To hedgers: Adding insult to injury

[jrz1972]

[ QUOTE ]
Suppose you believe the game is really 50:50. Imagine you start from the position of hedging 100%, with a sure win of about $475. Reducing your hedge bet by $110 is the equivalent of betting $100 to win $110. That's a good deal for you if your bankroll is at least $4500, assuming a conservative level of risk tolerance (Kelly fraction greater than 1/4), though it might not be optimal for you. With a bankroll over $450, you can at least afford to reduce the hedge by $11, risking $10 to win $11.

[/ QUOTE ]

For the record, this is where you're going wrong. You don't need to make up VNM utilty functions or anything to analyze this. You can get a good handle on the situation just by looking at that last, marginal hedge decision.

Like you said, if I decided to forgo that last $10 of hedge, I would be risking $10 to win $11. Is it rational for me to take that wager? The answer (and there's no way for you to get around this point), is that it depends on my level of risk aversion.

I can tell you with 100% certainty that if somebody walked into my office right now and asked me if I would like to flip a coin and get paid $11 if I win while losing $10 if I lose, I would decline the offer. I would also not take this gamble for 1/10th the stakes.

There's nothing irrational about that. I'm just risk averse. When hedging my Mansion bet, I preferred to go ahead and eliminate the risk on that last $10, even though I gave up some EV (but not EU) to do so.

[econophile]

Here's a typical representation of reference dependent prefereces with loss aversion.

U(x) = g(x-R) if x >= R
U(x) = -L*g(R-x) if x < R

R is a reference point, g(.) is some function, and L is a parameter that corresponds to the degree of loss aversion.

Let's set up the problem of hedging. We have already bet $1,100 on the Steelers as a pk to win $1,000. We have the option of hedging any amount on the Dolpins at 1.962 to 1. To achieve a riskless outcome, we would wager $1,000/1.962 = 509.68 on the Dolphins. This guarantees a win of $491.32. This risk free win seems like a natural reference point. To compare a full hedge to smaller hedges, lets pick a simple form for our utility function, namely g(x) = x.

So our utility function is now
u(x) = x - 490.32 if x > = 490.32
u(x) = -L*(490.32 - x) if x < 490.32

There are two possible outcomes. If PITT wins, we win 1,000 minus H (the amount we bet on MIA). If MIA wins, we win 0.962*H.

So expected utility is
EU = .5*(1000 - H - 490.32) - .5*L*(490.32 - 0.962*H)

To find the optimal H, calculate
dEU/dH = (.481*L - .5)

if L > 1.039, then dEU/dH >0 and a 100% hedge is optimal.

You can try this out using other forms for g(x), but you will get similar answers for concave g(x).

[pzhon]

[ QUOTE ]
Here's a typical representation of reference dependent prefereces with loss aversion.

[/ QUOTE ]
You wrote the economics version of technobabble. Did you really think that would pass inspection by a mathematician? This was not a plausible set of preferences for an advantage gambler.

First, reference dependent preferences are a descriptive tool for explaining the actions of apparently irrational agents, not a normative tool for saying how rational advantage gamblers should behave. Why not say that hedging is right if it leaves you with $8888, because eights are lucky?

Such a nondifferentiable point in your utility function is a sign your preferences are time-inconsistent, and that you are making choices you will regret later, choices inconsistent with your other choices. As advantage gamblers, we are not done. We will continue to find more advantage wagers in the future. So, whether we are above or below this reference point will not be determined only by this wager. Even if for some strange reason we really care extra about ending up below +$475, the fact that we may play another session of poker soon can easily push us above or below the reference point, which smooths out the kink you forced at the reference point unless the preferences are inconsistent over time.

Second, it is absurd to set the reference point to be $475 above the advantage gambler's starting point. Even in the descriptive theories, there is usually only one point where the utility function is not differentiable, and you are trying to say that by some unbelievable coincidence, that point is not at the starting point, but at the value of the 100% hedged position. That is not a plausible description of why advantage gamblers hedged 100%. It would say that people would make at least small wagers, betting 10 to win 11, at almost every point except after gaining the magic $475. You are saying that if people won $10 elsewhere or lost $100 elsewhere, they would hedge less than 100% since they no longer have the magic +$475. This is absurd.

Is it so hard for you to believe that people made a mistake? I often err, and I see other people make mistakes more frequently. It's part of being a winning poker and backgammon player. The key is to learn from your mistakes, not to attempt to rationalize them and ridicule anyone who dares to tell you that you made a mistake.

[econophile]

pzhon,

i don't understand why you are so dogmatic. economics certainly is not. i could write other preferences that result in 100% hedges, even ones that are continuously differentiable, but i fear that they would elicit more long and boring responses from you.

i anticipated your objection about the preferences i wrote not being consistent with "advantage gamblers." have you considered that many people would not play poker were it not for the added benefits of rakeback and bonuses? the preferences i wrote would be perfectly consistent with such behavior.

[pzhon]

[ QUOTE ]
i could write other preferences that result in 100% hedges, even ones that are continuously differentiable,


[/ QUOTE ]
Really? I doubt you can come up with anything reasonable, since with a smooth utility function with positive value for money, the derivative of the expected utility of hedging is not 0 at a 100% hedge, and the derivative should be 0 at even a local optimum. Do you think it is a good idea for you to make such statments before actually trying?

Your first attempt was to assume that there is a kink at +$475. Was that reasonable? No. Imagine that just before the Mansion promotion, an interviewer was asking someone about their levels of risk aversion:
<ul type="square">Suppose we give you $n, and then let you bet on a coinflip, risking 10 to win 11. Please state how much would you wager at each level of N.

0: Maybe $5.

10: Still $5.

100: Maybe $10.

1000: I think I'd bet $20.

10000: Gamboool! I'd bet $1000, then.

500: Maybe $15.

475: Not a single penny. The difference between gaining $475 and gaining $474.90 is greater than the difference between gaining $475.11 and gaining $475.

450: Then I'd bet $10.[/list]Is this really supposed to be a plausible pre-Mansion interview of an advantage gambler who hedged 100%? It's absurdly contrived.

[ QUOTE ]

but i fear that they would elicit more long and boring responses from you.


[/ QUOTE ]
You are bored when people point out that you are both rude and wrong? Does this mean it happens frequently?

Despite having "econ" in both of your names, you have shown no command of basic economic theory, only irrelevant statements, empty boasts, and condescension. (Meanwhile, I've made a point many people have stated is interesting.) I will ignore you henceforth.

[jrz1972]

[ QUOTE ]
First, reference dependent preferences are a descriptive tool for explaining the actions of apparently irrational agents, not a normative tool for saying how rational advantage gamblers should behave.

[/ QUOTE ]

Yep. You're basically telling people that their utility functions are "wrong." You're advancing a "HEDGERS ARE ALL PU$$IES!!11!" argument and dressing it up in academic-sounding language.

[Izverg04]

All advantage gamblers have two things in common: they are money-loving and risk-averse. We are not argue this, right? When such a person is offered to make a 2% +EV or a 5% +EV bet, and choose the stakes, he will find the stakes at which this bet is optimal for him. (Optimal bet will be Kelly bankroll * percent advantage). The answer will never be: "you can take the bet and shove it", unless making the bet is inconvenient in some way.

[ QUOTE ]
Yep. You're basically telling people that their utility functions are "wrong." You're advancing a "HEDGERS ARE ALL PU$$IES!!11!" argument and dressing it up in academic-sounding language.

[/ QUOTE ]

I don't think pzhon called you a pu$$y, more like an irrational [censored].

[pzhon]

[ QUOTE ]

I can tell you with 100% certainty that if somebody walked into my office right now and asked me if I would like to flip a coin and get paid $11 if I win while losing $10 if I lose, I would decline the offer. I would also not take this gamble for 1/10th the stakes.


[/ QUOTE ]
If you believe the situation, then that is either a mistake, or you are not an advantage gambler.

I see that most of your recent strategy posts are in the microlimits forum, so perhaps for you, risking $10 to win $11 still looks pretty risky. However, you also have posts in the small stakes short-handed forum. I'm not familiar with that forum, but normally small stakes is at least $2-$4, 4 BB/100 would be an amazing win rate, and 20 BB/100 would be slightly high for a standard deviation. Playing a single orbit of 6-handed $2-$4 with these stats has an EV of about $1 with a SD of about $20, which is a lot like betting $20 to win $22.

Even if you are not ordinarily bankrolled to risk $10 to win $11, you first receive about $475 for free. That should give you a large enough bankroll to take that risk, even if you are quite conservative in managing your bankroll. Not being willing to wager $1 to win $1.10 on a $475+ bankroll is ridiculous for an advantage gambler. The odds against ruin are astronomical.

Do you really think avoiding betting $10 to win $11 (or even $1 to win $1.10!) is a reflection of your preferences on the outcomes, or is it an irrational desire to say that you don't gamble, or is it that you don't appreciate the difference between $11 and $10, but do understand the value of the much riskier activity of playing winning poker?

People who are not advantage gamblers do all sorts of nonsensical crap because it makes them feel better. They make sucker bets at casinos, claim mechanical devices have memories or respect hunches, cold-call with garbage, overcall on the river with bluff-catchers, etc. These don't have to make sense, since they are spending money on entertainment. Don't pretend these behaviors are rational for an advantage gambler, or that so many advantage gamblers hedged 100% was a reflection of their desire not to wager even $10 to win $11, even with at least $475 extra in their bankrolls.

As JPTT III said, "...doubtless the wager is something most [advantage gamblers] would take, in some amount or another. This is why OP can say with certainty that to hedge the whole amount is almost always wrong (although not knowing what amount would make it right). "

If you read a poker book, and the author explains that calling a river bet in a certain situation is a good gamble, as you are getting 20:1 odds and are only 15:1 against having the best hand, do you similarly say, "He can't say it's a good gamble for me. I'm risk averse. It's not right for me to call there?"

[uDevil]

[ QUOTE ]
[ QUOTE ]

I can tell you with 100% certainty that if somebody walked into my office right now and asked me if I would like to flip a coin and get paid $11 if I win while losing $10 if I lose, I would decline the offer. I would also not take this gamble for 1/10th the stakes.


[/ QUOTE ]
If you believe the situation, then that is either a mistake, or you are not an advantage gambler.

I see that most of your recent strategy posts are in the microlimits forum, so perhaps for you, risking $10 to win $11 still looks pretty risky....

[/ QUOTE ]

Odd that poker players have this view because the same idea was well received when you made this post:

[ QUOTE ]

I tell my family that poker is like a game of chess for $1 followed by a coin-toss for $10. Whether I win or lose in a night is determined by the coin-toss, but at the end of the year, what matters most is how well I have been playing in the chess games.

[/ QUOTE ]

[E.Z.]

[ QUOTE ]
[ QUOTE ]


Do you really think avoiding betting $10 to win $11 (or even $1 to win $1.10!) is a reflection of your preferences on the outcomes, or is it an irrational desire to say that you don't gamble, or is it that you don't appreciate the difference between $11 and $10, but do understand the value of the much riskier activity of playing winning poker?



[/ QUOTE ]

i really wish you would quit using your 11-10 ratio. a full hedger lost $25 which with a 3rd grade ed. you would realize this does not reflect 11-10 on $475 bet.

i wish you would respond to me OP.

although i will stick to my original reason for hedgers being +ev i would like to offer an explanation that the bet was not a coin flip as well.

if you got in with mansion at -5
you lose the coin flip if pitt wins by 4 or less
you win coin flip if wins by 6 or more

and since this was a free bet and were risking nothing you also LOSE if pitt wins by 5.

that sure is a funny looking coin. but i'll post my last one for ya so you can respond <font color="red"> </font> <font color="black"> </font>