Expected Value and Risk Aversion

Author: Poker Penguin

Summary: Expected Value isn’t as simple as we think

Welcome to my second article on economics / game theory.

People value money. Everyone who hasn’t taken a vow of poverty would rather have more money than less (Duhh). However, people value risk differently. For example. Let’s say you won a magical prize from happyland. Would you rather have $100 or a 50% shot at 200? The Expected Value (EV) is the same, so if you’re a good poker player, you would say “it depends”.

Actually, good poker players seem to say “it depends” a lot. But anyway. That example isn’t very interesting. But what if the choice was between $100 and a 50% shot at S201? Technically, the EV is marginally higher if you take the coin toss. However, many people will give up the 50 cents in order to get a sure $100. They are what we would call risk averse.
The opposite is of course, risk seeking. These people would take a coin toss for $199 over a sure $100. Well, in theory anyway.

This is all very well, but as one of my lecturors used to ask whenever people brought up interesting facts that didn’t support their argument – so what?

Well, people’s attitude to risk directly affects their poker play on both the big and the small levels. For example. I love fishy gambooling, raise it up, what-the-hell games. I positively grin when I see a capped family pot. Why? Because I know that long term, my solid play will get me the money. A friend of mine, on the other hand, can’t stand it. He starts complaining and table-hopping when the maniacs start taking over. Why? Because he prefers not losing big to potentially winning big.

There is a trade off between EV and risk. Where we individually consider the optimum point to be depends on how we value money. Unfortunately, for most of us, this isn’t a straight line.

Consider if you personally would accept the following (no welching, or fat Tony cuts off your thumbs and other dangly bits). Be honest now.

Coin toss, win $11 versus lose $10
Coin toss, win $110 versus lose $100
Coin toss, win $1100 versus lose $1000
Coin toss, win $110,000 versus lose $100,000
Coin toss, win $1,100,000 versus lose $1,000,000

At a guess, even with a 10% advantage, there were few of you with the stones to wager a cool million. The scale of the advantage is the same, but the consequences of losing are bigger. However, if you were a very big bankroll player, you might well be willing to wager a million, or more at these odds. In the large scale (life altering sums of money), we’re almost all somewhat risk averse. One of the keys in bankroll attitude I think is knowing when to ignore the risk aversion and focus on the EV, and when to ignore the EV and focus on safety.

For example, in Lock Stock and Two Smoking Barrels the boys ignore safety in the pursuit of one shot at what they thought would be a +EV game. It didn’t work out for them (and in real life that could just as easily have been a run of bad cards as a crooked gangster that did them in). Were they right in pursuing EV and ignoring the risk? I think that they were. On a related note, they didn’t accurately assess the EV of the game (which is becomes more and more important as the stakes get higher).

An example of EV over risk that worked out is Benny Binion (of the Horseshoe and WSOP fame). He made the famous offer of allowing people to bet as much as they wanted at his casino. Multiple exposures reduce the variance / risk. A million coin tosses at $1.1 to $1 is far far better for the house than one at $1.1Million to $1Million, but Benny didn’t care. He wanted the greater EV

The core attitude here is the same – get as much money in the pot as you can when you’re the favourite. But in the long term, you might get more money in the pot as a favourite if you do NOT push small edges. For example, you’re playing a NL freezeout heads up with 25cent and 50 cent blinds and a $10,000 stack each. Your opponent pushes allin, every single time no matter what you do. Would you call him with QJ? This might just be the time he’s got KK. I’ve made this point before, but I think that against really bad opponents, small edges are worth much less. Would you rather have 100 coin tosses at $1.1 to 1, or one at $400 to zero?

Obviously that isn’t an exact parallel to a tournament game (since one loss pushing a small edge in the NL tournament situation is all you get), but the point is the same. If you are a better player than your opponent, then in marginal sitations, I believe that the money in your stack could almost be considered to pay interest. This means that the cost of getting involved in a coin toss should be raised to include lost chances in the future. In other words, it’s a bit like Phil Helmuth. He is willing to lay down even a slight edge in coin toss situations because he feels that he can get the better of you in the long run. Which is the exact opposite of Doyle Brunson’s attitude that his aggression at stealing blinds allowed him to take the marginally unfavourable coin tosses.

Who am I to say which is better? Well, I’m a jumped up little punk with delusions of competance. I’m also arrogant and opinionated. I think, that as much as you might dislike Phil Helmuth’s attitude, he’s got the right policy on risk avoidance here. If you think you’re enough better than your opponents, maybe laying down percieved coin-tosses might be right?

Obviously a lot of this assumes a limited bankroll. If you’re a billionaire (US or European) playing $1-2, then sure, play strictly mathematically correct poker, and run those 1% edges. But, if you’re an average guy, with a non-infinite bankroll, then I suggest some edges are too small to be worthwhile.

On a related note, in my view, many of the poker writers are assuming that you are playing another reasonably good player. In reality, at the limit most of us play, I think that sometimes even assuming our opponents are sane is stretching it. Unfortunately, I often remember this at the wrong times – just after they’ve shown some bizzare hand.