Sunday, February 20, 2011

How Much Ya Wanna Bet?

This phrase tilts the living Bejesus out of me, and a fair number of you know exactly why. People who don't gamble for a living get this one wrong all the time; even really, really smart people. How much you'd be willing to bet on a certain thing isn't the real defining measure of how sure you are of it's truth. The odds you'd be willing to lay are. Just last night Danielle and I were out to a fancy dinner at this high end establishment (to be quite honest the pizza was pretty good, and the 30 or so odd tokens I found in my drawer bought quite a bit of skee ball, pop a shot, and Mario Kart) and she said to me "How much would you bet we're the only party here without any kids?" I just kind of laughed at first, then realized that was rude and explained to her how exactly I'd think about the problem.

1. Define kid. Later in the evening we realized there was a serious edge case; 17 year old couples. Do they count as kids? In the spirit of the rule, no. But maybe they were there with their parents and siblings and were actually 15. We'd need a definition of both kid and party before such a bet could be undertaken. Let's go with the loosest definition of party (just people who met, not necessarily came in the same car) and kid (anyone under the age of 18).

2. After that, I wouldn't think about it in terms of how much I'd be willing to bet. I'd think about it in terms of trying to determine what I thought a fair price was (just glancing around the restaurant I thought a fair price was something like 9:1 against there being another kidless party).

Proud of myself for explaining something, I sat back and sipped on my caffeine free diet pepsi (partyer that I am). Danielle simply restated her question, as is her custom and that of some other intelligent people I know after someone has explained that a problem is a lot deeper than what she originally posed, but also still very answerable. I thought about trying to discuss it in terms of the Kelley Criterion, but I assume that would have to be flawed as this is obviously an edge case. The answer to such a question relies on many things, all of which are inherit to a given person and therefore likely can't be decided with a formula:

1. How much money do you have?
2. How much money do you make?
3. How safe is that income stream?
4. How much money do you spend? Could you reduce it?
5. Will you kill yourself if you lose a certain number of dollars?
6. How old are you?

After thinking about these questions, and declaring that I thought the fair price of the bet was something like 9:1, I asked Danielle how much she would bet at even odds that we were in fact the only kidless party at the fine eatery. Her immediate response was....$500. I restated the question and this time told her not that I felt the odds were 9:1, that they actually were...simplifying the problem. You have an event that will happen with 90% certainty, and someone is willing to give you unlimited action at even money that it will not. How much will you bet? I managed to get her to say $5000, but that was as high as she'd go. Now I'm not going to go into her personal finances at any length here, but she does work at Google. That is all.

She wanted to know what my answer was, and after a few seconds of reflection I responded "I think all of it." She was stunned. Now I have money in a Roth IRA, and she argued that it's tax sheltered status meant I should never touch it, not even with a 90% chance of doubling it. And she might be right, I suppose. 30 years of untaxed growth is nothing to sneeze at. And I don't have a job, so betting every penny I could get my hands on would be perhaps a little silly. But even so, I think my personal answer is "the vast majority of my liquidatable net worth," and some of my answers to the questions above are decided "wrong." I don't have a job, and my income stream isn't very safe, specifically. Were I in her situation, I literally think I might bet every penny I could get my hands on.

So anyway, I thought that was interesting, and am curious as to what other people think. How much of your net worth would you bet at even money on a 9:1 favorite?


bellatrix78 said...

Listen to the 2p2 Pokercast this week. They ponder something like that at the end of it. They ask the guys if they'd be willing to bet 90% of their net worth on 90% bet. Mike says he might have been at some poit, now he isn't willing, because it's like 100k (I think they were going by liquid assets), he says it's the amount that's important. Adam (the actual poker player) laughs and says, gimme the bet, I bet 90% everytime, no matter the height of the bet. Mike answers: "Don't let your wife listen in to this week!".

The blindman said...

I wonder if that discussion has made Danielle think about your poker playing in a different light?

To answer this question, you would have to ask what *losing* a bet of that size would actually cost. Would it cost your marriage/relationship? Your house? Would it cause a severe bout of depression?

Speaking as someone with a large mortgage, wife and kids, I'd certainly bet 90% of my *poker bankroll* (which is kept completely separate from my life bankroll and in which I have never invested any of my "real" money). But I don't think I would touch my "life" bankroll, because even at 9:1 the downside is just not worth it.

If I was childless and worked at Google, however, that's a whole different game.

David said...

The previous commenters I think have this right - the problem is that the utility of money isn't linear. Let's say we could map money to happiness units (I will call them smileys). What would that curve look like? Let me throw out some data points:

We need a range of smileys. I'll say that 0 = borderline suicidal, and 100 = as happy as you could be. Note that this may be asymptotic - you may never be so miserable due to money that you would be suicide, and so happy due to money that you would have reached your maximum happiness.

10 smileys (very unhappy) = 0 $ (flat broke)
30 smileys (unhappy) = $20K (around poverty line)
50 smileys (neutral) = $40K (can pay bills, no luxury)
70 smileys (happy) = $100K (can afford most of what you want)
90 smileys (very happy) = >$2M (can afford almost all of what you want)

Of course, all of this is arbitrary, but it looks like smileys increase roughly as log (money). I'd posit that inflection point where doubling your money actually doubles your smileys is at quite a low number.

Oh, and you and Danielle are gigantic nerds.

Wacky said...

Dave said...

I prefer "smileys".

jesse8888 said...

So further reflection on this has lead me to believe that your current absolute net worth has the biggest impact on how changes in it affect your utility. Basically you have to know where you are on your personal curve to decide how to proceed.

And yes we are gigantic dorks. I'd argue that you'd be hard pressed to find a more attractive couple that is also smarter. To qoute Pete, I'm dating up several levels.

jesse8888 said...

But if given the option to bet 90% of my net worth at even money on a 9:1 favorite I'd fist pump ship it in.

Dave said...

Maybe you'd do that now. Would you do it if your net worth were $2M? $10M? What's your limit?

Also, I said you were gigantic nerds, not dorks. Big difference.

If Danielle were really that smart, wouldn't she be dating someone else?

The blindman said...

It's easier in some ways if your net worth is more. If you bet 90% of $100M, then if you lose you are still filthy rich. Isn't this how hedge funds work?

Yodaman said...

I wonder how much flipping the bet changes how much we should be betting. Say it was getting 9:1 on a coin flip?

Also, I like blindman's point about the 100M networth.

jesse8888 said...

I think Kelley would feel the same way about getting 9:1 on a coin flip as getting even money on a 9:1 favorite, but I could be mistaken. I'd certainly feel different about the situation, and it definitely would affect my answer because the expected value of my utility (or smileys....) changes drastically.

I'm pretty sure nobody with a $100M net worth would bet $90M on either case, but I'd think they'd be even more hesitant to take the 9:1 odds on the coin flip.

Wacky said...

I think you happen to be thinking about this the wrong way. A person with little net worth should be willing to bet all of his money, because the increase in utility is much greater for him. As your current utility increases, the marginal increase in utility decreases exponentially. From a pure EV perspective, the correct answer is to kelly bet, but from a utility point of view, there's come a point at which if you have enough money, there is no point in making the bet at all.

The blindman said...

Different people have often dramatically different utility functions. It obviously depends a lot on your income and how secure your job is, as well as your dependents and your financial obligations.

I would not wager my house at 9:1 on a coinflip, because the downside would be devastating. I probably wouldn't wager it at 19:1. I might think about it at 99:1 if I was certain the coin was legit.

I suspect that most utility functions look like a logarithm close to the origin, but flatten into something more or less linear at high values. I imagine that a billion dollars to someone like Warren Buffet is a billion dollars, whether it's his third billion or his 30th.

Steve said...

"from a utility point of view, there's come a point at which if you have enough money, there is no point in making the bet at all."

This is the dumbest thing I've ever heard. I'm amazed that you stated the correct facts and came up with the exact opposite of the correct conclusion.

Wacky said...

Maybe I didn't articulate my point clearly enough. If you plot "a" normal utility curve is reaches a point at which it becomes very flat. In other words the first derivative becomes 0. So from the perspective of increasing or decreasing your utility, even if you kelly bet, win or lose, you are at essentially the same point on the utility curve. So what's the point in betting in the first place? Where's the utility gain?

I guess another way of looking at this, is to ask yourself, should I lose this bet, what remaining utility am I willing to retain? This is more on the lines of bayesian risk analysis, where define a cost function for each of the outcomes.

I think it's an interesting question, which mixes a variety of interesting topics ranging from optimization theory, economic theory, and psychology.

And if you still think this is the "dumbest" thing you have ever said, I can live with that. I've been called worse.

The blindman said...

It's quite wrong to assume that marginal utility approaches zero in the limit. While marginal utility is generally assumed to decrease as money increases, it's perfectly reasonable for it to approach a nonzero finite (positive) limit.

Maybe talking about limits is a bit pointless when it comes to money though, since it is necessarily a finite quantity.

RFC said...

grammar nittery: inherent