by Filip Palda
Once again we see that the narrow market equilibrium of demand and supply is nestled within a broader logic. The conditions have to be right for property rights to be created. Just what these conditions are and how a society might acquire them is an ongoing topic of research.
In his book, Plagues and Peoples, William H. McNeill hypothesizes that markets emerge from a lengthy evolutionary process that culminates in a symbiosis between human predators and their human prey. At first, roving bands raid villages for food and put them to the torch. Over time some bands learn that they can extract more wealth by a less destructive and more systematic form of extraction. Thus taxes are born. Then the rulers may learn that by providing services such as education and health care, their subjects become richer and so better able to pay tax. Through symbiosis, governments and the markets they foster may develop.
However, despite a great deal of fancy intellectualizing, as yet we have no way of understanding why only a very few societies make the transition to a market economy. Nor do we understand why most of humanity is stuck in other forms of equilibria that have little to do with markets: internecine strife; clan alliances; foraging and self-sufficiency; communities living in harmony without courts of law or formal property. These phenomena are not easily categorized within a standard equilibrium framework. Think of them as the dark matter of economics. Integrating them into a model of social accounts that is as yet a feat of grand unification we are far from performing.
References
Becker, Gary S. 1983. “A theory of competition among pressure groups for political influence.” Quarterly Journal of Economics, volume. 98: 371–400.
Black, Duncan. 1948. “On the rationale of group decision-making.” Journal of Political Economy, volume 56: 23–34.
Coase, Ronald H. 1960. “The problem of social cost.” Journal of Law and Economics, volume 3: 1–44.
Demsetz, Harold. 2002. “Toward a theory of property rights II: the competition between private and collective ownership.” Journal of Legal Studies, volume 31: s653–s672.
Knight, Frank H. 1936. “The place of marginal economics in a collectivist system.” The American Economic Review, volume 26: 255–266.
Lange, Oscar. 1936. “On the economic theory of socialism: Part one.” The Review of Economic Studies, volume 4: 53–71.
Makowski, Louis and Joseph M. Ostroy. 1992. “General equilibrium and market socialism: clarifying the logic of competitive markets.” UCLA Economics Working Papers 672.
McNeill, William H. 1976. Plagues and Peoples. Anchor Press/Doubleday.
Myerson, Roger. 2008. “Perspective on mechanism design in economic theory.” American Economic Review, volume 98: 586–603.
Palda, Filip. 2011. Pareto’s Republic and the New Science of Peace. Cooper-Wolfling Press.
Pigou, Arthur Cecil. 1920. The Economics of Welfare. Macmillan.
Stiglitz, Joseph E. 1991. “The invisible hand and modern welfare economics.” NBER working papers series 3641.
Varian, Hal. 1984. Microeconomic Analysis, 2nd Edition. W.W. Norton.
GAMES 7
GAME THEORY IS THE EVEREST of economics. It daunts us with its seemingly bizarre view of how people interact and its irrefutable but mind-bending proofs of how so much can go wrong in human relations. But to the few who manage to understand its logic the theory makes sense of so many seemingly chaotic behaviors. In his Essay on Man, the late 1700s British poet Alexander Pope wrote, “All nature is but art unknownst to thee; all chance direction which thou canst not see; all chaos harmony not understood; all partial evil, universal good.” Those were the mysteries game theory sought to penetrate. Its inventors felt it was needed because of a shortcoming in classical economic theory. As John Harsanyi explained in his Nobel lecture
In principle, every social situation involves strategic interaction among the participants. Thus, one might argue that proper understanding of any social situation would require game-theoretic analysis. But in actual fact, classical economic theory did manage to sidestep the game-theoretic aspects of economic behavior by postulating perfect competition, i.e., by assuming that every buyer and every seller is very small as compared with the size of the relevant markets, so that nobody can significantly affect the existing market prices by his actions. Accordingly, for each economic agent, the prices at which he can buy his inputs (including labor) and at which he can sell his outputs are essentially given to him. This will make his choice of inputs and of outputs into a one-person simple maximization problem, which can be solved without game-theoretic analysis.(1995, 291).
The classical economics Harsanyi was talking about is based on the notion that people try to do as well as they can with what they have. The formal phraseology is that people maximize some objective, such as profit, or pleasure, subject to material constraints. For the consumer, income is usually a big constraint, and so are prices. For most of us, income and prices are fixed quantities. We have to take them pretty much as given. Most economists are content to see the world in this manner because this view makes sense of so much of what we see in markets and even in politics. But there are a few who believe that material constraints are not the whole story. They call themselves game theorists and what they call strategic interaction is really a form of decision making under a very special sort of constraint.
The constraint central to game theory is not material but rather mental. In 1953, von Neumann and Morgenstern illustrated these mental constraints by retelling the famous death chase between Sherlock Holmes and his arch-enemy, Professor Moriarty, the Napoleon of crime:
Sherlock Holmes desires to proceed from London to Dover and hence to the Continent in order to escape from Professor Moriarty who pursues him. Having boarded the train he observes, as the train pulls out, the appearance of Professor Moriarty on the platform. Sherlock Holmes takes it for granted and in this he is assumed to be fully justified that his adversary, who has seen him, might secure a special train and overtake him. Sherlock Holmes is faced with the alternative of going to Dover or of leaving the train at Canterbury, the only intermediate station. His adversary whose intelligence is assumed to be fully adequate to visualize these possibilities has the same choice. Both opponents must choose the place of their detrainment in ignorance of the other’s corresponding decision. If, as a result of these measures, they should find themselves, in fine, on the same platform, Sherlock Holmes may with certainty expect to be killed by Moriarty. If Sherlock Holmes reaches Dover unharmed he can make good his escape (1953, 177).
In this death chase Holmes’ best choice of an exit station depends on the station at which he expects Moriarty to disembark. This expectation in turn depends on the station Moriarty divines as Holmes’ escape. The story has the structure of an economic problem in that Holmes must attain some objective (in this case preserving his life), but he is not like the consumer of products who must passively accept the constraints imposed upon him by the economic environment. In this example, the constraint Holmes faces is Moriarty’s mind. That mind will conceive a strategy based on its anticipation of Holmes’ strategy. This means that Holmes’ own mind is shaping to some degree the constraints he faces. Unlike the passive consumer taking prices and making choices, Holmes’ choices can influence his possibilities. Holmes is forced into a “strategic interaction” with his environment. He does not choose his outcome, but rather a strategy that may or may not produce the outcome he wants. By taking charge of his environment he also determines in large part what the “equilibrium” outcome of the game is.
Obviously Holmes had his brain-work cut out for him. Before game theory, so did researchers wishing to analyze similar situations in which people constrain each other through their anticipated behavior. Today, the analytical apparatus these researchers developed in order to crack the mystery of strategic interactions is gradually becoming a standard part of economic thinking. Some go further in their assessment of the importance of game theory. Nobellist Roger Myerson claims that game theory holds the key
s to understanding how political institutions should be designed, and to answering whether capitalism is better or worse than socialism. Others, such as Bates Medal winner David Kreps are more tempered in their evaluation of the field. Despite these differences it is clear that here is a form of reasoning that needs to be learned by anyone wishing to embark on a mastery economics. The Holmes-Moriarty conflict captures the essential spirit of game theory and thus can be our low door in the wall to what you will come to understand is the wonderland of game theory.
The essence of games
THE PUTATIVE CONFRONTATION between Holmes and Moriarty can be boiled down to a few essential elements. Both are players with strategies that depend on the payoffs of any possible choice, and the way each player values these payoffs. Both must move “simultaneously”, as in the game of rock-paper-scissors, or matching pennies. That is, each must disembark from their respective trains at the same instant, whether they are in the same station, or in different stations.
More generally, game theorists take this to mean that even before the game starts all players will have devised their strategies and will not deviate from these. The players might as easily stay at home and mail their moves to a referee who opens the envelopes to determine who won and so administer the prescribed rewards and punishments. The prize in question is Holmes’ life, and his loss of it is as painful a prospect to him as it is pleasurable to Moriarty. This is called a zero-sum payoff structure because the gain to one player is exactly matched by the loss to the other. You can think of a zero-sum payoff as the case where a pickpocket removes a $100 bill from your back pocket. His or her gain is exactly your loss. The game is “one-shot” in that it will be played only once. This means that some sort of reciprocity cannot develop and neither player can punish the other in the future for vicious attacks in the past. Neither player can communicate with the other. No contract can be written between Holmes and Moriarty to limit the damages to each other. The conflict between them is total and unconstrained.
If these seem painfully unrealistic assumptions then have patience. Understanding so simply contrived an interaction flummoxed some of the finest minds of the 20th century. There is ample opportunity after studying this example to very quickly complicate the picture.
Superior intellect of no use
SO AT WHICH stations does each disembark? Conan Doyle has Holmes getting off at the first station (Canterbury), while Moriarty speeds by that station to alight at Dover, fooled by the maneuver. Conan Doyle’s reasoning for Holmes’ victory is absent. Game theory can fill the gap.
The first thing to note is that there is no way either antagonist can out-think the other to come up with a certain strategy that dominates every other possibility. If Holmes reasons that Moriarty will guess he wishes to escape at the first station, then Holmes must get off at the last station. But Holmes knows that Moriarty will know that Holmes knows, and so the right choice is to get off at the first station. Both antagonists can go on reasoning like this until both trains run off the end of the earth. Neither will be able to settle on a strategy, also called an “equilibrium”, if they are engaged in an “infinite regress of reciprocal expectations”.
There is no strategy Holmes can choose given his anticipation of Moriarty’s strategy that Holmes would not wish instantly to change. Children know this game as “matching pennies”. The idea is for each player to reveal either heads or tails to the other. One player wins if the faces match and the other wins if the faces do not match. As in the Holmes-Moriarty game, there is no certain choice any player can settle on in matching pennies. Game theorists call the absence of a complete, certain plan of action a “pure strategy”. There is no fixed pure strategy in an infinite regress.
There is, however, a way out of the death-chase regress. That is for each opponent to flip a coin, or roll a die, to stop at Canterbury with some probability, and Dover with the residue of that probability. If you flip a coin, then you will get off at either station with a chance of fifty per cent. If you roll a die with Canterbury on one side, and Dover written on the five remaining sides, then you will get off at Canterbury with a one-sixth chance, and off at Dover with a five-sixth chance. Many such random possibilities are open to the players.
Why can playing randomly lead to a definite strategy whereas playing with a sure strategy leads to an indeterminate infinite regress? You are about to learn one of the greatest theorems of economics, so kindly put your seat backs up and remove all sharp objects from pant and shirt pockets.
Put yourself in Holmes’ shoes with the broadened knowledge that you can pick a probability of getting off at Canterbury, with the residue of the probability being the chance you will get off at Dover. Now a vista of possible strategies opens. If you think that Moriarty will get off at Canterbury for certain and you chose a “degenerate” random strategy of setting your probability of getting off at Canterbury as zero then you can get off at Dover and you survive. But there is a potential downside. If Moriarty decides to get off at Dover you die. So your strategy of setting the probability of Canterbury as zero has a very wide range of possible rewards ranging from 100% certain death to 100% certain life. Now suppose instead you decide to get off at Canterbury with a 50% chance. If Moriarty’s strategy is to get off at Canterbury this means your chance of death is 50%. If he gets off at Dover, once again your die with 50% chance.
So the two extremes possible to you are actually the same. Your 50% strategy gives you a 50% chance of survival no matter what Moriarty does. Now suppose you decide to get off at Canterbury with 100% certainty. If Moriarty gets off there then you are dead and if at Dover you live. What I have just done is examine three mixed strategies. The two extreme strategies of certain disembarkation at one station or the other give the widest range of possible “payoffs” to the game from Holmes’ perspective. The middle strategy of flipping a coin gives the narrowest range of outcomes in an average or “expected value” sense.
In fact, I could have described a strategy of either 40% or 70% chance of getting off at Canterbury and would have found a range of expected values for Holmes somewhere in between his middle strategy and the two extreme strategies.
One can carry on this exercise until on a graph one sketches a “butterfly tie” of the range of possible payoffs to Holmes. Where Holmes chooses a chance of getting off at Dover of 100% the butterfly tie is at its widest since the certain strategies of Moriarty disembarking at either port present the stark possibilities of certain life, or certain death. The tie narrows to a point where Holmes chooses to flip a coin. At that point his expected value of the game does not depend on what strategy Moriarty chooses.
Of course in this exercise I have assumed Moriarty only has the possibility of choosing a pure, or rather non-mixed strategy of certain disembarkation at one station or the other. It is easy to show that if he plays various mixed strategies the butterfly tie retains its general shape, but the range of expected payoffs changes, except at one crucial point. The strategy of flipping a coin still gives Holmes the single expected value of 50% of surviving. And it is tedious but straightforward to show that if Moriarty adopts a coin flip he too will have a 50% chance of killing Holmes.
Is the 50% coin flip an equilibrium strategy for the two? It might be. What is clear is that with the coin flip there is no possibility of an infinite regress for Holmes because if he flips the coin there is no range of expected payoffs. A coin flip leaves Holmes indifferent to the strategy Moriarty adopts. Similarly the coin flip strategy leaves Moriarty indifferent to an infinite regress. We still do not have an answer. We need to ask whether either can do better than the coin flip knowing that the other is flipping. That is the test of whether an equilibrium is stable.
The answer depends on what “doing better” means. Which in turn depends on their preferences. Holmes might be extremely daring and not mind the risk of an extreme strategy of getting off for certain at Dover. In that case the coin flip is certainly not an equilibrium because a more daring strate
gy would seem more attractive. But what if Holmes is morbidly afraid of risk—so much so that he always tries to minimize the maximal possible loss that he can expect to realize?
We have all acted like this in our lives, though perhaps not always with pride. There are moments when we say it is better to take a course of action that will not give us the prospect of a brilliant success at the risk of great failure, but at least will put a floor on our losses. Some people approach marriage in this manner. Game theorists call these “minimax” preferences. They are also known as “absolute risk aversion” preferences.
If Holmes has minimax preferences then a coin flip is the mixed strategy he was born to play. It has the highest possible downside of all other possible degrees of randomness he might choose. Think of the bow tie. All strategies except at the knot offer higher possible expected gains but also lower possible gains than the knot. Moriarty faces a similar bow tie of expected returns on an up-down scale. He also faces a left-right scale of his chosen probability of getting off at Canterbury.
If you put the two horizontal probability scales and the unique expected value scale together (you can do that because this is a zero-sum game where returns are perfectly symmetric) then you get a three dimensional picture akin to a saddle. In fact the minimax solution of chosen probabilities and expected value can be found at the lowest part on the saddle where your rump would settle if you chose to mount the otherworldly beast that is the zero-sum game.
