Tuesday, 13 September 2011


There are 3 rough categories of "games". Games here in the sense of game theory, a branch of maths dealing with massively simplified and anaemic economic situations. In a game there are some number of players, they all make a decision and walk away with some amount of happiness depending on what everyone decided. Playing roulette is such a game. You turn up and call out a number and leave with (on average) less money than you started with. My question is why people play each category of game.

First, the 3 categories. The key quantity here is the "sum" of a game. Ie how much the game itself adds to the total happiness of everyone playing. There are three types:

  • Positive sum: on average normal rational players leave the game more happy in total than they arrived. Some individuals may be less happy but on average more happiness has entered. 
  • Zero sum: when normal rational players leave the game they have in total as much happiness as they stared with. Happiness has been transferred around but there's as much of it as the start.
  • Negative sum: normal rational players end up with less happiness on average. Some individuals may be more happy but on average happiness has been destroyed.
Notice that I'm talking about happiness or utility here. A lot of people get confused, look at roulette and say that the house always wins money so it is a negative sum game, this would be true if it weren't for adrenaline. Some people enjoy the mere act of gambling regardless of the money they win. They may be just as happy gambling for 10 mins as having $100 in the bank, if so the fact that they loose only $50 every 10 mins means they have increased their happiness, they are playing a positive sum game.

 A huge number of failures of thinking come about by people confusing money and utility/happiness. I care here (and in general) only about happiness. Money is not (outside of the mint and weird art projects) created or destroyed. And so people get a false idea if they are used to equating money and happiness, or they have been confused into thinking that the acquisition of money per se is the end-in-itself we ought to strive to. They get the idea that all exchanges are zero sum. That anything one person gains the other parties must have lost. From this some conclude that in some situations where one party can guarantee a profit or positive return then logically the other players must be in a negative sum game. This confuses me. As I dont see why a sane person would consent to play a zero-sum game. Much less a negative sum game.

Some examples of each type of game.
  • Negative sum. Roulette with a fixed "playing charge" at least as valuable as how much you enjoy the act of gambling. A "manliness contest" where the person who gets beaten up the most wins esteem less valuable than the beatings everyone got. 
  • Zero sum. Poker with a fixed "playing charge" exactly as valuable as how much you enjoy the act of gambling. Cutting a cake that everyone wants the same amount. Elections.
  • Positive sum. Gambling if the enjoyment is greater than the losses. Employment, bartering, friendship, public policy.
It is strange to think that anyone would willingly play the first type of game. I can see no reason why you would willingly make yourself less happy with with hope of an eventual payoff. The people who think anyone would do this or think they should campaign against real-life instances of it have to ask themselves, do you honestly think people willingly do this? Because you must either think people are insane or coerced, or you must just be wrong.

One reason why people would play zero or negative sum games is straight ignorance, people who are bad at gambling may not realise there're going to loose a lot playing roulette. Another is the reason most people play poker, they believe they have a skill, they believe they will win on average and others will loose. This is strange. To believe this you must believe that either other people are ignorant, which wont last long, or they know you're a better poker player and will play you anyway. This is a rather strange belief to have about people we assume to have no enjoyment in the playing per se. 

So what about real examples outside casinos? What about obviously zero-sum things like barter? I give you two apples you give me an egg, you win what I loose. This is the most obviously zero-sum thing ever. And anyone who's read a bit of second-hand Marx and not thought since then can tell you why. Trade exchanges "value" for "value", "value" as in work done. Obviously it's only fair to exchange things of equal "value", and trade doesn't put extra labour into our commodities, so obviously trade will keep "value" constant. ... Except that if our idiot Marxist had thought for 30 seconds and realised that "value" is totally worthless and that the thing that is actually an end-in-itself is happiness or human satisfaction he wouldn't care about such metaphysical fluff.  He would care about the happiness.

I have apples, lots of them. I like apples as much as the next guy, but that doesn't mean I want 500 apples 50 times as much as I want 10 apples. 500 apples are actually not as nice to own, I cant eat them all before they rot and they'd be hard to get rid of and would smell. So in fact when I give up my apples I'm not giving up "value", I'm giving up negative happiness. And the guy who has eggs is in the same boat. We have a trade that makes up both happy. It's never true to say that people exchange things of equal worth. If I have apples and so do you we dont trade them. Nobody actually barters things of the same worth, what would be the point?

Once you actually think about things that matter (like happiness) and not about things that dont matter (like "value") or things that only matter instrumentally to get things that do matter (like money) things become clearer. It becomes a lot harder to get outraged that society has set up situations that rob people if in fact it hasn't, and it becomes a lot easier to point out that people are being conned or are just plain confused when they are. Luckily, almost all human interactions are positive sum, so lets carry on interacting.


  1. So here's why it's very difficult to talk about 'positive sum', etc when talking about utility.

    To talk about games, you need to have something constant to measure against. It's very difficult to find the right constant when it comes to utility, as it relies on so many factors.

    One obvious constant is that sum of utility on the planet = 1. Obviously with this, everything is zero sum.

    A more contrived constant is that *amount of happiness you would gain if you had 1000 apples*= + or -1 (you seem to think having more than 500 apples would be a bad thing, I think there's some happiness in knowing you could spend them all on a binge egg fest if you wanted). With this, your transaction is negative sum, as '1' gets bigger.

    You seem to want to say that your constant is something like *the utility felt by Queen Victoria on the 12th June 1895 at 2pm*=1. With this, yes, your transaction is positive sum, but I'm not convinced this is the right constant, I think humans do feel jealousy and so if some people prosper, others will suffer because of it.

    I think the real constant is somewhere between Victoria and sum=1, but not clear where, so it's very difficult to be sure things are positive sum.

    'the utility felt by possessing $1'=1 is a very convenient, easily measured thing to choose, as well as being a fairly good estimate. In the real world, it's a very good choice for measuring happiness.

    On the whole, a very good post, but you can't really band about mathematical terms like that :P

  2. I agree utility is very complicated and depends on many things. But I'm not sure that should stop us from drawing the conclusions I propose.

    Minor point, the utility of owning 500 apples if there is a market willing to buy them (ie the utility of anticipating having things you want) is very different from the utility of having 500 apples if there is no such market.

    Sum=1 utility seems obviously insane to me. The thought experiment that springs to mind is 2 people left on earth. Alice is not a sadist, but has a weak conscience, inflicting pain on Bob is less painful for her as it is for Bob. So obviously it's in her interest (to maximise sum1 utility) to torture Bob as badly as she humanly can, even if it makes her miserable to get the greatest proportion of the 1 util. ... This obviously doesn't make any sense. It also stops us wanting to engage in any kind of social or ethical project to make the world better, because we cant by definition. I dont see how a sum1 utilitarian could ever be convinced to act in any way that didn't make others unhappy.

    I'm not sure Queen Victoria utility is so flawed as you suggest. Yes there is such a thing as envy, but I'm not convinced it will automatically balance everything. If so then I agree with you. But that a very bold empirical claim that needs a lot of evidence to back it up.

    I agree 1 util=$1 is very convenient, for normal situations it's a very good approximation to Victoria utility. But it does break down in strange situations. For large sums it breaks. Few people would give up a million dollars to get a 1/million chance of wining a trillion. It also breaks down if you've not already got a stable market to make the utility of $1 well defined.


Feedback always welcome.