- Email: [email protected]
The minority game with different payoff functions: crowd–anticrowd theory
Available online at www.sciencedirect.com
Physica A 321 (2003) 309 – 317
www.elsevier.com/locate/physa
The minority game with di!erent payo! functions: crowd–anticrowd theory Kuen Leea , P.M. Huia;∗ , N.F. Johnsonb a Department
of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China b Department of Physics, Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, UK
Abstract The crowd–anticrowd theory is applied to explain the features observed in a class of the minority game using di!erent payo! functions. Simulations results using both the full strategy space and a reduced strategy space reveal that the standard deviation (SD) in the number of agents making a particular decision over time as a function of the agents’ memory size m does not depend on the explicit form of the payo! function. The robustness of the results is explained in terms of the general features in strategy selection among the agents in di!erent regimes of m. While di!erent payo! functions may a!ect the popularity of a particular strategy, the strategy selection is found to be insensitive to the choice of payo! functions. The crowd–anticrowd cancellation e!ect leads to a minimum in SD at an intermediate value of m separating the small m regime characterized by crowd behavior and the large m regime characterized by random coin-toss behavior of the agents. c 2002 Elsevier Science B.V. All rights reserved. PACS: 01.75.+m; 02.50.Le; 05.40.+j Keywords: Complex adaptive systems; Econophysics; Agent-based models
1. Introduction The minority game (MG) [1–6] has become a standard toy-model for studying a wide range of interesting phenomena in dynamical, complex adaptive systems [7], just as the role played by the Ising model in the investigation of phase transitions and critical phenomena. The MG aims at modelling a competing population in which there ∗
Corresponding author. Tel.: +852-26096351; fax: +852-26035204. E-mail address: [email protected] (P.M. Hui).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/03/$ - see front matter doi:10.1016/S0378-4371(02)01786-7
310
K. Lee et al. / Physica A 321 (2003) 309 – 317
are always more losers than winners. It is not unusual a situation when a population of agents are continuously competing for limited resources. As a simpliGcation of the bar-attendance model of Arthur [8] in which a population of agents decide whether to go to a bar for entertainment and at the same time to avoid a big crowd [9], the MG is a binary game in which selGsh agents repeatedly compete to be in a minority. The agents adapt by switching from one strategy to another, based on the past performance of their strategies. The performance of these strategies is, in turn, determined by the collective actions of the agents. Thus, the MG constitutes a simple, yet non-trivial, model with strong feedback and internal frustration. The MG also o!ers a simple paradigm for the complicated decision dynamics of ‘real-world’ agents in everyday situations. An example is that of traders in a market trying to decide whether to buy or to sell—more buyers than sellers implies higher prices and hence more beneGt to the sellers. Subsequent to the pioneering work [1–3] of Challet and Zhang, the MG has been extensively studied [5,6,4,10–13]. In particular, Johnson et al. [11–13] proposed a crowd–anticrowd theory of the MG in which features in the MG are explained in terms of the collective actions of a crowd of agents using similar strategies and an anticrowd using strategies opposite to that of the crowd. Li et al. [14] recently proposed and studied numerically variations of the MG in which the performance of the strategies is quantiGed by a payo! function depending on the size of the minority group (and hence the size of the majority group) in each turn. Interestingly, the main features in the modiGed models are found to be similar to that of the basic MG. In particular, the Luctuations in the number of agents making a particular decision in time characterized by a standard deviation is found to have a robust dependence on the agents’ capability. In the present work, we aim to explain the observed robustness when di!erent payo! functions are used. By invoking the idea of a reduced strategy space, we Grst establish that the robustness as seen in Ref. [14] for simulations in the so-called full strategy space carries over to numerical studies within a reduced strategy space. The reduced strategy space is a representative collection of strategies in the full strategy space. Strategy selection can be conveniently studied within the reduced strategy space. Interestingly, the robustness of the features can be explained in terms of popularity ranking of the strategies and strategy-play among the agents [11–13]. While di!erent payo! functions alter the cumulative performance of each of the strategies in the strategy space and thus the relative popularity ranking of the strategies, it is found that the underlying general features of the popularity ranking are unaltered by the choice of payo! functions, and the robustness in the dependence of the Luctuation on the agents’ capability follows from the general features of the popularity ranking. The plan of the paper is as follows. In Section 2, we deGne the MG and introduce di!erent payo! functions for assigning virtual points as an indicator of the performance of the strategies. In Section 3, we show that the results of numerical simulations on the models using the reduced strategy space [1–3,5,6] are in general the same as those obtained by using the full strategy space. Within the reduced strategy space, we then apply the analytical expression within the crowd–anticrowd theory to evaluate the standard deviation [12,13]. The evaluation relies on the di!erence in the number of agents using anti-correlated pairs of strategies and thus depends on the strategy
K. Lee et al. / Physica A 321 (2003) 309 – 317
311
selection among the agents. The results of the crowd–anticrowd theory are shown to be in good agreement with numerical results, hence supporting the idea of explaining general features in the MG and its variations in terms of the crowd e!ects. 2. Model The basic minority game consisting of an odd number N of agents repeatedly competing to be in a minority group. In each turn of the game, each agent decides between two possible options, say 0 and 1. The winners are those in the side with fewer agents. The outcome, i.e., the winning side, is made known to all agents. The agents have similar capability and they all decide based on the m most recent outcomes. Since there are a total of 2m possible bit-strings of length m constituting all the possible history bit-strings and there are two possible options for each history bit-string, there m are a total of 22 strategies in the full strategy space. At the beginning of the game, each agent picks s strategies from the pool of strategies, with repetitions allowed. At each turn, every agent uses the best performing strategy among his s strategies to decide. An agent records the performance of his strategies in the basic MG as follows. After each turn, the agent assigns one (virtual) point to each of his strategies which would have predicted the correct outcome. The most striking feature in the MG is that the standard deviation (SD) in the number of agents making a particular decision, say 0, is a non-monotonic function of the memory size m, with a minimum in the SD occurs at the value of m satisfying 2 · 2m ∼ N · s. The full strategy space grows rapidly with m. Fortunately, it has been shown that the SD in the basic MG is essentially unchanged if a ‘reduced’ strategy space is used instead of the full strategy space [1–3,12]. The reduced strategy space only contains 2m+1 strategies or equivalently 2m pairs of strategies G. The two strategies within a given pair G are anti-correlated, i.e., they di!er by the maximum Hamming distance dH = 2m and they make opposite predictions for a given history bit-string. Strategies between any two pairs G and G are uncorrelated, i.e., they di!er by dH = 2m−1 . In the present work, numerical simulations on the MG with di!erent payo! functions are carried out using the reduced strategy space. In the basic MG, one virtual point is assigned to each strategy that would have predicted the correct outcome regardless of the size of the minority group. However, the distribution of the whole population in the two possible options reLects how successful the strategies are in their predictions. For example, when the population splits itself into (N − 1)=2 and (N + 1)=2 in the two options, the successful strategies have only barely predicted the winning side. To reLect better the performance of the strategies, Li et al. [14] proposed modiGed models of MG in which the virtual points are assigned according to some payo! functions A(r), where r = n=N is the ratio of the number of agents in the winning (minority) side to the total number of agents in the population. In particular, they studied A(r) of the forms A(r) ∼ r − and A(r) ∼ e− r numerically using the full strategy pool. Here, and are constants setting the scale of reward to the successful strategies. The most interesting result is that the Luctuation in the number of agents making a particular choice, characterized by the standard deviation
312
K. Lee et al. / Physica A 321 (2003) 309 – 317
(SD) in the distribution of agents over time, shows the same dependence on m as in the basic MG, regardless of the form of the payo! functions. Note that the case of A(r) = 1 reduces to the basic MG. In what follows, we Grst establish that simulation results obtained using the reduced strategy space are the same as those obtained using the full strategy space. We then apply the crowd–anticrowd theory [12,13] to show that the robustness of the dependence of the SD on m is a result of the crowd e!ect in the strategy-play among the agents. 3. Results and discussion As it is more convenient to apply the crowd–anticrowd theory within the context of the reduced strategy space, we have carried out detailed numerical simulations for the payo! functions studied in Ref. [14] using the reduced strategy space. Fig. 1 shows the numerical results (symbols) for the SD in the number of agents choosing a particular option, say 0, as a function of m for the four di!erent payo! functions e−4r ; e0:1r ; r −2 and r −1 studied in Ref. [14]. Our results show the same features as those reported in Ref. [14] obtained using the full strategy space. Thus, as in the basic MG [2–4], we found that the reduced strategy space and the full strategy space give the same dependence of the SD on m when di!erent payo! functions are used. The dependence of SD on m, as pointed out in Ref. [14], is also the same as the basic MG. This robustness indicates that there are some intrinsic features common to the basic MG and the modiGed MG that are independent of the choice of the payo! function. For the basic MG, the crowd–anticrowd theory gives a physically transparent explanation of the m-dependence of the SD. Qualitatively, the physics behind the crowd–anticrowd e!ect is the following [11–13]. In the small m regime referred to as the ‘eQcient’ or ‘crowded’ phase, the strategy-play among the agents is that there exists crowd of agents using a strategy in the absence of a corresponding anticrowd using the anti-correlated partner of the strategy. This leads to a crowd of agents choosing a particular option and thus a large SD. For large m, the total number of possible strategies is much larger than the number of strategies actually being picked in the game. Therefore, crowd formation is unlikely and the agents e!ectively behave as if they are making their decision randomly. This limit is referred to as the random coin-toss limit. For some intermediate value of m, the strategy pool size and the number of strategies in play are comparable. This leads to a delicate cancellation e!ect between the action of a crowd using a particular strategy and that of an anticrowd using the anti-correlated strategy. Since an anti-correlated pair of strategies make opposite predictions, crowd and anticrowd of comparable sizes lead to a nearly even distribution of the agents in the two options and thus a small SD. The crowd–anticrowd theory can also be extended to study the dynamics of the MG [13]. An important implication of the crowd–anticrowd theory is that the m-dependence of SD in MG, and hence in the modiGed MG with di!erent payo! functions, is intrinsically related to the strategy-play or strategy selection among the agents. It should be noted that the validity of the crowd–anticrowd theory does not depend on which strategy
K. Lee et al. / Physica A 321 (2003) 309 – 317 A(r)= e
-4r
313 0.1r
A(r)= e
15
15 simulation theory
simulation theory
10 SD
SD
10
5
0
5
0
5
10
0
15
0
5
m A(r)= r
10
15
m -1
-2
A(r)= r
20 20 simulation theory
15
simulation theory
SD
SD
15 10
5
10 5
0 0
5
10 m
15
0 0
5
10
15
m
Fig. 1. The standard derivation (SD) in the number of agents making a particular decision as a function of the memory size m for four di!erent payo! functions A(r) = e−4r ; e0:1r ; r −1 , and r −2 . The symbols are numerical results obtained using a reduced strategy space. The solid lines are results of the crowd–anticrowd theory. Other parameters are: N = 101 and s = 2.
being the most popular, i.e., being used by more agents than the other strategies. As it is expected that di!erent payo! functions only a!ect the popularity ranking among the strategies, the crowd–anticrowd theory should remain applicable and the same scenario on strategy-play occurs as in the basic MG. Within the crowd–anticrowd theory, the SD can be expressed in terms of the di!erence in the numbers of agents using a strategy and its anti-correlated partner. Let nR be the number of agents using a strategy labelled R and nRR be the number of agents using the strategy RR that is anti-correlated to R. It is the di!erence |nR − nRR | in each crowd–anticrowd pair (nR ; nRR ) that contributes to the SD. Following Refs. [12,13], the SD can be expressed as 1=2 2 1 = : (1) 4 |nR − nRR | R (R;R)
314
K. Lee et al. / Physica A 321 (2003) 309 – 317 4 m=2 3
A(r) = e
-4r
m=5
P(k’=k)*2
m+1
m=10 2
1
0 0
0.2
0.4
0.6
k’/2
0.8
1
0.8
1
m+1
4 m=2
A(r) = e
3
0.1r
m=5
P(k’=k)*2
m+1
m=10 2
1
0 0
0.2
0.4
0.6
k’/2
m+1
R for k = 1 evaluated numerically for payo! functions of the form Fig. 2. The probability function P(k = k) A(r) = e− r with = 4 and = −0:1. Results for three di!erent values of m characterizing three di!erent regimes are shown.
Implicit in Eq. (1) is a conGguration average taken over di!erent initial strategy distributions among the agents in addition to a time average in obtaining the SD [12,13]. For each of the payo! functions, we run the numerical simulations and read out the number of agents playing each of the 2 · 2m strategies for every time step within a time window. Eq. (1) is then used to evaluate the SD. A conGguration average over 32 di!erent initial strategy distributions is taken. Results (solid lines) are shown in Fig. 1 and they are in very good agreement with simulation results.
K. Lee et al. / Physica A 321 (2003) 309 – 317
315
4 m=2
A(r) = r -2
3
m=5
P(k’=k)*2
m+1
m=10 2
1
0 0
0.2
0.4
0.6
k’/2
0.8
1
0.8
1
m+1
4 m=2 3
A(r) = r
-1
m=5
P(k’=k)*2
m+1
m=10 2
1
0 0
0.2
0.4
0.6
k’/2
m+1
R for k = 1 evaluated numerically for payo! functions of the form Fig. 3. The probability function P(k = k) A(r) = r − with = 1 and 2. Results for three di!erent values of m characterizing three di!erent regimes are shown.
To investigate the change in the strategy-play among the agents for di!erent regimes of m, we further study the popularity ordered list {nk } of the strategies, where the strategy k = 1 is that used by the largest number of agents, strategy k = 2 is that used by the second largest number of agents, etc. The subtlety in strategy-play is best R for k = 1, i.e., the probability that illustrated by the probability function P(k = k) the anti-correlated partner of the most popular strategy (k = 1) has a ranking of k R for k = 1 for the in the popularity ordered list [12]. Fig. 2 (Fig. 3) shows P(k = k)
316
K. Lee et al. / Physica A 321 (2003) 309 – 317
two payo! functions of exponential (power-law) form for three di!erent values of m characterizing di!erent behavior. The horizontal axis is the ranking k normalized by the total number of strategies 2·2m . Results are obtained by numerical simulations using the reduced strategy space. The following general features, similar to those in the basic MG, are observed. For small m (e.g., m = 2), the anti-correlated partner of the most popular strategy is likely to be the least popular. Thus, there is a crowd using the most popular strategy. The anticrowd using the least popular strategy is small, by deGnition. This crowd e!ect leads to the large SD in the small m regime. A striking behavior for intermediate values of m (e.g., m = 5) is that the anti-correlated partner of the most popular strategy is also highly popular. In this case, n1 and n1R are approximately the same and the contribution to the SD is small. This is the crowd–anticrowd cancellation e!ect which leads to a suppression in SD. For large m (e.g., m = 10), the agents R for k = 1 results. The robustness in behave as random agents and a Lat P(k = k) the m-dependence of SD in the modiGed MG thus stems from the general features in the strategy-play among agents in di!erent regimes of m which are independent of the payo! functions. While the speciGc form of the payo! function may alter the strategy which is the most popular, the general features in the relative ranking of the anti-correlated partner to the most popular strategy remains unchanged for di!erent payo! functions. In summary, we have shown that the m-dependence of the SD in a class of MG with di!erent payo! functions can be explained in terms of the formation of crowd and anticrowd. The minimum in the SD is a result of the formation of crowd and anticrowd of comparable sizes. The robustness of the m-dependence of the SD is due to the invariance in the nature of strategy selection among the agents. The crowd– anticrowd theory therefore captures the essential physics of the crowd or herd behavior in a competing population as modelled by the MG and its variations.
Acknowledgements This work was supported in part by the Research Grants Council of the Hong Kong SAR Government under Grant No. CUHK4241/01P. We thank T.S. Lo for useful discussions.
References [1] [2] [3] [4] [5] [6] [7] [8]
D. Challet, Y.C. Zhang, Physica A 246 (1997) 407. D. Challet, Y.C. Zhang, Physica A 256 (1998) 514. D. Challet, Y.C. Zhang, Physica A 269 (1999) 30. D. Challet, M. Marsili, R. Zecchina, Phys. Rev. Lett. 84 (2000) 1824. R. Savit, R. Manuca, R. Riolo, Phys. Rev. Lett. 82 (1999) 2203. R. Manuca, Y. Li, R. Riolo, R. Savit, Physica A 282 (2000) 559. J.H. Holland, Emergence, Oxford University Press, Oxford, 1998. W.B. Arthur, Amer. Econ. Assoc. Papers Proc. 84 (1994) 406.
K. Lee et al. / Physica A 321 (2003) 309 – 317 [9] [10] [11] [12] [13] [14]
317
N.F. Johnson, S. Jarvis, R. Jonson, P. Cheung, Y.R. Kwong, P.M. Hui, Physica A 256 (1998) 230. D. Challet, M. Marsili, Phys. Rev. E 62 (2000) 1862. N.F. Johnson, M. Hart, P.M. Hui, Physica A 269 (1999) 1. M. Hart, P. Je!eries, N.F. Johnson, P.M. Hui, Physica A 298 (2001) 537. M. Hart, P. Je!eries, P.M. Hui, N.F. Johnson, Eur. Phys. J. B 20 (2001) 547. Y. Li, A. Vandeemen, R. Savit, Physica A 284 (2000) 461.
Physica A 321 (2003) 309 – 317
www.elsevier.com/locate/physa
The minority game with di!erent payo! functions: crowd–anticrowd theory Kuen Leea , P.M. Huia;∗ , N.F. Johnsonb a Department
of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China b Department of Physics, Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, UK
Abstract The crowd–anticrowd theory is applied to explain the features observed in a class of the minority game using di!erent payo! functions. Simulations results using both the full strategy space and a reduced strategy space reveal that the standard deviation (SD) in the number of agents making a particular decision over time as a function of the agents’ memory size m does not depend on the explicit form of the payo! function. The robustness of the results is explained in terms of the general features in strategy selection among the agents in di!erent regimes of m. While di!erent payo! functions may a!ect the popularity of a particular strategy, the strategy selection is found to be insensitive to the choice of payo! functions. The crowd–anticrowd cancellation e!ect leads to a minimum in SD at an intermediate value of m separating the small m regime characterized by crowd behavior and the large m regime characterized by random coin-toss behavior of the agents. c 2002 Elsevier Science B.V. All rights reserved. PACS: 01.75.+m; 02.50.Le; 05.40.+j Keywords: Complex adaptive systems; Econophysics; Agent-based models
1. Introduction The minority game (MG) [1–6] has become a standard toy-model for studying a wide range of interesting phenomena in dynamical, complex adaptive systems [7], just as the role played by the Ising model in the investigation of phase transitions and critical phenomena. The MG aims at modelling a competing population in which there ∗
Corresponding author. Tel.: +852-26096351; fax: +852-26035204. E-mail address: [email protected] (P.M. Hui).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/03/$ - see front matter doi:10.1016/S0378-4371(02)01786-7
310
K. Lee et al. / Physica A 321 (2003) 309 – 317
are always more losers than winners. It is not unusual a situation when a population of agents are continuously competing for limited resources. As a simpliGcation of the bar-attendance model of Arthur [8] in which a population of agents decide whether to go to a bar for entertainment and at the same time to avoid a big crowd [9], the MG is a binary game in which selGsh agents repeatedly compete to be in a minority. The agents adapt by switching from one strategy to another, based on the past performance of their strategies. The performance of these strategies is, in turn, determined by the collective actions of the agents. Thus, the MG constitutes a simple, yet non-trivial, model with strong feedback and internal frustration. The MG also o!ers a simple paradigm for the complicated decision dynamics of ‘real-world’ agents in everyday situations. An example is that of traders in a market trying to decide whether to buy or to sell—more buyers than sellers implies higher prices and hence more beneGt to the sellers. Subsequent to the pioneering work [1–3] of Challet and Zhang, the MG has been extensively studied [5,6,4,10–13]. In particular, Johnson et al. [11–13] proposed a crowd–anticrowd theory of the MG in which features in the MG are explained in terms of the collective actions of a crowd of agents using similar strategies and an anticrowd using strategies opposite to that of the crowd. Li et al. [14] recently proposed and studied numerically variations of the MG in which the performance of the strategies is quantiGed by a payo! function depending on the size of the minority group (and hence the size of the majority group) in each turn. Interestingly, the main features in the modiGed models are found to be similar to that of the basic MG. In particular, the Luctuations in the number of agents making a particular decision in time characterized by a standard deviation is found to have a robust dependence on the agents’ capability. In the present work, we aim to explain the observed robustness when di!erent payo! functions are used. By invoking the idea of a reduced strategy space, we Grst establish that the robustness as seen in Ref. [14] for simulations in the so-called full strategy space carries over to numerical studies within a reduced strategy space. The reduced strategy space is a representative collection of strategies in the full strategy space. Strategy selection can be conveniently studied within the reduced strategy space. Interestingly, the robustness of the features can be explained in terms of popularity ranking of the strategies and strategy-play among the agents [11–13]. While di!erent payo! functions alter the cumulative performance of each of the strategies in the strategy space and thus the relative popularity ranking of the strategies, it is found that the underlying general features of the popularity ranking are unaltered by the choice of payo! functions, and the robustness in the dependence of the Luctuation on the agents’ capability follows from the general features of the popularity ranking. The plan of the paper is as follows. In Section 2, we deGne the MG and introduce di!erent payo! functions for assigning virtual points as an indicator of the performance of the strategies. In Section 3, we show that the results of numerical simulations on the models using the reduced strategy space [1–3,5,6] are in general the same as those obtained by using the full strategy space. Within the reduced strategy space, we then apply the analytical expression within the crowd–anticrowd theory to evaluate the standard deviation [12,13]. The evaluation relies on the di!erence in the number of agents using anti-correlated pairs of strategies and thus depends on the strategy
K. Lee et al. / Physica A 321 (2003) 309 – 317
311
selection among the agents. The results of the crowd–anticrowd theory are shown to be in good agreement with numerical results, hence supporting the idea of explaining general features in the MG and its variations in terms of the crowd e!ects. 2. Model The basic minority game consisting of an odd number N of agents repeatedly competing to be in a minority group. In each turn of the game, each agent decides between two possible options, say 0 and 1. The winners are those in the side with fewer agents. The outcome, i.e., the winning side, is made known to all agents. The agents have similar capability and they all decide based on the m most recent outcomes. Since there are a total of 2m possible bit-strings of length m constituting all the possible history bit-strings and there are two possible options for each history bit-string, there m are a total of 22 strategies in the full strategy space. At the beginning of the game, each agent picks s strategies from the pool of strategies, with repetitions allowed. At each turn, every agent uses the best performing strategy among his s strategies to decide. An agent records the performance of his strategies in the basic MG as follows. After each turn, the agent assigns one (virtual) point to each of his strategies which would have predicted the correct outcome. The most striking feature in the MG is that the standard deviation (SD) in the number of agents making a particular decision, say 0, is a non-monotonic function of the memory size m, with a minimum in the SD occurs at the value of m satisfying 2 · 2m ∼ N · s. The full strategy space grows rapidly with m. Fortunately, it has been shown that the SD in the basic MG is essentially unchanged if a ‘reduced’ strategy space is used instead of the full strategy space [1–3,12]. The reduced strategy space only contains 2m+1 strategies or equivalently 2m pairs of strategies G. The two strategies within a given pair G are anti-correlated, i.e., they di!er by the maximum Hamming distance dH = 2m and they make opposite predictions for a given history bit-string. Strategies between any two pairs G and G are uncorrelated, i.e., they di!er by dH = 2m−1 . In the present work, numerical simulations on the MG with di!erent payo! functions are carried out using the reduced strategy space. In the basic MG, one virtual point is assigned to each strategy that would have predicted the correct outcome regardless of the size of the minority group. However, the distribution of the whole population in the two possible options reLects how successful the strategies are in their predictions. For example, when the population splits itself into (N − 1)=2 and (N + 1)=2 in the two options, the successful strategies have only barely predicted the winning side. To reLect better the performance of the strategies, Li et al. [14] proposed modiGed models of MG in which the virtual points are assigned according to some payo! functions A(r), where r = n=N is the ratio of the number of agents in the winning (minority) side to the total number of agents in the population. In particular, they studied A(r) of the forms A(r) ∼ r − and A(r) ∼ e− r numerically using the full strategy pool. Here, and are constants setting the scale of reward to the successful strategies. The most interesting result is that the Luctuation in the number of agents making a particular choice, characterized by the standard deviation
312
K. Lee et al. / Physica A 321 (2003) 309 – 317
(SD) in the distribution of agents over time, shows the same dependence on m as in the basic MG, regardless of the form of the payo! functions. Note that the case of A(r) = 1 reduces to the basic MG. In what follows, we Grst establish that simulation results obtained using the reduced strategy space are the same as those obtained using the full strategy space. We then apply the crowd–anticrowd theory [12,13] to show that the robustness of the dependence of the SD on m is a result of the crowd e!ect in the strategy-play among the agents. 3. Results and discussion As it is more convenient to apply the crowd–anticrowd theory within the context of the reduced strategy space, we have carried out detailed numerical simulations for the payo! functions studied in Ref. [14] using the reduced strategy space. Fig. 1 shows the numerical results (symbols) for the SD in the number of agents choosing a particular option, say 0, as a function of m for the four di!erent payo! functions e−4r ; e0:1r ; r −2 and r −1 studied in Ref. [14]. Our results show the same features as those reported in Ref. [14] obtained using the full strategy space. Thus, as in the basic MG [2–4], we found that the reduced strategy space and the full strategy space give the same dependence of the SD on m when di!erent payo! functions are used. The dependence of SD on m, as pointed out in Ref. [14], is also the same as the basic MG. This robustness indicates that there are some intrinsic features common to the basic MG and the modiGed MG that are independent of the choice of the payo! function. For the basic MG, the crowd–anticrowd theory gives a physically transparent explanation of the m-dependence of the SD. Qualitatively, the physics behind the crowd–anticrowd e!ect is the following [11–13]. In the small m regime referred to as the ‘eQcient’ or ‘crowded’ phase, the strategy-play among the agents is that there exists crowd of agents using a strategy in the absence of a corresponding anticrowd using the anti-correlated partner of the strategy. This leads to a crowd of agents choosing a particular option and thus a large SD. For large m, the total number of possible strategies is much larger than the number of strategies actually being picked in the game. Therefore, crowd formation is unlikely and the agents e!ectively behave as if they are making their decision randomly. This limit is referred to as the random coin-toss limit. For some intermediate value of m, the strategy pool size and the number of strategies in play are comparable. This leads to a delicate cancellation e!ect between the action of a crowd using a particular strategy and that of an anticrowd using the anti-correlated strategy. Since an anti-correlated pair of strategies make opposite predictions, crowd and anticrowd of comparable sizes lead to a nearly even distribution of the agents in the two options and thus a small SD. The crowd–anticrowd theory can also be extended to study the dynamics of the MG [13]. An important implication of the crowd–anticrowd theory is that the m-dependence of SD in MG, and hence in the modiGed MG with di!erent payo! functions, is intrinsically related to the strategy-play or strategy selection among the agents. It should be noted that the validity of the crowd–anticrowd theory does not depend on which strategy
K. Lee et al. / Physica A 321 (2003) 309 – 317 A(r)= e
-4r
313 0.1r
A(r)= e
15
15 simulation theory
simulation theory
10 SD
SD
10
5
0
5
0
5
10
0
15
0
5
m A(r)= r
10
15
m -1
-2
A(r)= r
20 20 simulation theory
15
simulation theory
SD
SD
15 10
5
10 5
0 0
5
10 m
15
0 0
5
10
15
m
Fig. 1. The standard derivation (SD) in the number of agents making a particular decision as a function of the memory size m for four di!erent payo! functions A(r) = e−4r ; e0:1r ; r −1 , and r −2 . The symbols are numerical results obtained using a reduced strategy space. The solid lines are results of the crowd–anticrowd theory. Other parameters are: N = 101 and s = 2.
being the most popular, i.e., being used by more agents than the other strategies. As it is expected that di!erent payo! functions only a!ect the popularity ranking among the strategies, the crowd–anticrowd theory should remain applicable and the same scenario on strategy-play occurs as in the basic MG. Within the crowd–anticrowd theory, the SD can be expressed in terms of the di!erence in the numbers of agents using a strategy and its anti-correlated partner. Let nR be the number of agents using a strategy labelled R and nRR be the number of agents using the strategy RR that is anti-correlated to R. It is the di!erence |nR − nRR | in each crowd–anticrowd pair (nR ; nRR ) that contributes to the SD. Following Refs. [12,13], the SD can be expressed as 1=2 2 1 = : (1) 4 |nR − nRR | R (R;R)
314
K. Lee et al. / Physica A 321 (2003) 309 – 317 4 m=2 3
A(r) = e
-4r
m=5
P(k’=k)*2
m+1
m=10 2
1
0 0
0.2
0.4
0.6
k’/2
0.8
1
0.8
1
m+1
4 m=2
A(r) = e
3
0.1r
m=5
P(k’=k)*2
m+1
m=10 2
1
0 0
0.2
0.4
0.6
k’/2
m+1
R for k = 1 evaluated numerically for payo! functions of the form Fig. 2. The probability function P(k = k) A(r) = e− r with = 4 and = −0:1. Results for three di!erent values of m characterizing three di!erent regimes are shown.
Implicit in Eq. (1) is a conGguration average taken over di!erent initial strategy distributions among the agents in addition to a time average in obtaining the SD [12,13]. For each of the payo! functions, we run the numerical simulations and read out the number of agents playing each of the 2 · 2m strategies for every time step within a time window. Eq. (1) is then used to evaluate the SD. A conGguration average over 32 di!erent initial strategy distributions is taken. Results (solid lines) are shown in Fig. 1 and they are in very good agreement with simulation results.
K. Lee et al. / Physica A 321 (2003) 309 – 317
315
4 m=2
A(r) = r -2
3
m=5
P(k’=k)*2
m+1
m=10 2
1
0 0
0.2
0.4
0.6
k’/2
0.8
1
0.8
1
m+1
4 m=2 3
A(r) = r
-1
m=5
P(k’=k)*2
m+1
m=10 2
1
0 0
0.2
0.4
0.6
k’/2
m+1
R for k = 1 evaluated numerically for payo! functions of the form Fig. 3. The probability function P(k = k) A(r) = r − with = 1 and 2. Results for three di!erent values of m characterizing three di!erent regimes are shown.
To investigate the change in the strategy-play among the agents for di!erent regimes of m, we further study the popularity ordered list {nk } of the strategies, where the strategy k = 1 is that used by the largest number of agents, strategy k = 2 is that used by the second largest number of agents, etc. The subtlety in strategy-play is best R for k = 1, i.e., the probability that illustrated by the probability function P(k = k) the anti-correlated partner of the most popular strategy (k = 1) has a ranking of k R for k = 1 for the in the popularity ordered list [12]. Fig. 2 (Fig. 3) shows P(k = k)
316
K. Lee et al. / Physica A 321 (2003) 309 – 317
two payo! functions of exponential (power-law) form for three di!erent values of m characterizing di!erent behavior. The horizontal axis is the ranking k normalized by the total number of strategies 2·2m . Results are obtained by numerical simulations using the reduced strategy space. The following general features, similar to those in the basic MG, are observed. For small m (e.g., m = 2), the anti-correlated partner of the most popular strategy is likely to be the least popular. Thus, there is a crowd using the most popular strategy. The anticrowd using the least popular strategy is small, by deGnition. This crowd e!ect leads to the large SD in the small m regime. A striking behavior for intermediate values of m (e.g., m = 5) is that the anti-correlated partner of the most popular strategy is also highly popular. In this case, n1 and n1R are approximately the same and the contribution to the SD is small. This is the crowd–anticrowd cancellation e!ect which leads to a suppression in SD. For large m (e.g., m = 10), the agents R for k = 1 results. The robustness in behave as random agents and a Lat P(k = k) the m-dependence of SD in the modiGed MG thus stems from the general features in the strategy-play among agents in di!erent regimes of m which are independent of the payo! functions. While the speciGc form of the payo! function may alter the strategy which is the most popular, the general features in the relative ranking of the anti-correlated partner to the most popular strategy remains unchanged for di!erent payo! functions. In summary, we have shown that the m-dependence of the SD in a class of MG with di!erent payo! functions can be explained in terms of the formation of crowd and anticrowd. The minimum in the SD is a result of the formation of crowd and anticrowd of comparable sizes. The robustness of the m-dependence of the SD is due to the invariance in the nature of strategy selection among the agents. The crowd– anticrowd theory therefore captures the essential physics of the crowd or herd behavior in a competing population as modelled by the MG and its variations.
Acknowledgements This work was supported in part by the Research Grants Council of the Hong Kong SAR Government under Grant No. CUHK4241/01P. We thank T.S. Lo for useful discussions.
References [1] [2] [3] [4] [5] [6] [7] [8]
D. Challet, Y.C. Zhang, Physica A 246 (1997) 407. D. Challet, Y.C. Zhang, Physica A 256 (1998) 514. D. Challet, Y.C. Zhang, Physica A 269 (1999) 30. D. Challet, M. Marsili, R. Zecchina, Phys. Rev. Lett. 84 (2000) 1824. R. Savit, R. Manuca, R. Riolo, Phys. Rev. Lett. 82 (1999) 2203. R. Manuca, Y. Li, R. Riolo, R. Savit, Physica A 282 (2000) 559. J.H. Holland, Emergence, Oxford University Press, Oxford, 1998. W.B. Arthur, Amer. Econ. Assoc. Papers Proc. 84 (1994) 406.
K. Lee et al. / Physica A 321 (2003) 309 – 317 [9] [10] [11] [12] [13] [14]
317
N.F. Johnson, S. Jarvis, R. Jonson, P. Cheung, Y.R. Kwong, P.M. Hui, Physica A 256 (1998) 230. D. Challet, M. Marsili, Phys. Rev. E 62 (2000) 1862. N.F. Johnson, M. Hart, P.M. Hui, Physica A 269 (1999) 1. M. Hart, P. Je!eries, N.F. Johnson, P.M. Hui, Physica A 298 (2001) 537. M. Hart, P. Je!eries, P.M. Hui, N.F. Johnson, Eur. Phys. J. B 20 (2001) 547. Y. Li, A. Vandeemen, R. Savit, Physica A 284 (2000) 461.