The horizon is not so far as we can see, but as far as we can imagine

An Introduction to Game Theory


John Von Neumann

A great deal has been written about the Theory of Games, but unfortunately, most of it is wrong: it doesn’t accurately represent what the theory means for economic and biological behavior. Some who write about game theory have taken a course in game theory and are honestly trying and are honestly trying to explain the idea that everything can be explained by game theory. They just don’t understand it well.

But some are more pernicious in their intent: they know that Game Theory does not actually make the point they want to make, but they know very few people will be able to catch the rather subtle intricacies on which their misleading theory rests.

There are also blunders that great human will make because of a blind spot in their thinking: Linus Pauling is one behind step behind us, laughing as he does so. (He made some errors in the structure of DNA, which amused everyone, especially when Watson and Crick’s very soon afterward got the structure right, with a great deal of help from Franklin’s x-ray pictures).

Game theory started with John von Neumann’s proof that mixed strategy equilibria could be to collapse down to at least one point. He originally did this using using a topographical proof know as the Brouwer fixed-point theorem. Eventually Nash expanded the theory include any number of players and describe what is now known as a Nash equilibrium.

This much you can learn with a few Wikipedia searches.

If one starts reading, one is soon introduced to the “Prisoner’s Dilemma”, which was one of the early games because of its application to real-world problems. In the simplest case, there are 2 players and they are caught, but the prosecutors do not have enough evidence to convict them with a confession. So the prisoners are separated and each of the same deal: if you rat on the other person, he will take the blame. What the prosecutors are hoping for is that each one will rat on the other, and they can get both.

So there is a two-person game, but which is set up by a 3rd person to get the information that they want.

But if one thinks about it, there is a fourth looking at the running of both games, and they will find out that loosely connected criminals will do as they expect: the two finger the other person. But in tightly knit communities with a deep commitment to crime, the 2 players will shut up, serve the much lesser sentence, and go free. This is because one of the goals of the prisoner’s dilemma does not apply: the 2 criminals had already said, and were committed to, not talking if they are captured.

So a 2 player game, becomes a 3 player game (prosecutors), becomes a 4 player game (politicians) becomes a 5 player game (criminal underworld world). This is the part of being a real game theorist: imagining a game, and asked whether a new dimension could be introduced by taking a set of people who are betting on the game itself.

It also points out a very necessary thing about game theory: it often does not produce the optimal set of outputs, but the pessimal series, the least worst position that a player can get. This is why, in game theory, one usually solves backward – take the last step, and then the next to last step, and so on.

One of the other core concepts of game theory is also on display here: to find out if one player has a dominant strategy, that is one play which will lead to the best outcomes whatever choice his opponent makes. The dominant strategy does not produce the best outcome, merely the best outcome given that the other player will accept. That is why the word “pessimalization” (not pessimization) is used certain groups of people, to emphasize that it is the best possible worst outcome.

Going forward, for future posts, I will post on real topics, and I will imagine that the people involved are expert players, or they can get hold of some. These topics will include Brexit, Climate Change, Trump and other problems with rational outcomes, though a great number of people do not want the rational outcome to happen. This too has a game theory solution: change the rules.


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  1. someofparts

    Stirling, do you have a new website somewhere? I tried to visit your old blog this weekend (the one Ian links to on this page) and did not find any content.


    Hey Sterling, good to digitally see you again.

  3. Stirling S Newberry

    No link, I am busy with college.

  4. ponderer

    This sounds very interesting. I’ve not studied game theory at all but have heard of the prisoner’s dilemma problem. The framing of the problem is interesting because if the prisoners are not connected by a crime, everything changes from the observers perspective but not for the prisoners. They still have the choice of lying to get a reduced sentence and screwing the other guy or stay quiet and get punished worse. Our view of the outcome radically changes, but the game remains the same. I look forward to your next article.

  5. Tom

    Ah the prisoner’s dilemma.

    None of them talks, they both walk.

    None of them signs, everything is fine.

    Only if they were two random drunks who met recently would they both rat each other out.

    It is a self-deception that is all too common:

    During WWII the RAF was looking at where its planes were getting shot at and demanded those areas get armored up. One guy spoke up that they should ignore the places shot full of holes and armor the spot that weren’t as planes shot there didn’t come back. It took two years for his advice to be followed and plane losses declined once he was heeded.

    Details and results matter when devising theories. Simplistic statements of X means Y get you nowhere, unless you understand the underlying details as to why things happen.

  6. jeff wegerson

    I have always enjoyed reading your take on things. It is good to be able to do so again.

  7. Hugh

    Yes, re climate change, what happens if the game is being played on the Titanic, or as you point out that regardless of the fact that a certain roulette wheel always comes up odd, or does so most of the time, some people will always bet even, just as some people will bet on the gray horse even though the gray horse almost never wins.

  8. NR

    An interesting example of game theory was the game show Golden Balls, a British show that aired back in the late 2000s. A group of contestants would accumulate money in the pot over the course of the show, and at the end there would be two contestants left and a pot of money available. The last round was called “Split or Steal,” and the contestants would face each other and choose a ball that said either “Split” or Steal.” If both contestants chose “Split,” they would divide the money in half, if one chose “Split” and the other chose “Steal,” the one who chose “Steal” would get all the money and the one who chose “Split” would get nothing, and if they both chose “Steal” both would leave with nothing.

    The problem with this setup is that, discounting the social pressure of being on TV, there’s no incentive whatsoever for a contestant to pick “Split.” In this system, your winnings are dependent on what the other player picks, and you can only hurt yourself by picking “Split.” If people understand this, everyone will always pick “Steal” and the players will leave with nothing.

    But there was one contestant who came up with a very ingenious solution to this problem, and you can see it here:

  9. Anthony K Wikrent

    It has been much too long, Stirling! I am thrilled and delighted to see you posting online again!

  10. I read about a stellar political game theorist, Bueno de Mesquita, some years ago, who regularly outperformed the CIA’s own analysts. Unsurprisingly, the CIA was/is one of his clients.

    Almost immediately, it occurred to me that citizens looking to take back their government from plutocratic control would do well to hire an accomplished political game theorist, to advise their electoral strategies. As I was commenting mostly in progressive blogs, back them, that was the sort of audience I challenged.

    To me, this is a NO BRAINER. Yet, the only person I encountered that ever showed any interest was Kevin Zeese, of

    Although I don’t explicitly raise the issue of hiring a political game theorist in conservative fora, where I’m nowadays more likely to roam, I do often point out that the mostly populist 5 Star Movement, founded by a comedian, has achieved something that the the Trump amen corner can only dream of. Hence, there are lessons to be learned…. There’s been a few upvotes, and a few signs that a handful of other commenters get it, but most seem quite content to just wallow in verbal polemics, and organize absolutely nothing.

    So, my assumption is that conservatives are no more likely to hire political game theorists than progressives.

    Although I don’t know of any evidence for it, I’d be utterly shocked if Big Business did NOT hire folks like de Mequita. Unlike us little people, they seem to get everything they want from Congress.

    So, I have 2 questions:
    1) who, if anybody, is hiring political game theorists
    2) why don’t progressives (or other disgruntled groups, such as populist conservatives) hire political game theorists



  11. Mark Pontin

    Look. You’re missing a very important point here in your urge to simplify. You allude to that point, sure. But you don’t spell it out and you need to because it’s basic.

    Game theory says that in _repeated_ or iterated games of Prisoner’s Dilemma each party’s optimal choice is NOT to defect on (rat out/lie about) the other player, but to cooperate.'s_dilemma#The_iterated_prisoner's_dilemma

    As Tom alludes to above me in the thread, “only … two random drunks who met recently” would be likely to automatically rat each other out.

  12. Mark Pontin

    “It also points out a very necessary thing about game theory: it often does not produce the optimal set of outputs, but the pessimal series, the least worst position that a player can get.”

    Yeah. This is the fundamental axiom of game theory, known as the minimax (or maximin) theorem and proved mathematically by John von Neumann in 1928.

    More here —

  13. Stirling S Newberry

    In general, a repeated version is called “Peace versus War” and your solution only works in certain circumstances. I will talk about and other variants in due course, because in most cases, war is very different when the two sides have the capacity to annihilate each other. There are also the variants were a limited war is on the table (e.g. Vietnam).

  14. Stirling S Newberry

    “Yeah. This is the fundamental axiom of game theory, known as the minimax (or maximin) theorem and proved mathematically by John von Neumann in 1928.”

    I believe I stated that one could do a few Wikipedia search to get familiar with the topic, and JvN was mention as the key thinker in 2×2 game theory. This is basic game theory, which is gotten wrong in illustrious journals. This has a point, which will be described later.

  15. Hugh

    I think game theory works best when choices are binary. But in politics there are usually far more possibilities. You can choose not to choose. You can choose to choose but not now. You can choose but do little or nothing. For example, you can call for a commission or task force to prepare a report issued sometime in the future with recommendations which may never be acted on or at a time long after the public outcry has died down and when it and its recommendations may be quietly forgotten. You can engage in cosmetic voting, which is, at least in the US Congress, what most votes are. For example, a Senator can vote for cloture on a bill ensuring its passage but then vote against it at the final vote and run on his/her no vote. A bad bill can be larded with sweeteners and a good bill can be filled with poison pills. A Congressperson can go their whole career without casting a single significant or telling vote. You only see this when someone is in a position to cast a make or break vote on a bill or somewhat more commonly where they will vote against their previous stated position, party, or base even though their vote wasn’t needed. Add in the degenerate state of our politics where politicians and their parties are bought and alternately pander to and betray their bases, I can’t help thinking it is less important to look at the choices and more important to look at the results.

  16. Tom

    They spent so much time betting on Guaido, but when the time came, they couldn’t move because Columbia didn’t want to be a springboard for what was an armed invasion.

    Never mind Latin American States decision to abide by US Sanctions caused the refugee crisis in the first place.

    Ah well, Maduro knows he just has to tough it out, let the dead weight flee, and restart at the local level like Cuba did when the USSR collapsed and it looked like Cuba would go under.

    Guaido can make all the noise he wants, but at the end of the day, he is a traitor and lacks the backing of the masses who know he naught but a puppet.

  17. Mark Pontin

    Stirling N: ‘I believe I stated that one could do a few Wikipedia search to get familiar with the topic, and JvN was mention as the key thinker in 2×2 game theory. This is basic game theory, which is gotten wrong in illustrious journals. This has a point, which will be described later.’

    I was affirming and agreeing w. your point (and clarifying a detail — everyone used to know about JvN and minimax years ago) here. Not being mildly contentious as in my previous comment.

    Sorry if you misread me or this wasn’t clear.

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