs. You should even derive satisfaction from a losing session when you know that other players would have lost much more with your cards.

Hourly Rate

As suggested in the coin-flip example at the opening of this chapter, hourly rate is closely related to expectation, and it is a concept especially important to the professional player. When you go into a poker game, you should try to assess what you think you can earn per hour. For the most part you will have to base your assessment on your judgment and experience, but you can use certain mathematical guidelines. For instance, if you are playing draw lowball and you see three players calling $10 and then drawing two cards, which is a very bad play, you can say to yourself that each time they put in $10 they are losing an average of about $2. They are each doing it eight times an hour, which means those three players figure to lose about $48 an hour. You are one of four other players who are approximately equal, and therefore you four players figure to split up that $48 an hour, which gives you $12 an hour apiece. Your hourly rate in this instance is simply your share of the total hourly loss of the three bad players in the game.

Of course, in most games you can’t be that precise. Even in the example just given, other variables would affect your hourly rate. Additionally, when you are playing in a public card room or in some private games where the operator cuts the pot. you need to deduct either (he house rake or the hourly seat charge. In Las Vegas card rooms the rake is usually 10 percent of each pot up to a maximum of $4 in the smaller seven-card stud games and 5 percent of each pot to a maximum of $3 in the larger seven-card stud games, in the Texas hold ‘em games, and in most other games.

In the long run a poker player’s overall win is the sum of his mathematical expectations in individual situations. The more plays you make with a positive expectation, the bigger winner you stand to be. The more plays you make with a negative expectation, the bigger loser you stand to be. Therefore, you should almost always try to make the play that will maximize your positive expectation or minimize your negative expectation in order to maximize your hourly rate.

Once you have decided what your hourly rate is, you should realize that what you are doing is earning. You are no longer gambling in the traditional sense. You should no longer be anxious to have a good day or upset when you have a bad day. If you play regularly, you should simply feel that it is better to be playing poker making $20 an hour, able to come and go as you please, than to be working an eight-hour shift making $15 an hour. To think of poker as something glamorous is very bad. You must think that you are just working as a poker player and that you are not particularly anxious about making a big score, If it comes, it comes. Conversely, you won’t be so upset if you have a big loss. If one comes, it comes. You are just playing for a certain hourly rate.

If you have estimated your hourly rate correctly, your eventual winnings will approximate your projected hourly rate multiplied by the total hours played. Your edge comes not from holding better cards, but from play in situations where your opponents would play incorrectly if they had your hand and you had theirs. The total amount of money they cost themselves in incorrect play, assuming you play perfectly, minus the rake, is the amount of money you will win. Your opponents various mistakes per hour will cost them various amounts of money. If the hands were reversed, you wouldn’t make these mistakes, and this difference is your hourly rate. That’s all there is to it. If they play a hand against you differently from the way you would play it five times an hour, and if it’s a $2 mistake on average, that’s a $l0-an-hour gain for you.

To assume you play perfectly is, of course, a big assumption. Few if any of us play perfectly all of the time, but that is what we strive for. Furthermore, it is important to realize that there is not one particular correct way to play a poker hand as there is in most bridge hands. On the contrary, you must adjust to your opponents and mix up your play, even against the same opponents, as we shall explain in later chapters.

Furthermore, it is sometimes correct to play incorrectly! You may, for example, purposely make an inferior play to gain in a future hand or future round of betting. You also may play less than optimally against weak opponents who have only a limited amount to lose or when you yourself are on a short bankroll. In these cases it is not correct to push small edges, You should not put in the maximum raises as a small favorite. You should fold hands that are marginally worth calling. You have reduced your hourly rate but have ensured yourself a win. Why give weaker players any chance to get lucky and quit big winners or get lucky and bust you if you arc on a short bankroll? You’ll still get the money playing less than optimally. It will just take a few more hours.

You should try to assess most poker games in terms of your expected hourly rate by noticing what mistakes your opponents are making and how much these mistakes are costing them. Don’t sit in a game with an insufficient hourly rate projection unless you think the game will become better – either because you expect some weaker players to arrive soon or because some good players in the game have a tendency to start playing badly when they are losing. If these good players jump off winners, you should quit if you can. However, it is sometimes good to continue in a game with a low hourly rate projection for political reasons – you do not want to get a reputation for gambling only when you have much the best of it. Such a reputation can make enemies, cost you money in the long run, and even get you barred from some games.David Sklansky ne arata in acest capitol cat de important este pentru un jucator de poker sa stie sa foloseasca statisticile matematice si evaluarea corecta a sanselor de castig.

Aprecierea matematica este suma pe care un pariu il va avea in medie de a castiga sau pierde. Este un concept extrem de important pentru un jucator, pentru ca ii arata cum sa evalueze cele mai multe probleme de joc. Utilizand aprecierea matematica este totodata cea mai buna cale de a analiza cele mai multe situatii din poker.

Aprecierea matematica in poker

Jocurile de poker pot fi analizate si prin prisma aprecierii. Poate va ganditi ca un anumit joc e profitabil, dar cateodata poate ca nu este asa deoarece un alt joc alternativ e mult mai profitabil. Sa zicem ca aveti un full house intr-o mana de cinci carti. Jucatorul de dinainte pariaza. Stiti ca daca plusati, acel jucator va plati. Deci plusarea pare a fi cea mai buna solutie. Insa, cand plusati, ceilalti doi jucatori de dupa dumneavoastra se vor retrage cu siguranta. Pe de alta parte, daca platiti cu sec, sunt sanse bune ca ceilalti doi jucatori de dupa dumneavostra sa plateasca si ei. Deci prin plusare, castigati o suma, dar doar prin plata cu sec castigati dublu. In concluzie, platirea cu sec are cea mai pozitiva apreciere de castig, deci este cel mai bun mod de a juca.

Iata o situatie similara, dar putin mai complicata. Cu ultima carte dintr-o mana de stud de sapte carti, faceti culoare. Jucatorul de dinainte, pe care il cititi ca are doua perechi, pariaza, si mai este un jucator dupa dumneavoastra, pe care stiti ca l-ati batut in trecut. Daca plusati, urmatorul jucator se va retrage. Mai mult decat atat, chiar si cel care a pariat se poate retrage daca are intr-adevar doua perechi; dar daca el ar face un full house, va plusa. In acest moment deci, a plusa nu numai ca nu va aduce o apreciere pozitiva, ci din contra, e de fapt un joc cu apreciere negativa. Pentrru ca daca jucatorul care a pariat initial are un full house si pluseaza, jocul va costa dublu decat daca platiti cu sec re-plusarea lui si pe jumatate daca aruncati cartile.

Mergand mai departe cu acest exemplu: daca nu va iese culoare pe ultima carte si jucatorul de dinainte pariaza, puteti da raise catorva oponenti! Urmarind logica situatiei daca a-ti face  intr-adevar culoare, urmatorul jucator va da fold, iar daca primul jucator are doar doua perechi, si el se va arunca. Chiar daca jocul are apreciere pozitiva (sau mai putin negativa decat a da fold) depinde de sansele pe care le primiti pentru banii dumneavoastra – adica marimea pariului mainii – si de estimarea sanselor ca primul jucator sa n-aiba un full house ci doar doua perechi. Facand aceasta din urma estimare cere, desigur, abilitatea de a citi mainile si pe jucatori, pe care le voi discuta in capitolele de mai tarziu. La acest nivel, aprecierea devine mult mai complicata decat cand doar dadeati cu banul.

Aprecierea matematica poate totodata sa arate ca un joc de poker este mai putin neprofitabil decat altul. Daca, de exemplu, credeti ca veti avea o medie de 75 de centi de pierdut, inclusiv cel de dinainte, jucand o mana, ar trebuie sa continuati sa jucati pentru ca e mai bine decat sa va aruncati cartile daca cel de dinainte e un dolar.

Un alt motiv important pentru a invata aprecierea este pentru ca va da un sentiment de calm si seninatate spre a castiga sau a pierde un pariu. Atunci cand faceti un pariu bun sau un fold bun, veti stii ca ati castigat sau salvat o anumita suma pe care un jucator inferior n-ar fi castigat-o sau salvat-o. Este mult mai greu sa faci acel fold daca esti suparat ca ati fost depasit la o mana. Totusi, banii pe care-i salvati printr-un fold sau sec se adauga la castigurile serii sau lunii. Eu chiar imi obtin placere dintr-un fold bun, chiar daca am pierdut potul.

Doar tineti minte ca daca mana a fost inversata, oponentul va va plati cu sec si, dupa cum vom vedea cand vom discuta despre Teorema Fundamentala a Pokerului in capitolul urmator, aceasta este un avantaj. Ar trebui sa va bucurati cand apare situatia aceasta. Ar trebui chiar sa dobanditi satisfactie din pierderea unei sesiuni cand stiti ca ceilalti jucatori ar fi pierdut mult mai mult cu cartile dumneavoastra.

Rata orara de castig

Dupa cum este sugerat in exemplul cu aruncatul banului la inceputul acestui capitol, rata orara este strans legata de apreciere si este un concept extrem de important pentru jucatorul profesionist. Cand intrati intr-un joc de poker, ar trebui sa incercati sa evaluati cam cat credeti ca veti castiga pe ora. In general va trebui sa va bazati evaluarea pe judecata proprie si experienta, dar puteti utiliza si cateva metode matematice. De exemplu, daca jucati la egalitate si vedeti ca trei jucatori pun 10$ si apoi cer doua carti, ceea ce este un joc foarte slab, va puteti spune ca de fiecare data cand ei joaca 10$, pierd in medie 2$. Ei fac asta cam de opt ori pe ora, ceea ce inseamna ca acei trei jucatori planuiesc sa piarda cam 48$ pe ora. Dumneavoastra sunteti unul dintre cei patru jucatori care sunt cu apoximatie egali si drept urmare voi cei patru jucatori va planuiti sa impartiti cei 48$ pe ora, ceea ce va aduca cam 12$ pe ora de fiecare. Rata dumneavostra orara in acest caz este pur si simplu partea dumneavoastra din pierdea totala per ora a celor trei jucatori slabi din joc.

Bineinteles, in cele mai multe dintre jocuri nu puteti fi atat de precis. Chiar si exemplul de mai sus, alte variabile pot afecta rata dumneavoastra orara. In plus, cand jucati intr-o zona publica sau privata unde dealerul intrerupe potul, trebuie sa deduceti fie regula casei sau costul pe ora de scaun. In salile de jocuri de carti din Las Vegas regula este de obicei 10% din fiecare pot pana la un maxim de 4$ in jocurile mai mici de stud cu sapte carti si 5% din fiecare pot pana la un maxim de 3$ pentu jocurile mai mari de stud cu sapte carti, pentru jocurile de Texas Hold ‘em, ca si in cele mai multe dintre celelalte jocuri.

Pe termen lung, castigul de ansamblu al unui jucator este suma aprecierilor sale matematice in situatii individuale. Cu cat mai multe jocuri efectuati cu o apreciere pozitiva, cu atat mai mare este potentialul castig. Cu cat mai multe jocuri efecutati cu o apreciere negativa, cu atat mai mult veti pierde. Drept urmare, trebuie aproape de fiecare data sa incercati sa faceti jocul care va maximizeaza aprecierea pozitiva sau minimeaza aprecierea negativa pentru a maximiza rata dumneavoastra orara.

De indata ce v-ati decis care este rata dumneavoastra orara, ar trebui sa realizati ca ceea ce faceti este un castig. Nu mai jucati in maniera traditionala. Nu ar mai trebui sa fiti nerabdator sa aveti o zi buna sau suparat cand aveti una rea. Daca jucati regulat, ar trebui sa simtiti ca este mai bine sa jucati poker facand 20$ pe ora, capabil sa intrati si sa iesiti cand doriti, decat sa munciti opt ore sa faceti 15$ pe ora. Sa va ganditi la poker ca la ceva stralucitor e foarte rau. Trebuie sa va ganditi ca doar muncti ca un jucator de poker care nu este neaparat nerabdator sa castige gramezi de bani. Daca castigi, castigi. In schimb, nu veti fi suparat daca pierdeti mai mult. Daca se intampla, se intampla. Dumneavoastra jucati doar pentru o anumita rata orara de castig.

Daca ati estimat corect rata dumneavostra orara, castigurile dumneavoastra eventuale vor aproxima ratele dumneavoastra orare proiectate multiplicate de numarul total de ore jucate.

Avantajul dumneavoastra vine nu numai din a detine carti mai bune, ci si din a juca in situatii cand oponentii dumneavoastra ar juca incorect daca ar avea cartile dumneavastra si invers. Suma totala de bani care-i costa intr-un joc incorect, in conditiile in care dumneavoastra jucati pefect, minus costul mesei, este suma de bani pe care-o veti castiga. Diversele greseli pe ora facute de oponentii dumneavoastra ii vor costa o anumita suma de bani. Daca s-ar inversa mainile, n-ati face aceste greseli, si diferenta consta in rata dumneavoastra orara. Asta este tot. Daca ei ar juca o mana impotriva dumneavoastra intr-un mod diferit de felul in care dumneavoastra ati fi jucat-o de cinci ori intr-o ora, si daca este o greseala in medie de 2$, aceasta reprezinta un castig de 10$ pe ora pentru dumneavoastra.

Asumand ca jucati perfect este, desigur, o ipoteza. Putini jucam perfect tot timpul, daca asta ne dorim. Mai mult decat atat, e important sa realizam ca nu este un mod in particular sa jucam o mana de poker asa cum se intampla in jocul de bridge. Din contra, trebuie sa va reglati in functie de oponenti si sa aveti un joc variabil, chiar si in fata acelorasi oponenti, dupa cum vom explica in capitolele urmatoare.

In plus, cateodata e corect sa joi incorect! Puteti, de exemplu, sa jucati din adins slab un joc doar pentru a castiga intr-o mana viitoare sau runda viitoare de pariuri. Puteti totodata sa jucati intr-o maniera mai putin optima contra unor jucatori mai slabi care au numai o suma limitata pentru a o pierde sau cand aveti fonduri limitate. In aceste cazuri nu e corect sa va jucati cu maruntis. N-ar trebui sa plusati la maxim ca o favoare mica. Ar trebui sa va aruncati la maini care sunt la limita platii cu sec. V-ati redus rata orara dar v-ati asigurat victoria. De ce sa dati jucatorilor mai slabi sanse sa devina norocosi si sa renunte la castiguri mari sau sa devina norocosi si sa va „arda” daca aveti un buget limitat? Veti castiga in continuare jucand mai putin optim. Doar ca va dura mai multe ore.

Ar trebui sa incercati sa evaluati majoritatea jucatorilor de poker in termenii ratei dumneavoastra orare anticipate prin a observa ce greseli fac oponentii dumneavoastra si cat ii costa greselile acestora. Nu stati intr-un joc cu o rata orara proiectata insuficienta decat daca credeti ca jocul se va imbunatati – fie pentru ca asteptati cativa jucatori mai slabi sa soseasca in curand sau pentru ca unii jucatori buni din joc au o tendinta sa joace mai prost dupa ce pierd. Daca acesti jucatori buni castiga oricum, ar trebui sa va retrageti daca puteti. Totodata, cateodata e bina sa continui intr-un joc cu o rata orara proiectata slaba din motive politice – nu vreti sa primiti o reputatie cum ca jucati doar cand castigati cel mai bine. O astfel de reputatie va poate face dusmani, sa va coste bani pe termeni lung sau chiar sa va fie interzis sa participati la anumite jocuri.


Author: Editor

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