An Overview Of Game Theory

An Experiment in Game Theory

Game Theory provides a formal treatment of well-specified situations in which the outcome depends on the choices of several agents who may have conflicting interests.

Abstractly, a player chooses a strategy, where a strategy specifies the player's move in every situation that may arise in the game. For example, a strategy for white in chess specifies, roughly white's move for every board configuration in which it is his turn. This example is rough because white's play will prevent certain configurations from arising, and his strategy need not provide a move for those unreachable configurations. A game tree is a useful representation for a game in extensive form. I think this definition of a strategy elides important issues of algorithms and computational complexity.

A game in normal form lists the players, the strategies for each player, and the expected payoffs to each player for each combination of strategies (the payoff matrix). Table 1 gives an example for what may be the most famous game designed by game theorists. The first entry in each ordered pair is the payoff to player A when A plays the strategy indicated by the row label and B plays the strategy indicated by the column label. The second entry shows the payoff to player B.

Table 1: A Prisoner's Dilemma

Player A's Strategy	Player B's Strategy
	Cooperate	Defect
Cooperate	(1/2, 1)	(-1, 2)
Defect	(1, -1)	(0, 1/2)

Suppose the payoffs, in each entry in the payoff matrix, sum over all players to zero. Then the game is a zero sum game. The prisoner's dilemma is not a zero-sum game.

Consider simple two-person zero-sum games like "Odds and Evens" or "Rock, Scissors, Paper". The best strategy is not to play the same simple strategy over and over, but to randomly mix strategies. This is an interesting insight from game theory - that randomness in economics can come from optimal choices even in games with completely deterministic rules. The probabilities that the players should choose depend on the payoff matrix. One can formulate a Linear Program for each player to solve for these probabilities. Each player assumes that the other player chooses his probabilities to minimize the other player's loss, given the first player's probabilities. A minimax problem arises. The neat thing about the two Linear Programs is that they are dual problems. Although von Neumann helped develop Linear Programming, vN and Morgenstern don't point out this connection. However, both vN's paper on activity analysis and vN and M's book used a fixed point theorem in the proof of the most important relevant theorems.

How to extend the concept of a solution to more than two players, or to non-constant sum games, is an interesting question. vN and M introduced "fictional players", so to speak, to make the general game like a two-person zero-sum game. A dummy player with one strategy can absorb the losses and winnings in a non-zero sum game. Thus, the game, with this dummy appended, becomes a zero-sum game. The multiplayer game can be thought of as a two player game between a winning coalition and the remaining players, thus becoming equivalent to a two-person game. vN and M emphasize that how the players in a coalition will split up their winnings is indeterminate, in general. Threats of players to leave a coalition and join the other side, though, impose constraints on the range of variability in the set of solution imputations.

Economists nowadays say that the vN and M solution applies to what are known as cooperative games. Players can discuss how to share winnings beforehand, and agreements are enforcable by some external institution. vN and M, had a different perspective:

"21.2.3. If our theory were applied as a statistical analysis of a long series of plays of the same game - and not as the analysis of one isolated play - an alternative interpretation would suggest itself. We should then view agreements and all forms of cooperation as establishing themselves by repetition in such a long series of plays.

It would not be impossible to derive a mechanism of enforcement from the player's desire to maintain his record and to be able to rely on the on the record of his partner. However, we prefer to view our theory as applying to an individual play. But these considerations, nevertheless, possess a certain signiificance in a virtual sense. The situation is similar to the one we encountered in the analysis of the (mixed) strategies of a zero-sum two-person game. The reader should apply the discussions of 17.3 mutatis mutandis to the present situation." -- John Von Neumann and Oscar Morgenstern (1953) p. 254.

John Williams, one of the participants in Flood and Dresher's original experiment was puzzled why Armen Alchian did not behave according to this way of thinking. On the 50th iteration, he wrote, "He's a shady character and doesn't realize we are playing a 3rd party, not each other."

John Forbes Nash extended the two-person zero-sum solution in another manner. He defined the Nash equilibrium. In a Nash equilibrium each player's mixed strategy yields that player the maximum payoff, given that all other players are choosing their optimal strategy by the same rule. A Nash equilibrium is not necessarily unique for a given game. Nash also redefined vN and M's approach to be applied to cooperative games. The Nash equilibrium is said to apply to non-cooperative games.

Lots of questions arose from this work. How can the players decide on which Nash equilibria to choose? Can this indeterminacy be narrowed? Researchers have proposed a whole slew of refinements and variations - subgame perfect equilibria, trembling hand equilibria, etc. - the details of which I forget. This looks like a different approach to economics than Walrasian General Equilibrium theory. Are they related? Well, the proofs of the existence of Arrow-Debreu equilibria grew out of the mathematics of game theory. Furthermore, the equivalence principle, which M. never accepted, states that game theoretic solutions will approach Arrow-Debreu equilibria as the number of players increases.

It seems many mathematicians and economists have decided that, in practice, one can usually not set up the game and solve it. Nevertheless, game theory provides a language to talk about such situations. Discussions in this language have dissected "rationality" until, perhaps, the concept has fallen apart. You can view Survivor or the Weakest Link as laboratories to test game theory. In fact, experimental economics grew up with game theory, including experiments in which the players are computer code.

References

• Philip Mirowski. 2002. Machine Dreams: Economics Becomes a Cyborg Science. Cambridge University Press.
• John Von Neumann and Oscar Morgenstern. 1953. Theory of Games and Economic Behavior, 3rd ed. Princton University Press

A Short History

William Petty began classical political economy in the 17th century. Classical economics was developed through the work of the physiocrats and such writers as Adam Smith, David Ricardo, and Karl Marx. Marx was also a critic.

About a century and a half ago, economists mistakenly accepted the marginal revolution. Jevons, Menger, and Walras had precursors, but they were regarded as cranks. Marx, however, posed a political problem. Some might mention Henry George here, or maybe even Silvio Gesell. Better have an imitation of physics than talk about the ideas of those opposed to capitalists.

Lionel Robbins made clear in the 1930s, when the theory plainly did not apply, that marginalist economics is about the allocation of scarce resources. Land and labor can be taken as given at a moment in time, but capital cannot.

About half a century ago, economists came to recognize that they were mistaken. By the way, this mistake was also noted in the Heckscher–Ohlin-Samuelson (HOS) model of international trade. The demonstration that marginalism is fundamentally wrong was pushed by Joan Robinson and Piero Sraffa.

So some economists returned to elaborating classical economics. Even in the marginal interregnum, some, such as Leontief and Von Neumann, elaborated classical themes.

But most economists have been spinning in a widening gyre, ignoring the incoherence of their teaching. If a sufficient political movement develops outside of academic economics, maybe more economists will return to serious work. If so, they may even find elements to repurpose in this half century of dissolution and confusion.

Elsewhere

• Anton Pichler, Marco Pangallo, R. Maria del Rio-Chanona, François Lafond, and J. Doyne Farmer have an article Forecasting the propagation of pandemic shocks with a dynamic input-output model. This is a non-equilibrium, simulation model applying Leontief's input-output analysis. I suppose this is applied Sraffianism.
• For the use of Leontief input-output models in modeling natural disasters, one could do worse than look at the work of Adam Rose.
• Steve Keen responds to this year's Nobel prize. Diamond and Dybvig (1983) is a well-referenced thought experiment with mathematics. The lessons it teaches are wrong.
• Matt McManus writes about Ludwig von Mises. He does not say much about the socialist calculation debate.
• Jan Toporowski writes about Oscar Lange.

On Equilibrium

I have found a common misrepresentation from many, including mainstream economists, is that critics of their models do not understand them or the role of the assumptions. Those mainstream economists rely on an incoherent essay from Milton Friedman to dismiss criticism of the realism of assumptions.

My favorite criticism, though, is that their conclusions do not follow from their assumptions. I like to show this by constructing numerical examples that contradict their teaching.

On the other hand, some do criticize the realism of assumptions. I have seen some complain that the economy is never in equilibrium. It is unrealistic to assume equilibrium. I often find this unconvincing.

One can go back at least as far as Adam Smith, the distinction he draws between market prices and 'natural' prices, and his metaphor of a gravitational process. At any given time, some commodities may remain unsold on the market, and the quantity demanded for some may exceed the quantity supplied at a moment in time. The rate of profits may vary among firms and industries more than one might expect because of differences in risk, the desirability of certain industries, and so on. A leveling process that may never be completed exists at a given moment in time. Capitalists, reacting to price signals, will be disinvesting in some industries and expanding in other industries.

One who studies 'natural' prices, that is, the system of prices of production, is investigating tendencies, not making a claim that equilibrium exists. Even after the marginalists started constructing an incorrect theory, they kept this approach. Alfred Marshall wrote about, market prices, the short run, and the long run. Here is Walras:

Finally, in order to come still more closely to reality, we must drop the hypothesis of an annual market period and adopt in its place the hypothesis of a continuous market. Thus, we pass from the static to the dynamic state. For this purpose, we shall now suppose that the annual production and consumption, which we had hitherto represented as a constant magnitude for every moment of the year under consideration, change from instant to instant along with the basic data of the problem... Every hour, nay, every minute, portions of these different classes of circulating capital are disappearing and reappearing. Personal capital, capital goods proper and money also disappear and reappear, in a similar manner, but much more slowly. Only landed capital escapes this process of renewal. Such is the continuous market, which is perpetuating tending towards equilibrium without ever actually attaining it, because the market has no other way of approaching equilibrium except by groping, and, before the goal is reached, it has to renew its efforts and start over again, all the basic data of the problem, e.g. the initial quantities possessed, the utilities of goods and services, the technical coefficients, the excess of income over consumption, the working capital requirements, etc., having changed in the meantime. Viewed in this way, the market is like a lake agitated by the wind, where the water is incessantly seeking its level without ever reaching it. But whereas there are days when the surface of a lake is almost smooth, there never is a day when the effective demand for products and services equals their effective supply and when the selling price of products equals the cost of the productive services used in making them. The diversion of productive services from enterprises that are losing money to profitable enterprises takes place in various ways, the most important being through credit operations, but at best these ways are slow. It can happen and frequently does happen in the real world, that under some circumstantces a selling price will remain for long periods of time above the cost of production and continue to rise in spite of increases in output, while under other circumstances, a fall in price, following upon this rise, will suddenly bring the selling price below cost of production and force entrepreneurs to reverse their production policies. For, just as a lake is, at times, stirred to its very depths by a storm, so also the market is sometimes thrown into violent confusion by crises, which are sudden and general disturbances of equilibrium. The more we know of the ideal conditions of equilibrium, the better we shall be able to control or prevent these crises." -- Walras (1954: Lesson 35, Section 322).

So Walras did not think any economy would ever be in equilibrium. On the other hand, many may incorrectly think Austrians, like Ludwig von Mises, dispensed with the assumption of equilibrium. But here he is asserting that the assumption of equilibrium is necessary for economic theory:

One must not commit the error of believing that the static method can be used only to explain the stationary state of an economy, which, by the way does not and never can exist in real life, and that the moving and changing economy can be dealt with only in terms of a dynamic theory. The static method is a method which is aimed at studying changes; it is designed to investigate the consequences of a change in one datum in an otherwise unchanged system. This is a procedure which we cannot dispense with." -- Ludwig von Mises, 1933. Intervention. (quoted by Kurz and Salvadori)

I do think, however, one can criticize the Arrow-Debreu model as not being consistent with this approach and always assuming that equilibrium exists. Any time to reach equilibrium in the Arrow-Debreu is too long. Any such equilbirum that might have a tendency to be approached cannot be expected to be consistent with the data. Many supposed dynamic models in economics are still subject to this old objection.

Many questions remain about how to analyze whatever tendencies to equilbrium that may exist. I have barely even touched on the distinction between logical and historical time, a distinction commmon to Joan Robinson and Ludwig Lachmann.