29 May 2015

John Nash’s Contribution to Game Theory

Game Theory, Shameless Self-Promotion 32 Comments

I wrote a pretty long post at Mises Canada trying to give the average person a real sense of what Nash did in his dissertation. Ron Howard is not the hero of my post. An excerpt:

The central result from the work of vNM was the minimax theorem. The full details are here, but the intuition is: In a finite two-person zero-sum game, there is a value V for the game such that one player can guarantee himself a payoff of at least V while the other player can limit his losses to V. The name comes from the fact that each player thinks, “Given what I do, what will the other guy do to maximize his payoff in response? Now, having computed my opponent’s best-response for every strategy I might pick, I want to pick my own strategy to minimize that value.” Since we are dealing with a zero-sum game, each player does best for himself by minimizing the other guy’s payoff.

This was a pretty neat result. However, even though plenty of games–especially the ones we have in mind with the term “game”–are two-person zero-sum, there are many strategic interactions where this is not the case. This is where John Nash came in. He invented a solution concept that would work for the entire class of non-cooperative games–meaning those with n players and where the game could be negative-sum, zero-sum, or positive-sum. Then he showed the broad conditions under which his equilibrium would exist. (In other words, it would not have been as impressive or useful if Nash had defined an equilibrium concept for these games, if it rarely existed for a particular n-person positive-sum game.)

32 Responses to “John Nash’s Contribution to Game Theory”

  1. guest says:

    Jennifer Connelly is the hero of my post …

  2. Tel says:

    Zero sum games, I get that bit.

    However “positive sum” where does that come from? I mean, suppose you that the Rock/Scissors/Paper game and you add 1 to all the payoffs. I mean you get two points for a win, one point for a draw and zero for a loss… that’s now “Positive Sum” right?

    You can’t actually lose any points, with the new scoring system; if you play badly you just don’t get as many points as you might have done. It doesn’t change the strategy you are going to play though… seems like you are still going to play for a win if you can. Well, I know I would.

    • Daniel Kuehn says:

      In zero sum Bob’s max is my min, so my min of Bob’s max is my max. “Strategy” encompasses knowing Bob’s actions and responding accordingly.

      In positive sum games that doesn’t necessarily apply. Just adding one to everyone’s score doesn’t make it positive sum, you’re just rescaling the payoffs. Positive sum in that case would (I guess) be something like if I do rock and Bob does paper, we both get 2.

      “Positive sum” doesn’t mean all payoffs are non-negative – particularly since we’re often talking utility and the numbers don’t matter. “Positive sum” is about how my payoffs relate to Bob’s payoffs – whether one person’s gain is the other person’s loss or not.

      • Harold says:

        As I understand it, any voluntary market transaction is a positive sum game, since both parties gain from it.

        • Daniel Kuehn says:

          These interactions are not always transactions. Think of one of the more notable applications of game theory: explaining imperfect competition. The strategies have nothing to do with the oligopolists transacting with each other. The transactions (with consumers) stay in the background.

          • Tel says:

            Right, that’s an example where “not playing” is not available as an option to choose. An individual of course can decide to sell their shares in Corp-X and walk away, but the corporation itself cannot walk away.

            Or at least, in theory the corporation should be wound up and the assets liquidated and redistributed to the shareholders… and in theory that should happen ahead of time once the management understand that whatever Corp-X does is no longer a good idea to keep doing. However, in practice no management will ever do that, and corporations never choose the “don’t play” option.

            So what happens is that corporate competition (between other corporates) is no longer voluntary, it is survival (for the corporation). They can still choose different strategies how they want to compete, but they have to stay in the game somehow.

        • Gene Callahan says:

          Harold, that is incorrect: any voluntary market transaction is *suspected in advance* by both parties to be positive sum. Whether or not it *actually* turns out to be so cannot be assumed from its voluntary character.

          • Harold says:

            Surely at the moment it takes place there is gain on both sides -at least from the Austrian perspective. Otherwise it would not take place. Either party may later regret it, but that is a different matter. If I trade my cow for magic beans, I have gained from the trade because I have obtained what I desire more than the cow.

          • Grane Peer says:

            Gene, I think it has to be positive sum insofar as it could be positive, neutral, or negative. What it turns out to be would have to be determined by a future transaction in which case the sum of the initial transaction is indeterminate and remains so even with respect to a future transaction because that is also indeterminate and so on to infinity. As Harold states the determination is immediate not prior or after the fact.

            If you trade $300 for the mystery box which turns out to contain a deadly snake bite the initial trade is still a positive sum transaction. Who is improved by a deadly snake bite? The transaction was for a mystery box if it was directly for a deadly snake bite it wouldn’t happen unless you thought you were better off with the deadly snake bite than $300. An interloper’s valuation of the transaction is irrelevant.

          • Major_Freedom says:

            Callahan you have a very common misunderstanding of the “mutual gain” aspect of trading in a free market.

            It is the trading activity itself that serves as the mutual gain. Since it is voluntary, that activity is in itself mutually gainful.

            What you are talking about when separating trade into ex ante and ex post valuations, is also important, yes, but, for the ex post valuation of a past trade that consists of less utility than expected, is now a different, new set of circumstances.

            Now, the person who experienced less utility the. expected, is again faced with making gains by virtue of the next trade.

            There are three “dimensions” to trading. The past, the present, which is where praxeology comes into play, and the future.

            The past and the future do not “exist”. The present actions of people in an absence of coercion are where gains are being made by virtue of being able to trade the way one wants.

            The theory of mutual gains in free trade was never about an insistence that people never have regrets, nor was it ever an approximation or estimate of a percentage of trades that are ex post viewed as garnering an increase in utility as opposed to a decrease.

            If you and I trade, I am making an absolute gain by virtue of doing so. Whatever I think later on after I come to own the thing, is a different but related question.

          • guest says:

            “Whether or not it *actually* turns out to be so cannot be assumed from its voluntary character.”

            For the record, I agree with this.

            The reason the voluntary trade takes place is the end that the trade serves.

            It is possible for the trade to fail to satisfy the end.

            In fact, it is such failures which constitute the malinvestments of the boom phase of the business cycle.

          • Major_Freedom says:

            Perhaps a clearer way to put it…

            Suppose a person got shot in the head despite them not threatening the shooter or anyone else. Clear violation of liberty, and no clearer example of an action reducing utility.


            But now suppose that the victim had an inoperable tumor that was terminal, and the shot was a one in a million shot that knocked out the tumor, and the one in a billion shot did not end up killing the victim. They lived.

            This is a future with new circumstances and new opportunities for valuation.

            Suppose after being in a coma for a year, the victim wakes up and after learning what happened, made a new valuation.

            Regardless of what their new valuation is, if they preferred to die because now their quality of life is too low, or they preferred to live no matter if they are wheelchair bound or whatever, these new circumstances do not affect the actual event of the person getting shot without their consent which was in itself not a mutual gain.

            The moral of the story here is that the orthodox, and for lack of a better term “statist” way of how to treat other people, consists in the notion that it is moral to hurt people and reduce their utility now, if at some later date there is an argument to be made that “it was for their own good”.

            The flaw here of course is that it takes a historical view of what happened, forming a bunch of what if scenarios that would have hurt people but had a different outcome that is viewed as superior by the statist, and then from that complex history of unexpected twists and turns, then tries to make absolute principles thst are like a hammer for every construction problem.

            So then we are told that some people stealing everyone else’s money, i.e. taxation, is a ” good thing” all the time as a permanent and universal activity, because heck, during such and such a period of time in the past, in this and that specific circumstances, there appears to be a good argument that a lack of taxation at that time, or the presence of taxation at that time, seems to have had an inferior (in the case of no taxation) or superior (in the case of taxation) outcome (according to the statist of course).

            Then slyly but ever so necessary does the statist then pretend or presume that these outcomes are objectively inferior or superior and there is no possible argument against it. It just is or isn’t, period. Any attempt to argue against it is either viewed as a vain attempt to change the inevitable, or go against humanity in the course of history.

            The statist can never give an answer for what principles underlie their universal desire to have some people hurt by aggression on purpose. If they are repeatedly asked why, they will invariably find nothing underlies what they are saying other than “this is what I want for your person and property no matter what”.

            I could ask Callahan why he would not think it is moral for someone to punch him repeatedly and threaten him with death so as to get him to stop eating fast food, or to get him to give more money to others who are in pain or who are hungry. I could use his “the ends justifies the means” set of haphazard unprincipled stories of history or thought experiments, and say “Yes yes, you can complain about being hurt by aggression, but the outcome is according to MY desired statist outcomes is superior. You’re not against humanity are you?”

            No statist can ever agree with another statist. It is why competition in politics always consists in deceit and aggression. No exceptions. It is because there is no way for statists to act according to their ideas, going forward, without the losers being aggressed against by the winners. Here, there can be no mutual gain. The activity of politics is itself exploitative.

            Free trade is inherently mutually gainful.

            That is the difference.

        • Tel says:

          As I understand it, any voluntary market transaction is a positive sum game, since both parties gain from it.

          But this is presuming that one of the possible strategies on offer in the game is not to play that game at all. That’s what the “voluntary” part is all about.

          Fine, so if the game has an option to not play then at least that option is not “positive sum”, it’s exactly zero I would have thought. All of the other options might look good in comparison with the zero (i.e. the “not play” option is your reference point), but seems strange to say that the game as a whole is “positive sum” just because some options look better than others.

  3. Harold says:

    If a revenue neutral carbon tax actually boosted the economy as well, that would be a positive sum game.

    • Daniel Kuehn says:

      John Nash taught us that instead of everyone fighting over a carbon tax we should go home with clean coal, Germany should go home with wind power, France should go home with nuclear and then everyone will be happy.

      • Carl says:

        But nuclear is bad because planet

        • Tel says:

          The bad thing about nuclear is the trust problem. If you look at Fukushima for example, the Unit 1 reactor had been running for more than 40 years, while the standard design-life of a reactor is only 30 years (but they often get extended, because every extra year you get out of it boosts the profitability, and dismantling the reactor is too expensive).

          There were a bunch of TEPCO issues found, they already knew that flooding and tsunami were vulnerabilities of their systems, but the operators were reluctant to make any changes. Partly it was the problem that always happens with putting resources towards backup and fail-safe systems it eats away at profit so no one wants to have that on their budget. The other problem is cultural resistance (both in Japan generally and in TEPCO management in particular) to sticking one’s head up and saying “hey this really needs to be fixed”.

          Here’s an example of the problems with disaster preparation: we do these silly fire drills where office workers walk down the fire stairs and someone wears a red hard hat. Totally unrealistic, for a start they always leave the main lighting circuits in the building switched on, so no one ever sees what the building looks like with emergency lighting only. Anyway the costs of these drills are carried by the companies losing work hours (the shareholders actually) and it’s mandated so they have no choice. Since office fires are extremely rare, the purpose of this is largely for government to be seen to be doing something important.

          In comparison, at Fukushima they apparently didn’t do regular exercises to run through how they would handle various types of disaster. There was limited availability of battery powered lighting and, insufficient redundancy in the basic instrumentation and control systems.

          Generally speaking, with nuclear power you have a very small risk of a big problem, but humans don’t handle that situation well. After all, there are millions of very small risks that we ignore… aliens could invade tomorrow, it’s not likely, but can’t entirely be ruled out either. A comet could land on your house and leave a giant smoking crater. Since there’s lots of these highly unlikely scenarios, it’s difficult to sort out the ones that are worth worrying about from the complete rubbish.

          I mean imagine some Native American standing around trying to convince his friends that huge boats are going to come and white people will bring guns, disease and bottles of debilitating cheap whisky … no one would have believed him.

  4. Harold says:

    My posts with links in them disappear – just checking it is because of the links

    • guest says:

      Barring any filtered text strings, your posts with links will appear if approved.

      The site is set to permit you to use one link without screening. Any more, and it needs approval.

      You can cheat, though, if you post a link that the site doesn’t recognize as a link. You can do this by prefacing the link with something.

      For example:


      This way, you can post as many links as you like.

      • guest says:

        Might as well make the example useful.

        Austrian resources:

        Free Stuff
        [Tons of resource pages]

        Lew Rockwell

        Videos: Youtube User, misesmedia

        Videos: Youtube User, LibertyInOurTime

        Tom Woods App Available for iPhone, iPad and Android

      • Harold says:

        Just tried prefacing as you did, and still post did not appear.

      • Harold says:

        Will copy and paste your Lew Rockwell disguised link in next post.

      • Harold says:

        Here is Guest’s link copied and pasted
        Lew Rockwell

      • Harold says:

        I copied my link added onto [www] as you did, and as worked for the link I copied from your message, yet it did not appear. It seems very strange. Will try one more time

      • Harold says:

        Different place to play guess 2/3 the average game, with disguised link, to see if gets through.

        Lets see if that one appears.

      • Harold says:

        Odd, one disguised link gets through, the other one does not. There seems to be something in the pother that is causing it to not appear, even though it is not recognised as a link.

      • Harold says:

        The other link did not get through when I placed spaces in the text. Will try posting it in sections to see if any are not arriving. Apologies for filling the blog with off topic stuff

  5. Harold says:

    Did not work if I removed http bit. Same link without the net bit

  6. Harold says:

    Same link with http, but without net bit

  7. Harold says:

    Will not work with .net at the end, either disguised or as link. .com seems to get through fine. Any idea’s what is happening?

  8. Harold says:

    what the heck is happening.net

  9. Harold says:

    Final analysis. If I type creativitygames and add .net to it, the post does not appear. If I type other words and add .net to the end, they do appear. I will stop now.

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