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Game Theory column (New York Times) by J. C. Herz
Game Theory by Anne Enthusiast
(You do not need to know this to play "6 Billion")
I have a theory that is mine. My theory is that academics that use Game Theory don't know anything about games, and game players and designers don't know anything about Game Theory. Books on Game Theory are full of excellent ideas which could be used in games, and yet nobody (not even my hero Reiner Knizia, a Doctor of Mathematics!) has designed a single decent game using Game Theory principles. If you can think of any I'd like to know about them - honest. I'm happy to be wrong here. Some books on Game Theory do, so I'm told, contain some quite interesting games - Playing Politics by Michael Laver ( Professor of Political Science at Trinity College, Dublin) is apparently an example.
OK - perhaps I should start at the beginning. Game Theory is a branch of mathematics that deals with strategic problems that relate to such things as politics, commerce, warfare and - more recently - biology and sociology. The theory was developed by Oscar Moregstern and John von Neumann during World War II, and published in 1947 in The Theory of Games and Economic Behaviour. In the political field, Oxford philosopher Michael Dummett has written a pretty readable book on voting procedures, called - imaginatively - Voting Procedures. Michael Laver has designed a game, called "Agenda", which draws on these ideas. There is apparently a vast literature devoted to the design of "fair" voting systems, and the strategic voting problems that can arise in the different possible voting systems one can choose. The absolutely seminal work in this field is Kenneth J. Arrow's Social Choice and Individual Values, in which Arrow's Impossibility Theorem is set out: five very plausible and desirable constraints about how any social choice system should use the preferences of the individuals in a group to calculate a preference for the group which, as a whole, turn out to be inconsistent with each other! Thus, if "fair" is defined to mean "satisfying Arrow's constraints", then a "fair" voting system is impossible.
The extension of the theory into biology and sociology is generally attributed to the likes of John Maynard Smith (and his concept of the Evolutionarily Stable Strategy, or ESS) and further expounded by the likes of Richard Dawkins in The Selfish Gene and Matt Ridley in The Origins Of Virtue. These two books I have read and can personally recommend. Whatever else I have read of Game Theory I have read on the Internet.
One of the most well-known expressions to come out of Game Theory is the Zero Sum Game. A Zero Sum Game is one where for any action, if you add the potential payout to the potential penalty, the sum is zero. Suppose you and I toss a coin to resolve a bet for $5. Heads I gain $5, and you lose $5 (you gave me $5). Tails you get $5, and I lose $5 (I gave you $5). In both cases, the payout added to the penalty produces a zero sum. This is neatly expressed as "one player's loss is another player's gain." If all possible cases in a game adhere to this rule then the whole game is a true Zero Sum Game game. This is often reflected in games with only one possible winner (such as chess), but it does not necessarily follow that a Zero Sum Game must have only one winner. Trade is a good example of a Non-Zero Sum Game as both parties in an exchange can win. Many games do incorporate game mechanics such as trade, and therefore incorporate Non-Zero Sum Game elements into the game (Settlers Of Catan is a good example).
The basic idea of Game Theory deals with how people employ strategies with the greatest payout and the minimum cost. Early "games" that use these theories are the "Prisoner's Dilemma". Here, two prisoners faced with a heavy sentence (say 10 years) are encouraged to give evidence against the other. Whoever does so gets a reduced sentence (say 5 years). It's a dilemma because, if neither defects on the other (they remain silent), they both get a reduced sentence (say 8 years) as the police do not have sufficient evidence for the heavier sentence to get passed. In theory they should co-operate and stay silent, but in practice one "player" usually defects. To create your own version of Prisoner's Dilemma is very easy. The situation, and the parameters, can be changed very easily so long as the result is that, regardless of what your opponent does, you are better off defecting. Interestingly, if the game is played repeatedly, co-operation becomes a more likely strategy. Or, if the game is played with various opponents, co-operators tend to quickly establish who the other co-operators are and play the game with them. Many excellent examples are detailed in Chapter 12, Nice Guys Finish First, in The Selfish Gene. This chapter explores concepts such as reciprocal altruism in real life, and uses variants of The Prisoner's Dilemma to do so. Chapter 9, Battle of The Sexes, deals with fun little issues like sexual fidelity.
In 1979, American political scientist Robert Axelrod challenged people to submit a strategy for a tournament which would pit these strategies against one another in Iterated Prisoner's Dilemma. Strategy is defined here as "blind unconscious behaviour program". The surprising winner from 14 submissions (plus a purely random strategy) was also the simplest, known as Tit-For-Tat submitted by a Canadian political scientist, Anatol Rapoport. Tit-For-Tat starts by co-operating, then copies whatever the opponent did last (co-operate, or defect). So, if the opposing strategy defects on one turn, Tit-For-Tat defects on the next turn (hence the name). A subsequent tournament with 64 submissions failed to beat Tit-For-Tat. Interestingly, Tit-For-Tat doesn't win many games (it depends upon the opposing strategy), but managed to accumulate the highest overall points in both tournaments.
This was the beginning of what is now known as Artificial Life, which started in a survival-of-the-fittest struggle on a scientist's computer. Such scientific models have subsequently been used to research selfishness versus co-operation in individuals and societies. Tit-For-Tat is regarded as a "nice" strategy because it discourages defections from co-operation, and does so consistently and clearly. However, nasty strategies which take advantage of such niceness can easily beat Tit-For-Tat in a one-on-one. And if two Tit-For-Tat players come face to face you could end up with real-life situations like World War I, or blood feuds. Once one defects, then each will engage in destructive mutual tit-for-tat defection for the rest of the 'game'.
More sophisticated games from the field of Game Theory are "games" like "Hawks & Doves", invented by John Maynard Smith. Hawks and Doves are terms used to describe 2 simple strategies employed by members of the same species. In its simplest form, Hawks & Doves assume that none of the members of the species can tell which strategy an opponent will employ until they fight the opponent (they also do not remember previous fights). Hawks beat Doves whenever they fight (& the Doves quickly run away), but Hawks are wounded by other Hawks. When a Dove fights a Dove the fighting is more prolonged & ritualised, and nobody gets hurt. Also, the odds for a win in a Dove versus Dove contest are 50-50. Taking my arbitrary scores from The Selfish Gene, allocate 50 points for a win, 0 for losing, -100 for getting injured and -10 for wasting time fighting.
Without getting into all the mathematics here, the average payout per fight in an all-Dove population is +15, so the average Dove does quite nicely. However, suppose a single Hawk is born into the population. Its payout is always +50 (it always wins without wasting time) so, in Dawkins' example, the Hawks genes will spread throughout the population. However, in an all-Hawk population, the average payout is -25. Supposing a single Dove is born into this population, its payout will always be 0 (it never wins, but it never wastes time). Hence, as the Dove strategy is now the best, its genes will spread through the population. Hence, the expectation is that populations will oscillate between the two extremes of all-Hawk or all-Dove population. However, for the scoring system used (and you can easily invent others), a population ratio of 5/12 Doves and 7/12 Hawks is proven to be stable - this is an ESS. Also, the average payout (irrespective of whether you are a Dove or a Hawk) is +6 1/4. This is not the optimum average payout, as you can see, but it is the most stable outcome. A similar Game Theory argument can explain why many sex ratios (not just for humans) are roughly 50-50.
More sophisticated strategies include the Retaliator (acts like a Dove to a Dove, and a Hawk to Hawk, and a Dove when it meets another Retaliator), the Bully (acts like a Hawk until attacked, then acts like a Dove), the Random Retaliator (50-50 chance of acting like a Dove, or a Hawk), and so on. Retaliator is a very successful strategy, much like Tit-For-Tat.
Hawks & Doves is one example of what John Maynard Smith calls a symmetric contest, where the contestants are equal except in the strategy they employ. Asymmetric contests between uneven contestants are also possible, and yet these also result in an ESS.
Anyway, back to my theory. Game Theory is used very seriously (usually in computer simulations) by scientists in fields such as economics, warfare, sociology and biology. There are lots of very serious web sites about Game Theory. Game Theory really does provide some very useful insights into real life. But all these scientists are not actually interested in games (not the way we are, anyway!), and game players are not overly interested in science.
One of many great examples of real-life strategies from The Origins Of Virtue is Blood Bat Brothers. Vampire bats, which nest in large groups, regularly display reciprocal behaviour by regurgitating blood for those who have not fed and so beg for food. Cheats, who simply beg for a living, should have an easy life. However, because the bats live for up to 18 years, and usually nest in the same place all that time, they get to know their neighbours. They know who the cheats are, and refuse them blood. On the other hand, they are proven to return favours. On the other other hand, new cheats are not known and should have some initial success. And what about occasional cheats? If they only cheat infrequently, how often should they get away with it? Couldn't this be a nice idea for a game? Perhaps a bit bizarre like Fresh Flesh, but it could also be fun. A similar theme which occurs to me, one perhaps more familiar to my fellow man or woman, is the concept of shouting rounds of drinks in a pub. "It's your shout", or "It's your round", we say. Being very social animals, with good memories, we know who to buy drinks (trusting them to return the favour) and who not to buy drinks (perhaps they always leave before its their round...).
Wake up you budding designers, and "shame on you" to the established game designers! I believe that thoroughly entertaining games (of the type we all love) could be created which actually have a useful bearing on reality. Why not have fun and learn something useful at the same time? Such games need not incorporate obvious, or limited, strategies for the players. Game Theory is full of subtlety.
What do I mean by useful? Wargames are very serious and, to my mind, are one of the most useful forms of gaming. Useful in bringing history alive with not just what happened, but what could have happened. They therefore increase our understanding. The lessons on how to wage war are perhaps debateable in their desirability or usefulness. Recent games in the excellent We The People series also add historical events which affect the political outcome. Business games (which I often find a bit dry for my own tastes) do teach some rudimentaries of economics and commerce. Abstract games (which I rarely play) teach abstract skills like recursive thinking (in chess - what's my best move?) and pattern recognition (you can't go past ... well, Japanese Go). And all games can teach us how to assimilate rules, how to win, and how to lose... Now, I'm not saying that every game should teach us how societies interact, or animals interact, or even how people interact. I've thoroughly enjoyed many a game of Bohnanza, for goodness sake! Silly game ideas are not at all instructive but huge fun. Quite right, too!
But surely some bright spark out there, hopefully someone eminently qualified in both fields (boardgames, and mathematics) such as Reiner Knizia, could design a game using some of the principles and ideas from Game Theory on things like the eternal struggle between co-operation and selfishness. Even novice game designers could draw some inspiration from learning how Game Theory applies to aspects of life. Something which illuminates, rather than just entertains. Perhaps somebody has done so, and I've missed it? Yes, I know many games employ concepts of balancing competition and co-operation (such as Diplomacy, or Cosmic Encounter). But the purpose of such games is to entertain. The purpose of games designed by those that study Game theory is to teach. I'm greedy - I am asking for a game designed for both purposes, and I'm saying that both fields (boardgames and game theory) could learn a thing or two from each other. The game must compare in playability and production with a typical German game, be useful as a demonstration of some Game Theory principle, and help our understanding of life. Perhaps it's an impossible ask? Well, then at least a game that educates and entertains, which appeals to both game players and game theorists.
6 Billion, my own design, was designed to make people think about our situation today (generally depressing) and our transition from Earth-bound species to a species which has broken into the huge arena of our solar system (which to me is a wonderfully uplifting idea). Thus, my design intent was to inspire a little optimism and hope. Not everyone loves 6 Billion (there, I admitted it!), and I'll be the first to state that 6 Billion was NOT designed with Game Theory principles in mind. However, cards which can be played to help an opponent are one unusual feature of 6 Billion which most reviewers have commented on favourably. The player of the card scores points for doing so, and must balance this with avoiding helping their opponent too much by playing the card. Novice and selfish players often fail to see how this can work in the game, but it does work in fun and surprising ways. I think, in a way, I unintentionally included a mechanism along the lines of selfish play versus co-operative play which also introduced an element of a Non-Zero Sum Game (both players can get something) into what is, essentially, a Zero Sum Game (with normally only one winner). Still, I didn't deliberately set out to design a game using Game Theory principles or subject-matter.
Imagine the possibilities if a game were designed intentionally to both entertain and educate, by deliberately incorporating recognisable aspects of Game Theory into the design. Hopefully, at the very least, we might introduce a whole new audience (game theorists from biology, sociology, computing, economics, political scientists and so on) to our wonderful hobby. It would be a bonus if such a game could both educate and entertain students learning Game Theory. If the game also demonstrated how vampire bats, or politicians (is there a difference, I wonder?) behave then so much the better! I think we undervalue our hobby if we insist it is only there to entertain. Like all of you, entertainment is probably my main reason for gaming. At the end of a long day, I don't necessarily want to think about the future of humanity (a game of 6 Billion anyone?), or the endless "game" of hunter and prey played out on the African plains. But I would like the choice, because some days that is exactly the sort of game that I do want to play. Such games, given respectful treatment of the subject, would earn respect for our hobby. Where are those games? They are too few, for me.
There have been some excellent attempts at modelling limited aspects of reality in playable games such as Britannia, History Of The World, Civilisation and so on. What do these games have in common? Two things - they are longer than your average German game, and they are all inspired by an historical perspective. German-style games have the edge these days because they are quicker and easier to play (games like Die Macher aside), well-produced and almost endlessly varied. Tigris & Euphrat is possibly my favourite game (pending release of Reiner's Lord Of The Rings game), which abstractly covers similar historical ground (but is easier and quicker to play, has original game mechanics, and is nicer to look at). But very few games today actually have anything to say - they are all too often empty baubles with little actual value except an almost trivial ability to entertain. Mere distractions, hugely entertaining (hopefully) which we all take rather seriously. No wonder our non-gaming friends don't take us seriously!
"So, let me get this right... It's a game about bean farming?" or "So I'm an Elf, and I use things such as a giant pig to get around?"
It's time for some new blood - games inspired by Game Theory studies which, because Game Theory is used to study aspects of real life from a relatively new perspective, will also be games about aspects of real life.
A game of Blood Bat Brothers, anyone?
I'd love to see a chapter on Game Theory in a book on games by someone like Reiner Knizia, who actually understands game design. Something equivalent to any of the afore-mentioned books, but with the emphasis on the games (preferably boardgames). Perhaps the kind people at Counter could persuade him? Please? Or, perhaps an article from another Counter reader with an analysis of current boardgames and how they do use Game Theory (but perhaps unintentionally...). But I won't be convinced until I read an article or book on Game Theory by an academic which mentions some well-known commercially available boardgames which demonstrate the various aspects of Game Theory. Perhaps the academics would care to design a decent boardgame? Ideally, the game should then prove its worth by winning the Spiel des Jahre, or some other major boardgames award!
You have just read a theory on Game Theory by a boardgames enthusiast.
I have another theory...
With invaluable assistance from fellow Billabong games club member, Steve Gardner (who, at least, has studied Game Theory!).
DAC: This article was also published in Counter Issue 11
For a list of articles by me, see the Articles page.
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