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Motivations

At the start econophysicists tried to use various models to simulate financial market mechanism. Independently, many groups proposed multi-agent trading models

like the Santa Fe stock market etc. These models shed light on the richness and complexity of the financial markets, but at the same time, it was felt that with so many parameters to describe a real market, the complexity level becomes quickly unmanageable. On the other extreme end, it was deemed desirable to have a prototype model, so simple as a kind of Ising model, yet complex enough to capture some essential features of financial markets. One has here in mind the Ising Model, or the BTW Self-Organized Criticality Model, supposedly applies only to ideal sand piles, but offer insight into many natural phenomena. Let us list a minimal set of ingredients that are indispensable for modeling markets:

1. A large number of independent agents participate in a market;
2. Each agent has some alternatives in making his decision;
3. The aggregate activity results in a market price, which is known to all;
4. Agents use the public price history to make their decisions.

This set of ingredients is quite arbitrary, we omit notably two important factors: a) no fundamental news in the market besides their own trading activity; b) agents do not believe in any theory, they only learn from their own experience and believe, that the price history contains information. We want to study the inherent dynamics of market, in the absence of external influences. Real economy has both internal and external contributes. B. Arthur has advocated the so-called 'inductive thinking' approach, which represents a minority opinion in economics. His idea is that since an agent cannot use the Theory to deduct his decision, his only choice is to learn from his own experience, as many a trader would attest. Our own model is inspired from Arthur's El Farol problem. Below we shall describe The Minority Model, inspired from B. Arthur's thinking.

Minority Game. The simplest model we can think of is defined in a form of evolutionary game. Let us consider a populationof N players. At each time step, everybody has to choose to be on side A or B (e.g. buy or sell). The payoff of the game is that after everybody has chosen side independently, those who are in the minority side win. In the simplest version, all winners collect a point. The players make decisions based on the common knowledge of the past record. Thus the time series can be represented by a binary sequence, 1 or 0 meaning A or B is the winning side. Let us assume that our players are quite limited in their analyzing power, they can only retain last M bits of the system's signal and make their next decision basing only on these M bits. Each player has a finite set S of strategies. A strategy is defined to be the next action (to be in A,B, or 1,0), given a specific signal's M bits. Take M=3 as example, there are 8 possible distinct price histories, and there are a total of 256 of possible distinct strategies. The latter number grows astronomically with M=3, 4, 5,... as 256, 65536, 4294967296,... .

We randomly draw a few (S) strategies for each player, from the pool. All the S strategies in a player's bag can collect points depending if they would win or not given the M past bits, and the actual outcome of the next play. However, these are only virtual points as they record the merit of a strategy as if it were used each time. The player uses the strategy having the highest accumulated points for his action, he gets a real point only if the strategy used happens to win in the next play. The fact of using alternative strategies makes the players adaptive to the market. A player thus tends to maximize his capital (cumulated points) and his performance is judged only on his time-averaged capital gain. Several remarks are in order. By the very definition of minority agents are not encouraged to form commonly- agreed views on the market. This is like in real markets bears and bulls live together. In real trading it is often observed that a minority of traders get into a trend (buying or selling) first, the majority get finally dragged in also. When the minority anticipates correctly and get out of the trend in time, they pocket the profit at the expense of the majority. There is limited resource available for competition. If the players manage to coordinate well, per play they can expect (N − 1)/2 points, the maximal gain possible. Since our players are selfish, no explicit coordination is imposed, their fate is left to the market. The important question is if they can somehow learn to spontaneously cooperate. S = 1 simplifies further the model in which instead of players the strategies compete directly. The outcome is trivial deterministic signal. It is the extra layer of complexity at the player's level ensures adaptability. (Remember the role of hidden layers in Neural Network?)

This binary model is very suitable for numerical experiments. The results show complex, rich consequences of the model beyond expectation. The parameters of the model are just three: N,M, S. However, there hidden parameters. The space of the strategies is most intriguing. Apparently this space is so huge, that for realistic parameter values say M = 10, this space should be regarded as infinite (10 followed by more than 300 0's) for all practical purposes. Numerical experiments show that the space is far from infinite. Depending on N the market has distinct behavior, M = 10 can be too large or too small for achieving coordination. How can such large number still be relevant in this model? A more refined analysis of the strategy space can solve this paradox of large number.

There is a striking similarity with Kauffman's Boolean NK network. Consider two neighboring strategies which differ only by one bit, in other words the Hamming distance between them is one. These two strategies predict almost always the same outcome acting on the past record. Therefore the distinct strategies are highly correlated. Players using correlated strategies tend to obtain the same decision, thus hindering their chance finding the minority side. Among the strategies there is huge redundancy. If two strategies are uncorrelated, their decision outcomes should match with 1/2 probability. This is possible if their Hamming distance is 1/2 of the maximal value. We are thus led to count mutually uncorrelated strategies. This count will provide a crucial measure of diversity (or independence) of the total number of strategies. From graph theory we learn that there is a subset of 2M pairs of points, within every pair the Hamming distance is maximal. Within the subset the strategies are completely independent. It is this reduced number which plays an important role in the model. If the number of strategies in the population is larger than this number, then the players have to use strategies which are positively correlated, the herd effect is unavoidable despite the adaptability of our players. The herd effect will result in fluctuations larger than random chance would warrant, thus leads to waste of the limited resources.

Evolution.

We would like to let the population to evolve to the cooperative region without imposing the fixed parameters. To this end we introduce another level of adaptation by adding the Darwinism to this competition game. Worst player is periodically wed out of the game; best playeris allowed to produce an offspring. The new born player is a clone copy of his parent with the virtual capital reset to zero. A small mutation rate is assumed to ensure genetic diversity, i.e. one of his inherited strategies is replaced. The total population is kept constant without loss of generality. The newly drawn strategy can have its M changed by one unit. This permits the players to find the best suitable M. The population can nevertheless reach a stationary state, clearly now with inhomogeneous M. In short, the population is able to evolve without outside intervention and self-organize themselves to find the critical region, where it is beneficial for everybody.

Market Efficiency. The debate of whether a competitive market is efficient has been a hot topic in the last few decades. Academics, practitioners and general public are divided roughly into two camps: believers and non-believers of the so-called Efficient Market Hypothesis (EMH). Both camps have piled up mountains of evidence, yet no consensus can be said to have ever reached. Efficient markets are a natural consequence of neoclassical economics, little wonder most believers of EMH come from mainstream economics departments. Paul Samuelson, the most influential living economist, claimed to have 'mathemathically proven' that competitive markets are efficient. Championing this camp now is the Chicago finance professor Eugene Fama. Nonbelievers, on the other hand, most are practitioners in the finance industry, and they make a living out of feeding on the residual inefficiency left in the market. Their very existence, they'd say, is proof enough of the contrary of EMH. One may get some idea from the recent book Alchemy of Finance by George Soros. Academic economics is not a tight block, forceful opposition to EMH can be found in the writings of well-known economists such like Colin Camerer, Robert Shiller and especially Richard Thaler. Their approach is labelled sometimes as 'behavorial economics'.

Marginally Inefficient Markets (MEM) Theory was recently introduced as an alternative approach and perspective. The theory maintains that financial markets are open systems, with unlimited potential entrants. Financial markets sport two main categories ofplayers: producers and speculators. Producers are the players who pay less attention than speculators to exploiting market inefficiency, their economic activities outside the financial markets provide them better opportunities for profit than exploiting the market inefficiency. Yet their normal business conduct depends on heavily using the financial markets and their participation inadvertently injets the the elememts of predictability into the markets. They do this not because ofgenerosality but of inevitability. And they care less since their core business is the outside economy. Speculators are the players whose speciality is in 'scavenging' financial markets for the tiny predictability (hence inefficiency) left by others. They do this purely for their own greed and inadvertently they render a social service: they provide liquidity so that the producers can use the market in a large scale under a relatively efficient conditions (compared to still larger opportunities in outside economy).Why don't our speculators arbitrage away the residual inefficiency in the financial markets? The key lies in that the predictability elements in the financial markets are of probabilistic nature, no one can take out of the money without making frequent bets and taking on substanstial risks.

Minority Game as a Test

It's hard to test the above theory directly, as we cannot halt the normal financial operations to see who played what role. However, powerful analytic tools are available to test the theory. Minority Game can be made convenient for our purpose. Minority Game is a platform which is flexible enough to model quite a large class of multi-agents problem. In MG model players have a finite number of alternative strategies, the simplest case being two alternatives. Players have the choice of using the alternatives according to the market fluctuations. In most cases, more alternatives, more average gain. If a player for some reason has less alternatives than the others, one can show that in general this handicap leads to relative loss.

From the standard MG model we can divide the players into two classes: producers and speculators. To model the real producers we impose on our MG producers the handicap of only a single strategy, i.e. no other alternatives and their operation in the MG market is like an automaton. Our MG speculators are normal MG players with two alternative strategies. They have the capability, as in the standard MG model, to adjust their use of alternative strategies according market fluctuations. With the handicap, it's is an extreme case to model the producers' inflexibility facing market fluctuations. As discussed above, producers get profit from the outside economic activities and they don't mind their possible relatively small loss due to market fluctuations. Left with only one strategy it is not hard for the speculators to detect that the market price has some regularity in it. But it's still highly non-trivial price signal since each producer's fixed strategy is randomly drawn, and in principle they are all different from each other. Moreover, the population is mixed with speculators and producers. Players with no alternative or less alternatives tend to be rigid or less agile than others. A price signal containing some regular elements implies to have some predictability in it. Any predictability in the price history is by definition market inefficiency. From information theoretic point of view we say that the conditional entropy of the price signal is not maximized. Or, there is some negative entropy left in the price signal that a savvy speculator can detect and act on it. It appears that the MEM theory outlined above is just part of a much large theoretical framework, which will be discussed in full in a forthcoming book.

MG players probe human intelligence

The Interactive Minority Game (IMG) on the internet proposes to further use the MG in modelling real markets by using the MG to try to gain insight into the behaviour of humans in a simplified market situation. Individual humans compete against the traditional computer-controlled agents, whose characteristics can be fine-tuned through our understanding of the traditional MG: the human player can be presented with a very inefficient and easy-to-predict market, or with one that is much more complex and difficult.

Fundamentally, the Interactive Minority Game is a minority game like any other. A group of N agents must choose between two actions, +1 ('buy') or -1 ('sell'), with the intention to make the opposite choice to the majority of agents; agents' decisions are based on the knowledge of which was the minority or majority action of the last M turns. However, in order to make the game a little easier to play, we have had to change the presentation (though not the underlying game!) a little for the sake of the human player. The human player has the advantage of a greater memory than the computer players. His opponents, i.e. our MG players are always ready, at pre-assigned values of M, S, and N. Since we know everything about our MG machine players, almost nothing about the human player, whatever difference the human player makes can be a posteriori deducted from our results. The experiments were done during 2003 and 2004, and it turns out that many players, not only physicists, scientists, but many total strangers signed up online to player the interactive game. These human players (about 500 in total), played 5 different difficuly-levels games, and in each category there are surprising winners. For example, the designers, i.e. the Fribourg team, are not always winners. There are players from as far as Brazil, Russia scored remarkably high, they surely have captured some information that our MG players, or ourselves were not able to detect!

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