Two days ago, NN [NiceNihilist?] made reference to a Nash
Equilibrium. Now I am going to be
perfectly honest. I vastly prefer
talking about things like that to talking about politics, so for all those
readers who are unfamiliar with the term, I am going to attempt to explain what
a Nash Equilibrium is. Think of this as
my version of sitting on a beach and sipping a drink that has a little umbrella
in it.
“Nash” here is John Nash [or Russell Crowe, as I prefer
to think of him], a brilliant but troubled mathematician who won the Nobel
Prize in Economics for this work.
Let me start by reminding you of John Von Neumann’s great
theorem on two person zero sum mixed strategy games. A pure strategy is a complete specification
of which move a player will make in any situation that can possibly arise in a
game. A mixed strategy is a probability
distribution over the set of pure strategies summing to one. In a two person game, if player A has m
strategies and player B has n strategies, then an (m + n – 2) vector space is
required in order to represent all the mixed strategy pairs available to the
two players. [ -2 because once all but one of the probability weights assigned to
the first m-1 or n-1 pure strategies have been specified, the weight assigned
to the remaining strategy is determined, since the weights must total to 1.] If the game is zero sum, then one more
dimension is required to represent the payoff to player A, since the payoff to player
B is simply the negative of player A’s payoff.
If the game is not zero sum, then two additional dimensions are
required.
A solution to a two-person game is defined as a pair of
strategies that has the following property: If A holds to his or her mixed strategy
choice, then B can only do worse by changing strategies, and if B holds to his
or her mixed strategy choice, then A can only do worse by changing
strategies. This is called an
equilibrium point in the mixed strategy space. [It is also sometimes called a saddle point, for reasons I will leave it to you to figure out.] Von Neumann proved that every two person zero sum game with mixed
strategies has a solution. What is more,
the solution is a strong solution in
the sense that if there is more than one equilibrium point, they all assign the
same payoffs to A and B.
Nash generalized Von Neumann’s result by showing that every
n-person game with mixed strategies has a solution. [If n is greater than 2, the game cannot be
zero-sum. Indeed, for n > 2, the concept
of the sum of a game is undefined.] A
solution in this case is a set of n mixed strategies with the property that for
any of the players, if all the other players hold their mixed strategy choices constant,
that player will only do worse by changing his or her mixed strategy choice. However,
in the general case, the solution is weak,
in the sense that that payoffs to the players of two equilibrium points may be
different from one another. Thus, they
can only be said to be local equilibria, not global equilibria.
So, is that clear?
5 comments:
John Forbes Nash.
Whoops, thanks.
Start, maybe, by explaining what Game theory is. Taking this to be an umbrella term for the science of logical decision making in humans, animals, and computers, well, originally, it addressed zero-sum games. I think you might have tarried a bit, though it may seem pretty basic, with what a zero-sum game is. Zero-sum games are most often solved with the minimax theorem which is closely related to linear programming duality, or with Nash equilibrium. That's a bunch of jargon, of course. But it's all more interesting stuff anyways, at this more basic and general level. Or at least, I regret getting into the 'Nash equilibrium' concept without being able to address the point that game theorists have discovered that it makes misleading predictions (or fails to make a unique prediction) in certain circumstances.
Danny, I created an entire separate multi-part blog on the subject: Formal Methods in Political Philosophy.
Many thanks for this. The criticism of formal methods in political philosophy is well worth reading, as is the criticism of Elster's methodological individualism and the illuminating exchanges with Roemer and Kliman--just to name a few examples. Roemer, for example, uses cooperative game theory to characterize the exploitation of one coalition by another--I wish I had thought of that!--but leaves domination undefined. (My guess--I haven't checked--is that it is possible to interpret Romer's coalition function within certain extensive form games in which domination is interpreted as the ability to block the present and future moves of other players. Since my mathematical intuition--that of the hobbyist whose enthusiasm far outruns his ability--is relatively weak, my guess is either false, uninteresting--worse than false--or "trivially obvious" to a sharper intellect.)
I still have a soft spot in my head for game theory in economics--when the assumptions of rationality work. David K. Levine has some instructive examples in "Is Behavioral Game Theory Doomed?" Levine and others are interested in learning in games. This refers to how quickly players converge on a Nash equilibrium (among other solution concepts).
I like Gintis on game theory as well. At some point Gintis seems to have abandoned socialism (I do not expect you to comment, for obvious reasons). The first edition of Game Theory Evolving includes a section on socialism (p 44) and a criticism of neoclassical economics. That section is removed from the second edition. Whether the editorial decision to remove this reflects a change of heart is difficult to say. Gintis observes that, "Surely [by now, one might think] someone has provided a plausible decentralized, market-oriented equilibrium mechanism to replace the auctioneer. But in fact, no one has succeeded in producing a plausible dynamic model of market interactions in which prices move toward their market-clearing levels."
On this point, David K. Levine writes in "Is Behavioral Economics Doomed?" that "[a]t one time a great deal of effort was expended by economists trying to understand the mechanism by which prices adjusted. Modern economic theory recognizes that the particular way in which prices are adjusted is not so important. An important modern branch of game theory is mechanism design theory. While game theory takes the game as given, mechanism design theory asks -- how might we design a game to achieve some desired social goal? To emphasize that the choice of the game is part of the problem, the way in which decisions of players are mapped to social outcomes is called a mechanism rather than a game. ... an auction ... is just one of many price setting mechanisms. ... There are many mechanisms that do this."
Now if only I could stop myself from pursuing technical subjects that lie beyond my ability and for which the professional opportunities are non-existent.
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