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by Carlos Pedro Gonçalves (Submitted: 04/23/2007)

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This is a evolutionary quantum game theoretical model of a financial market, introduced and tested empirically by C.P. Gonçalves and C. Gonçalves (2007).


The model works with a single agent type known as value investor. Unlike the artificial financial market models, there is no fixed number of agents here, the number of agents fluctuates in time.

The model is divided in trading rounds, each corresponding to an evolutionary quantum game. For each trading round, the social system processes the information so that there is a quantum fluctuation from a vacuum state to a coherent state for the two strategies’ occupation number states (the two strategies being either a buying strategy or a selling strategy).

The market’s returns are the result of the difference between the number of buyers and sellers at the end of each trading round (game) divided by a liquidity parameter (that can be controlled by the user). The occupation number state at the end of each trading round is the result of a probabilistic collapse that follows Born’s rule.


The user can control seven parameters (all represented by sliders).

The liquidity parameter (lambda) transforms the difference between the number of buyers and sellers into returns. The parameter allows the user to control the size of the market fluctuations, and it should be set so that the market fluctuations match those of the actual markets.

The user can also control the processing time, which determines the quantum unitary evolution operator. The higher the processing time the higher will be the average number of agents that tend to enter the game (because they have more time to process the information).

This parameter also determines the size of the fluctuations so that it should be set in conjunction with the liquidity parameter.

The parameter ‘miu’ reflects an average returns on value, which leads the investors to tend to buy shares and hold them. The average returns on value is equal to ‘miu * processing time’ (see the paper for the theoretical explanation of this matter and for the relation with market equilibrium and the EMH).

The news sensitivity parameter controls for how the news affects the overall market sentiment (playing a similar role as in the previous artificial financial market models).

The main idea is that the social evaluation of value determines the average number of buyers and sellers in accordance with the quantum unitary evolution defined in the paper.

The social sentiment regarding value, for the k-th trading round (the k-th quantum game) is defined by:

V(k) = propensity-to-sentiment-contagion(k) * (number of sellers(k-1) – number of buyers(k-1)) + news-sensitivity * news-eigenvalue(k)

(The difference between the number of sellers and the number of buyers is the previous' round quantity after the probabilistic collapse (which corresponds to the agents decisions - see also the discussion, in the paper, on issues of relative decoherence and density matrix decoherence)).

There are two news' eigenstates, mimicking the previous artificial financial market. Using Dirac’s ket notation we could write: |good news> and |bad news>. The news' eigenvalue is +1 if the news were good and -1 if the news were bad. The parameter ‘news-sensitivity’ corresponds to the ‘sigma’ parameter in the paper.

The propensity for sentiment contagion follows Sornette and Zhou’s modification of the rule we proposed for the artificial financial market, where gamma corresponds to Sornette and Zhou’s alpha and beta is Sornette and Zhou’s beta parameter. Although the interpretation is different from Sornette and Zhou’s model (see this model’s paper for explanation of the differences).

The main dynamics reproduces the previous artificial financial market’s rule for sentiment contagion. In this sense if the news for the trading round vary in the same direction than the previous trading round’s returns, then the social system of potential investors tends to give more weight to the previous market polarization (technical indicator) and the propensity for sentiment contagion increases.

In the artificial financial market model, beta was set equal to 1, and gamma was equal to zero, the gamma term accounts for a social inertia in the propensity for sentiment contagion.

Formally the rule, for the k-th trading round, is written as:

sentiment-contagion(k) = base-sentiment-contagion + gamma * sentiment-contagion(k-1) + news-eigenvalue(k) * beta * returns(k-1)


As can be seen in the paper, the market captures several of the multifractal signatures present in actual markets. You should change the model’s parameters and see how the market changes for different parameters.

There are several market observables’ plotted:

(1) The price logarithm’s series;
(2) The price returns’ series;
(3) the market activity (number of buyers + number of sellers), which coincides, in this case with transaction volume;
(4) The returns in intrinsic time, which corresponds to a plot of the returns against the aggregated volume (taking volume as defining an intrinsic time scale), this shows how an intrinsic time scale, reported by Mandelbrot, may emerge spontaneously from the market dynamics;
(5) The sentiment contagion parameter (you should notice the relation between this parameter’s dynamics and the market dynamics).


This model is different in terms of code because it solely uses global variables, we did not use any turtles or patch primitives.


This model is in continuity with the previous artificial financial market models.


Gonçalves, Carlos Pedro and Gonçalves, Carlos. (2007). "An Evolutionary Quantum Game Model of Financial Market Dynamics - Theory and Evidence".

Sornette, Didier and Zhou, Wei-Xing. (2006). "Importance of Positive Feedbacks and Over-confidence in a Self-Fulfilling Ising Model of Financial Markets". Physica A 370 (2), 704-726, arXiv:cond-mat/0503607

Zhou, Wei-Xing; Sornette, Didier. "Self-organizing Ising model of financial markets". The European Physical Journal B, vol.55, no.2, pp.175-181, (2007).

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