Is the Efficient Market Hypothesis a Nash Equilibrium ?

The efficient markets hypothesis (EMH) maintains that market prices always fully reflect all available information. Developed independently by Paul A. Samuelson and Eugene F. Fama in the 1960s, this idea has been applied extensively to theoretical models and empirical studies of financial securities prices, generating considerable controversy as well as fundamental insights into the price-discovery process. The most enduring critique are psychologists and behavioral economists who argue that the EMH is based on counterfactual assumptions of human behavior, that is, rationality. The recent advances in evolutionary psychology along with the cognitive neurosciences may be able to reconcile the EMH with behavioral anomalies. Behavioral scientists also emphasize that the arbitrage activities of rational professional investors, who might expect to bring stock prices back to fundamental values, are often impossible to execute. Mostly they are risky in any event, therefore limited.
Hedge funds and algorithmic traders in particular thrive on these imperfections. Driven by profit opportunities, an army of investors pounce on even the smallest informational advantages , and try to incorporate their information into market prices and quickly eliminate the profit opportunities that first motivated their trades. ‘The dilemma on how to define and measure the informational efficiency of actual markets and an optimistic view of market behavior (EMH, Surowiecki,2005) who claims that crowds especially when organized in markets and/or democratic institutions know more than individuals(“Wisdom of the crowds”) poses a classic prisoners dilemma framework for understanding how to strike a balance between cooperation and competition using the most famous game theory tool for strategic decision making’. Nash Theory of non cooperative game is one of the outstanding intellectual work. It reflects the fundamental impact in economics and social science. John Nash (1950) formally defined an equilibrium of a noncooperative game to be a ‘profile of strategies, one for each player in the game, such that each player’s strategy maximizes his expected utility payoff against the given strategies of the other players.’
In this article we will analyze and evaluate whether efficient market hypothesis is Nash Equilibrium in a Prisoner’s Dilemma game ? To do any kind of analytical process we must formulate a 2 player game theory model that includes both descriptive behavioral analysis of the players that we are studying and also prediction of the market motivated to maximize its own efficiency.
Game Model
Behavioral finance is one of the most sought after subject and so do game theory prisoner’s dilemma and Nash equilibrium. I’ll design a 2 person game theory to evaluate a hypothesis that efficient market hypothesis is the Nash Equilibrium. Lets consider a very simplistic model, we will analyze two players say hedge Fund A and hedge fund B and their arbitrage positions trying to merit over market inefficiency. We will take a very basic arbitrage approach where stock of a global company is selling at two different prices suppose $8 and $3 in New York Stock Exchange and London Stock Exchange respectively. The players are trying long and short stock market share in different trading exchanges to maximize their profit in iterated prisoners dilemma game playing multiple times in a market where all information are not readily available. So the hedge fund A & B can take action of the inefficient and efficient market and arbitrage the price difference. Arbitrage is the simultaneous purchase and sale of an asset in order to profit from a difference in the price. It is a trade that profits by exploiting price differences of identical or similar financial instruments, on different markets or in different forms. The two players has the option to take advantage of in-efficient market hypothesis(I-EMH) or efficient market hypothesis (EMH). The payoffs to a player affect its own and other’s reproductive success. EMH and I-EMH are the two strategies.
No matter, what the other player does, the selfish choice of defection yields a higher payoff than cooperation in prisoner’s dilemma. Similarly here the hedge funds are trying to outperform each other in an inefficient market to maximize their profit output, which is the temptation payoff. But in order for one hedge fund to maximize its profit means the other hedge fund payoff should be zero which means either the hedge fund has not arbitrage the stock price difference or has taken a market neutral position. Here the temptation pay off is profit of $8-$3= $5, considering the information is not available to anyone i.e. the other player payoff is zero. The reward pay off is when the both the players arbitrage the position, hence the value of the pay off is lesser than temptation payoff .The reason is reward strategy which is arbitrage action of both the players raises the stocks in less valued market. The reward payoff is $8-$5= $3, here the price of the lower stock is trying to catch up with the higher price, as both the hedge funds are trying to compete against each other.
Temptation pay off > Reward pay off.
Robert Axelrod (1980) played the iterated prisoner’s dilemma game where strategies were submitted by game theorists in economics, sociology, political science and mathematics. The result of the tournament was that the highest average score was attained by the simplest of all strategies, TIT FOR TAT(TFT). This strategy is very basic where the other players cooperate and then mirror whatever the other player did on the preceding move. TFT is a strategy of cooperation based on reciprocity.
In this game model in an iterated prisoner’s dilemma, when the hedge funds tries to mirror each other action to maximize their profit they drive the stock price to sucker payoff bringing both the stock price at the same value. The arbitrage squeezes profit from the market inefficiency. The value of sucker payoff is considered as $1 (nominal representation of the minimal payoff)
This prisoner’s dilemma game model is shown below in a simple diagram.

The only difference in this model is playing defection is taking advantage of inefficient market and getting temptation payoff, so the payoffs are switched from a prisoner’s dilemma. There are other shortcomings of this hypothesis game model, which will be discussed, in the next segment.
Discussion
The hypothesis game model discussed in this article is an iterated prisoner’s dilemma game played by 2 players. In real market, n players play such games in multiple variable conditions.
But before diving into various anomalies, I would discuss the hypothesis — how efficient market hypothesis is Nash equilibrium. Market adjusting to prices based on new information is instantaneous in today’s information overflowing era. EMH does not imply that prices will always be “correct” or that all market participants are always rational. There may be abundant evidences that many market participants are far from rational and suffer from systematic biases in their processing of information and their trading activities. But even if price setting were always determined by rational profit maximizing investors, prices can never be “correct”.
The key characteristic of this model is that players’ expectations do not follow a fixed rule such as rational expectations rule. Instead players choose among an evolving set of expectation rules depending on which ones have proved to be the most successful predictors of recent stock-price changes. Using this framework, we show first that in a market in which all other traders follow strictly fundamental rules of the kind that would characterize efficient-market equilibrium, an individual player might gain from adding arbitrage rules (or other complex pure play) to her repertoire of forecasting techniques. Second, using a game theoretic analysis, we show that while the use of arbitrage rules (in addition to fundamental rules) is the optimal strategy of a single player, the use of this strategy by all players in the market drives the market to a symmetric Nash equilibrium where all players use only fundamental rules. Adoption of various arbitrage other than pure play (alpha seeking focus) rules by all players in the market add to the noise in the market. Thus it makes more difficult for everyone to predict future stock-price movements than in a regime where only fundamental rules are used.
A common explanation of departures from EMH is that investors do not always react in proper proportion to new information. For examples, in some cases investors may overreact to performance, selling stocks that have experienced recent losses or buying stocks that have enjoyed recent gains. Such overreaction tend to push prices beyond their fair/rational market value, only to have rational investors take the other side of the trades and bring prices back in equilibrium eventually. An implication of the inefficient market phenomenon is price reversals: what goes up must come down.
The shortcomings of the hypothesis discussed in the article are multiple. Arbitrage discussed in this game model is very basic (hypothetical), in real world its very complicated and not risk free. There are multiple arbitrage like statistical arbitrage, convertible arbitrage, fixed income arbitrage, market neutral (bond, stock) strategy will not survive and cannot be explained in this simple 2x2 game prisoner’s dilemma model. The most common challenge to the EMH is the anomaly, a regular pattern in an asset’s returns which is reliable, widely known, and inexplicable. The fact that the pattern is regular and reliable implies a degree of predictability and the fact that the regularity is widely known implies that many investors can take advantage of it eg: “January Effect”. In a repeated Prisoners dilemma game for n periods, where n is commonly known to two players, a pure strategy in this game is a plan that prescribes which action is to be taken at each stage, contingent on every possible history of the game to that point. Nevertheless, all Nash equilibria of this finitely repeated game involve defections at every stage. When the number of stages n is large, equilibrium payoffs that could have been attained under mutual cooperation. It has sometimes been argued that the Nash prediction in the finitely repeated prisoner’s dilemma (and in many other environments) is counterintuitive and at odds with experimental evidence. However, experimental tests of the equilibrium hypothesis are typically conducted with monetary payoffs, which need not reflect the preferences of subjects over action profiles. In other words individual preferences over the distribution of monetary payoffs may not be exclusively self-interested. Furthermore, the equilibrium prediction relies on the hypothesis that all subjects commonly know these preferences, which is a likely similar challenge in the efficient market hypothesis.
Contrarian strategies citing momentum “to beat the market” identifies that the investors does the opposite of what other investors do. In other words, he buys/sells when others sell/buy. Their logic centers on the notion that they buy when others dump a stock because they think that its price might have fallen too far. Once the other investor realize that they have overreacted to the bad news , the stock’s price rebounds toward its fundamental value.
Conclusion
How efficient market theory goes wrong one clear example is the pricing of the shares of Royal Dutch and Shell, both are independently incorporated companies formed an alliance to merge their interests on a 60–40 basis while remaining separate and distinct entities. EMH reflects the law of one price, but the market deviated from the 60–40 parity of both of their stocks. Royal Dutch traded at an 8% to 10% premium relative to Shell and the parity only widened more with crisis. This bet against market inefficiency lost money, and a lot of money if leveraged. The inefficiency in the pricing of Royal Dutch and Shell is a fantastic embarrassment for the efficient market hypothesis; it shows that deviation from efficiency can be large and persistent, especially with no catalyst to bring markets back to efficiency. It shows market forces need not be strong enough to get prices in line even when many risks can be hedged, and that rational and sophisticated investors can lose money along the way, as mis pricing deepens. This is completely contrary to the assumption we had in our hypothesis.
To summarize the game theory model discussed here in this article, where pricing inefficiency drives the greed of two players to market efficiency is the Nash Equilibrium in a prisoner’s dilemma game is the hypothetical manifestation of efficient market hypothesis condition. There are various efficiency forms such as weak form, semi-strong form, strong form efficiency that have not been discussed in this paper, hence poses question, opening to evaluation and more criticism of the current proposal of Nash Equilibrium. Poundstone (1992) states that –“ The limits of an iterated prisoner’s dilemma may be unknown, but there are good limited strategies. Ignorance is bliss: TIT FOR TAT is better than Nash Equilibrium all-inclusive strategy of constant defection. Hence the Efficient Market Hypothesis model being the fundamental rule of the market poses question to the constant defect strategy (Nash Equilibrium) where it has been repeatedly shown that cooperation rules (Axelrod, tournament).
References:
The Economist Magazine — March 5th 2009 — The grand illusion
Andrew W Lo — EFFICIENT MARKETS HYPOTHESIS –
Burton G. Malkiel - — The Efffcient Market Hypothesis and Its Critics — Journal of Economic Perspectives — Volume 17, Number 1 — Winter 2003 —
Computational Finance 1999 by Yasar S. Abu-Mustafa
Bodie Kane Marcus — Investments 10th edition
The Wisdom of Crowds, James Surowiecki
Non-cooperative games John Nash (Oct 11, 1950) Annals of Mathematics
William Pounstone (1992) Prisoners Dilemma — Prisoners Dilemma Chapter
Robert Gibbons — Game Theory for Applied Economics- Basic Theory Page
William Poundstone (1992) Prisoners Dilemma
Adam Smith — Wealth of Nation , Invisible hand
William Poundstone (1992) Prisoner’s Dilemma
Robert Axelrod — Evolution of Cooperation, Chapter 2 — Success of TIT for TAT in computer tournaments
Understanding Investments : Theories and Strategies — Nikiforos Laopodis
William Poundstone(1992) — Prisoner’s Dilemma Page
