Applying Game Theory to Gambling is Silly

I was driving to work today and realized something kind of interesting.

People will talk about gambling in terms of game theory. Often times they will go as far as to claim that lotteries are for stupid people, and since casino odds always favor the house, gambling in a casino is universaly stupid too unless you’ve found a way to shift the odds in your favor.

However, I realized that a literal application of game theory to real life gambling is silly at best, and wrong at worst.

Here’s my reasoning:

a) Direct application of game theory assumes that monetary pay outs are linear, i.e. $1000 is 1000 times better than $1. However, this would only be true if your lifetime was infinite and nothing ever changed. First and foremost, growth rates on investments in real life are non-linear, and an inverstment of $1 would certainly have a growth rate lower than inflation. Second, since your lifetime is finite, you care about the gross growth amount of your investment almost more than the amount of investment itself. Having a $1,000,000,000 in general deposit accounts will yield a lot more interest than having $1 in a deposit account. Since you only have a few years to live anyway, $1,000,000,000 turns out to be a much better thing to have than something that’s a 1,000,000,000 times better than $1.

b) It also assumes that loss value is also linear. Losing $1 1000 times is the same as losing $1000. However, since life is finite, losing $1 1000 times is a much better loss than losing $1000 once. Interest and appreciation for one, emotional effect as well. Also wealth has uncertainty, and small amounts are within that uncertainty. In fact, it can be argued that spending less than a certain percentage of your net worth is equivalent to spending nothing. Since spending requires an action, an action that is going to psychologically be compensated somewhere else (a smaller donation, smaller tip, cheaper purchase option, shorter phone call, etc.)

c) It assumes time is worthless. When you gamble, you spend time doing so that could be spent making money (or not).

d) It assumes enjoyment derived from gambling is worthless. For a lot of people, gambling is the past time they need to get away from it all, without which they would require health and travel expenses that could be greater than that of gambling losses.


Cheap, fun gambling is profitable even if odds are in favor of the house. Yes, this even applies to lotteries, because the $1 spent on a lottery is essentially $0, yet the jackpot winnings will change your life.

Lotteries aren’t a tax on the stupid, they’re a license to dream.

Slotmachines lose a LOT of their luster when you own one. Run out of cash? grab the key and get another handful.

I’m fairly lucky in that if I lose a small amount of money ($80 or so). It HURTS. And I stop.

Then I go back to the three hand blackjack game on the cellphone and clean house.

Thsi really should have been Item A. No assessment of the value of gambling can possibly be correct without factoring in the utility derived from entertainment.

The casino is FUN. The track is FUN. I’m good at math, I know the odds.

Well, to be faaaair, it’s not so much game theory as it is utility theory and probability theory with some financial engineering in there somewhere. Game theory is more along decisions between multiple parties, and the lottery isn’t really deciding between anything.

But anyway, yes, every point you make is true, and can be explained using utility theory.

OR I could be a real nerd and explain how spending $1, 1000 times has a forward value greater than $1000, or how most people are risk adverse and whose second-order derivative of utility is negative, or how the reason why losing $1 seems so insignificant is because we are weighing it against our current total net worth, or how utterly boring this subject seems to be to everyone except me…

But don’t blame game theory, it was never mean to you. :slight_smile:

No math whiz me :dubious: but will you expand on this? If I don’t have $1000.00 but I do have $100.00 is spending $1.00 on the lottery a bad choice for the possible gain?

I know I’m seriously bad at math / game / logic theory. I’m the everyday, joe lottery guy you would need to explain this to in layman’s terms.

OK, utility theory sort of works like this:

If I start out with $0 and somehow come across $100,000, I’m pretty happy. Really happy, in fact; all my problems seem to be solved. Let’s say I find another $100,000. I’m up to $200,000. It’s good, I’m happy, but not as happy as when I was initially broke. The gain in “happiness points” is greater from $0 to $100,000 than from $100,000 to $200,000, even though the money gained is the same.

That’s when trip over a Chia pet and make $1,000,000,000 in the resulting lawsuit.

So I magically come across another $100,000. Ho hum, I think, that’s chump change, and hand it out to the next homeless guy on the street. Same amount of money, but worth way less to me. Practically worthless, in fact. The more money I have, the more I have to gain in order to feel the same effect in “happiness points,” or “utility.”

So let’s take this and show how this applies to gambling. So a random casino offers you this bet: 50-50 chance of either losing your billion, or gaining 3 billion.

Half the time, you’ll end up with $0. Bad deal, don’t want to go there. Half the time, you’ll end up with $4 billion. That’s good, I suppose. So on average, you’ll end up with $2 billion. It seems mathematically good, but we’re not gonna do it. Why?

Because going from $1 to $4 billion doesn’t really affect me that much. We gain 3 billion, but there are only so many yachts and Pacific islands we can buy. On the other hand, lost just $1 billion and we lose our entire way of life. And it really is the better plan to take. People tend to be risk adverse, meaning that sometimes they’ll avoid taking risks that on average, would be profitable. We tend to stick with the safe route.

So in your example, let’s say we have a 0.1% chance of winning $900, with a 1 dollar ticket. That means that, 999 times, we’ll end up with $99, and 1 time, we’ll end up with $999 dollars. So let’s see what this means in terms of happiness:

$99 is say… 50 happy points.
$100 is … 51 happy points.
$999 is … 300 happy points.

Play the lottery:
99.9% chance of ending up with 50 happy points,
0.1% chance of ending up with 300 happy points.

On average, we’ll have (0.999)(50) + (0.001)(300) = 50.25 happy points

On the other hand, we can have a guarenteed 51 happy points by sticking with the $100. If these values of money make me happy like I defined above, then we should stick with the $100. It all depends on how happy certain amounts of money make us. It turns out that if $999 gives us 1000 happy points, than we should do the lottery. However, that’s unlikely to happen because as we pointed out above, money tends to get less valuable the more you get it. If $100 is worth 51 happy points, $1000 is probably not going to be worth more than 51 x 10, or 510 happy points.

So for the most case, the lottery is a bad idea, money-wise. However, maybe it gives us a happy point to play the lottery in the first place. That would make playing the lottery give us 51.25 happy points on average, and would make it the smart choice. There are so many factors other than money that tie into our happiness, and we have to take those into account when finding our average gain, not just our cash gain.

That’s sorta kinda vaguely utility theory. In a nutshell.

Holy Dear God which I don’t believe in but I think may be real when faced with people who actually understand stuff like this: Thank You! Thank you for trying to educate me! Thank you for putting people like Turing Complete here to try to fight my ignorance. I followed for about half of the explanation and then pfffft my brain imploded.

Thanks man, I want to understand, just don’t know if I can.

This is the difference between why I don’t play the lottery (or really gamble) and why my friend does. I look at the poor probabilities and don’t get excited about the possibilities of winning. My friend enjoys playing the lottery, even though he and I both agree that the odds are stacked against us.

That said, I spend much more money in bars looking at “getting lucky” than he does on lottery. :smiley:

Wrong. You are free to model the positive payoffs so that they are not linear.

Wrong. You are free to model the negative payoffs so that they are not linear.

Wrong. You are free to put time into your model.

Wrong. You are free to include the utility of risk in your model.

That depends on how you’re defining your terms. I worked w/ a guy who would play nickle slots w/ his wife one night a week. They were members of some sort of players’ club, which meant that they got cheap dinners & drinks. Over his lifetime he could expect to lose, say, 20% of what he gambled; however, he did get cheap meals, a nice night out every week, and time w/ his wife. All that is not excluded from game theory, it just isn’t generally included because explaining things to undergrads involves simplifications.

VegemiteMoose, think of it like this: You’re a bum. You get $5. You’re happy because you can buy some food. That $5 is worth a lot to you. … Now you’re a college student. You get $5. You’re happy because you can buy a pitcher of beer. That $5 is worth quite a bit, but not nearly as much as it would be if you were a starving bum. … Now you’re a successful engineer. You get $5. You’re happy because you can buy dessert to go w/ your lunch. The $5 is nice, but no big deal. … Now you’re a prominent attorney. You get $5. You’re happy because you can give it to some bum. It’s nice to help out bums. The $5 means a lot more to the bum than it does to you.

That’s not utility theory proper, but the notion that utility is concave with respect to income. Do pictures help? If I can I’ll draw one to illustrate just what is going on (as best I can).

Or just run a GIS for “exponential decay”. Granted, you need to imagine the axes to be labeled differently (X=Your net worth, Y=utility of getting five bucks). As your net worth increases, the amount that five bucks holds utility (let’s call it usefulness or happiness) is decreasing. It decreases a lot when your net worth is five bucks as opposed to $105 (far left on the decay chart), but it decreases very little when your net worth is $100,005 as opposed to $100,105 (far right on the decay chart).

Turing Complete great post. I knew some of that intuitively, but never would have been able to make it so clear mathmatically.

Responding to those calling the lottery a ‘tax on the stupid’, I like to call it…

Paying my dream tax.

I really do that. I’ll pull up to get gas and while the pump is running, I’ll tell the wife that I’m running in to pay my dream tax. She’s learned to just smile and humor me.

Applying game theory to gambling is actually rather interesting if the questions you ask are interesting. When used to model aggregate behavior, you can learn a great deal about the preference and strategy profiles of gamblers and can test prospect theory and constrained rationality very nicely.

Applying game theoretical tools to guide your own behavior is indeed silly. The fault is yours, not the tool’s.

By the way, if you are interested in game theory but have little mathematical background, Martin Osborne’s An Introduction to Game Theory is not too bad. It does have formal proofs, but there are chapters that are pretty accessible to people who want to get the gist. It is possible to get some good intuition without delving into the proofs.

Well, yeah, you’re graphing marginal utility, but I suspect that’s not as clear as total utility w/ respect to income. A picture of utility w/ respect to quantity of a good consumed is here. Hopefully the explanation, as part of an online textbook, I think, will make sense to VegemiteMoose. If not, we’ll try a tifferent tack.


And I hesitate to mention it because 1) it’s quite dense, 2) it’s relatively specialized, and 3) it’s perpetually out of any major printing so it costs $50-60 for the 300 page paperback, but An Economic Theory of Democracy by Anthony Downs has pretty well been the bible on Utility and Game theories in the American political arena for the past fifty years.

Shepsle and Bonchek Analyzing Politics is a much more accessible introduction to formal theory in politics.

I prefer to work in the realm of marginal anyway, especially since I read your bum/beer/dessert/lawyer example in terms of marginal utility.

Not to say either of us is wrong. Heck, yours really is clearer (and illustrated, to boot). Just trying to give a different understandable tack (yup, tack) to increase the utility of the lesson learned, even if you do get the lion’s share of credit :wink:

Downs 1957 is one of the most cited and least read books in all of political science. Great book, massively out of date. Game theory has come a hell of a long way since the 50s.

Never read Shepsle and Bonchek, so I’ll have to check it out.

I re-read this thread (hey it rhymes!) and I now at least get the gist of the money / utility idea. It was trying to put happiness into a mathematical model that threw me off at first. My brain decided that it just wasn’t going to understand that and there was nothing I could say to talk it out of it’s intractabilty.

So thanks to all. Consider ignorance (semi) fought.

Also, this part of the online text made me laugh:

Yeah, I last read it for a college course three years ago or so, and it was tough for me then (with little to no grounding in the theory already), so I would not be surprised to find error in it now. Especially the last third, about the utility for the individual voter to seek out information and be informed. The individual’s desire toward civic engagement has drastically changed (cite: my current DC Metro reading, Robert Putnam “Bowling Alone”), which would almost definitely undermine that entire chunk of the book. And the rest is the Hotelling/Median Voter Theory, which has also changed as mainstream participation in politics has waned and the extreme pundits on either side grown more influential/rabid.

In other words, I can imagine you’re perfectly right without re-reading Downs.

But I did read it before ever trying to cite it. And I can attest that it has been taught as the Word as recently as 2002.

This thread has quickly digressed away from the OP’s (weak) intent, and become a prime example of what I love about this board. Digression becomes a geekfest on a neglected corner of academia.

Aha! Good eye, I had never thought of that! I am so used to thinking in terms of the cumulative utility graph that the one you presented threw me for a loop. My bad!

I don’t think I do; hell, you’ve even opened my eyes to something.

That is actually a good illustration of my economics education; periods of increasing confusion and distress, which end with a brilliant flash of insight and understanding. In retrospect, that may be one of the things that made econ so rewarding.