Posts Tagged ‘game theory’

On the Practical Uselessness of Minimax Prescriptions

Monday, 25 October 2021

[I posted the following as an entry to Facebook two years ago.]

In game theory, there's a proposal that, in selecting amongst options, a player should choose that option with the least-bad worst case. So, for example, if the worst that could happen if you go to the bistro is that they get your order wrong, while the worst that could happen if you go to the bar is that you get knifed, then you should go the bistro.

Such a strategy is usually called minimax. (I think that it should have been called maximin, and indeed some people call it that, but they're very much bucking convention.)

The proposal seems plausible, especially when measures of probabilities are unknown (so that expected values cannot be calculated). But I think that notions of probabilities as measures inform and thus disinform the apparent reasonableness of minimax strategy.

When one conceptualizes probability as a measure, it is all too easy to think that all events with probability measure 0 might as well be treated as impossible. But they're not quite the same thing if one has to deal with infinitely many possible cases; some precisely specified possibilities would each have to have a probability measure of 0.

Possible events with truly horrible consequences can have a probability measure of zero, and will be disregarded if one treats that probability as equivalent to impossibility.

And, in the absence of actual measures of probability, then from the habit of treating events with very low measures of probability as if impossible can slide into the practice of more generally disregarding events that have low probability rank.

In a textbook game of poker or of dice or whatever, the worst possible outcome in the model is most often a loss of funds. In a real-world game of poker or of dice or whatever, the worst possible outcome is something far more dire, and hard to identify; we can think of awful possibilities, and then think of something still worse. Puffy may get angry, and shoot us; or he may shoot not only us but in his rage go shoot our loved ones as well.


Sunday, 19 November 2017

It has famously been argued that the word game cannot be defined in a way that adequately captures the various senses in which it is used. I believe that, in everyday use, the term game most often means a system of contrived challenges properly imposed or undertaken for purposes of amusement. Hence, someone might assert something such as Love is not a game! But, even in lay-use, game can have other meanings. For example, when a person proceeds deceitfully or insincerely, he or she may be said to be making a game of things, without necessarily seeking amusement in proceeding in this way.

Economists and mathematicians applying themselves to problems of economics or proximate to those of economics can use the term so very broadly as to refer to any problem of optimization. But, most often, they mean a system in which multiple parties interact with the potential for one or more parties to advance an interest or something that is treated as an interest (such as reproduction). It is in this sense that I here use the term game.

The rules of games are often subject to to change, and those changes may be affected or effected by players of the governed game. There is thus a meta-game — a system in which multiple parties interact with the potential for one or more of them to advance an interest by changing the rules of the game; or, in the context of others trying to changing the rules, by preserving the rules. The concept of meta-games is hugely important for understanding social processes.

Of course, a meta-game might have its own meta-game — a meta-meta-game. For example, the determination of a legal frame-work might be the meta-game of the social processes that the frame-work governs, and a struggle over social values might be the meta-game of the determination of the frame-work and thus the meta-meta-game of those social processes. But it can be difficult — without necessarily being useful — to work-out an actual hierarchy.

Sometimes, all that we really need to recognize is that some activity is a meta-game of some other game, without concerning ourselves as to whether the other game is itself a meta-game. People might readily recognize meta-gaming in activities such as political lobbying, but they generally don't recognize it when it's effected by psychologists, by teachers, or by screen-writers.

I want to draw upon this notion of meta-games for at least one 'blog entry, but I will probably want to draw upon it for multiple entries, so I will leave this entry as infrastructure. And I may later and without notice rework it, in an attempt to improve it as infrastructure.

Prairie Dogs' Dilemma

Sunday, 26 April 2009

I have posted one entry to this 'blog that made reference to Cournot-Nash equilibria, and I expect to write another soon. I'm going to use this entry to explain the concept of a Cournot-Nash equilibrium, without resorting to mathematical formulæ.

First, let me give my favorite example of the idea, the behavior of prairie dog mothers in at least some towns. Prairie dogs are omnivores; they are primarily herbivorous, but will also consume small animals such as insects. If a prairie dog mother stays away from her litter of pups, they are liable to be eaten by something, so she will prefer food that is close at-hand — such as the pups of another mother who is away from her burrow. In fact, in some towns, when pups are eaten, it is usually by mothers trying to get home before their own pups are eaten. If any one prairie dog were to stop eating pups while the others continued, then her own pups would more likely be eaten because she'd be away from home for longer or more frequent periods. They eat each other's babies because they eat each other's babies.

Some of you may be thinking of the Prisoners' Dilemma, which, under classic assumptions, results in a similar mess. It too is an example of a Cournot-Nash equilibrium.

The essence of a Cournot-Nash equilibrium is that each participant has no incentive to change behavior unless other players change behavior, so each — and thus every — participant sticks with his or her established behavior. Although the prairie dog example and the classic telling of the Prisoners' Dilemma are sub-optimal equilibria, it could be the case that an equilibrium were the best-possible equilibrium, and no one had an incentive to change his or her behavior so long as no one else changed his or her behavior; so it's important to distinguish optimal Cournot-Nash equilibria from sub-optimal Cournot-Nash equlibria.

The Nash to whom the name refers is John Forbes Nash jr, whose life and work were grossly misrepresented in the movie A Beautiful Mind (2001). Nash's most famous accomplishment was explicitly generalizing and formalizing the idea of a Cournot-Nash equilibrium, which some simply call a Nash equilibrium. But there were famous antecedent uses of the idea, the best-known of which was by Antoine Augustin Cournot, in an 1838 model of oligopolistic competition.[1]

A less-often recognized antecedent use was by Thomas Hobbes in Leviathan (1651). Hobbes famously proposes that, in the absence of a State, life will be nasty, brutish, and short. More specifically, he said

Hereby it is manifest that during the time men live without a common power to keep them all in awe, they are in that condition which is called war; and such a war as is of every man against every man. For war consisteth not in battle only, or the act of fighting, but in a tract of time, wherein the will to contend by battle is sufficiently known: and therefore the notion of time is to be considered in the nature of war, as it is in the nature of weather. For as the nature of foul weather lieth not in a shower or two of rain, but in an inclination thereto of many days together: so the nature of war consisteth not in actual fighting, but in the known disposition thereto during all the time there is no assurance to the contrary. All other time is peace.

Whatsoever therefore is consequent to a time of war, where every man is enemy to every man, the same consequent to the time wherein men live without other security than what their own strength and their own invention shall furnish them withal. In such condition there is no place for industry, because the fruit thereof is uncertain: and consequently no culture of the earth; no navigation, nor use of the commodities that may be imported by sea; no commodious building; no instruments of moving and removing such things as require much force; no knowledge of the face of the earth; no account of time; no arts; no letters; no society; and which is worst of all, continual fear, and danger of violent death; and the life of man, solitary, poor, nasty, brutish, and short.[2]

According to Hobbes, without the State, production is subject to predation, so potential producers have less incentive to produce and everyone has incentive to prey upon everyone else.

But Hobbes has also identified a special case of one solution to what would otherwise be a sub-optimal Cournot-Nash equilibrium. In Leviathan, men end the war amongst them by explicitly agreeing to the creätion of an institution (the State) which will change the equilibrium. More generally, agreements need not be explicit or conscious, and the transforming institution could be a code of conduct. For example, the classic statement of the Prisoners' Dilemma treats the game as played in a social vacuum of a sort. In real life, people build reputations, reward desired behaviors, and punish the behaviors to which they object. Commitment mechanisms don't necessarily free us from every possible sub-optimal Cournot-Nash equlibrium, but naïve game theory too often fails to consider their possibility. (There was some perverse gloating in A Beautiful Mind about how Nash had somehow refuted Adam Smith, but the liberal order of which Smith wrote is filled with commitment mechanisms. Private property itself is an example of such a mechanism.)

Perhaps, in time, even the prairie dogs will evolve a mechanism such that eating each other's pups is no longer an equilibrium.

[1] Recherches sur les principes mathématiques de la théorie des richesses, Ch 7.

[2] Chapter XIII ¶ 8-9.