Game information dynamic models based on fluid mechanics

Entertainment Computing 3 (2012) 89–99

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Entertainment Computing journal homepage: ees.elsevier.com/entcom

Game information dynamic models based on fluid mechanics q Hiroyuki Iida, Takeo Nakagawa ⇑, Kristian Spoerer Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan

a r t i c l e

i n f o

Article history: Received 31 January 2011 Revised 16 March 2012 Accepted 8 April 2012 Available online 12 May 2012 Keywords: Game model Information dynamics Base Ball Fluid mechanics Game meandering

a b s t r a c t This paper is concerned with the proposal of two different kinds of novel information dynamic models based on fluid mechanics. These models are a series of approximate solutions for the flow past a flat plate at zero incidence. The five Base Ball games in the World Series 2010 have been analyzed using the models. It is found that the first model represents one game group where information of game outcome increases very rapidly with increasing the game length near the end and takes the full value at the end. The second model represents another game group where information gradually approaches to the full value at the end. Three game-progress patterns are identified according to information pattern in the five games, viz., balanced, seesaw and one-sided games. In a balanced game, both of the teams have no score during the game. In a seesaw game, one team leads score(s), then the other team leads score(s) and this may be repeated alternately. In a one-sided game, only one team gets score(s), but the other no score. It is suggested that the present models make it possible to discuss the information dynamics in games and/or practical problems such as projects starting from zero information and ending with full information. Ó 2012 International Federation for Information Processing Published by Elsevier B.V. All rights reserved.

1. Introduction Information is a long persisting enigma for human beings, because we still do not fully understand what it is, when it appears or disappears, where it comes from or goes, and how it behaves. Rauterberg [1] has made an extensive survey in the literature on information, and found that there exist at least six different interpretations; that is, (1) information as a message(syntax), (2) information as the meaning of a message(semantic), (3) information as the effect of a message(pragmatic), (4) information as a process, (5) information as knowledge, and (6) information as an entity of the world. In the present study, information of game outcome represents the data which is certainty of the game outcome, and so the present information might correspond to Rauterberg’s sixth interpretation. It may be evident that information before and after the reception of a message is not the same [1]. There are two possible dynamical views of information (incorporating force or energy); firstly, information flows, secondly, information is entropy [2]. The gap between the two viewpoints is not small in practice: the former considers that information is tractable within physics, but the latter views that information is beyond

q Part of this paper has been published in Proceedings of the Second International Multi-Conference on Complexity Informatics and Cybernetics (IMCIC 2011), 134– 139, 2011. ⇑ Corresponding author. Tel.: +81 761 51 1292; fax: +81 761 51 1149. E-mail addresses: [email protected] (H. Iida), [email protected] (T. Nakagawa), [email protected] (K. Spoerer).

physics, even though there is some relation to it. If information flows, it may be natural to consider that motion of information particles having mass is governed by the basic equations for fluid mechanics. The dependent variables in fluid mechanics are velocity, pressure, temperature and density, all of which depend on the position and time [3]. These are considered to be related to information in games for example. We consider that information is produced as the result of motion of information particles. Solso [4] shows that motion of visualized fluid particles is detected by the eye through light, having enormous high speed (3  1010 cm/s), and is almost instantaneously mapped on the retina of the eye. We consider that during this process, through fluid particles are transformed into information particles. The eye and brain work together in collecting light and in processing the information particles. The eye collects light and passes it to the neural network in the brain, which processes it and then redirects the eye to scan elsewhere. Visual signals from the physical dimensions enter the eye as light and are normally recorded on the retina as twodimensional images. The brain interprets two-dimensional visual images as having three-dimensions by use of contextual cues and knowledge of the world as gained through a lifetime of experience [4]. Electromagnetic signals or photons due to light carrying the information on the motion of visualized fluid particles are converted to electrochemical signals and passed along the visual cortex for further processing in other parts of the cerebral cortex. It is, therefore, considered that flow in physical world can be faithfully transformed to that in eye and brain (referred to informatical world here after). If a flow phenomenon due to the motion of fluid

1875-9521/$ - see front matter Ó 2012 International Federation for Information Processing Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.entcom.2012.04.002

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particles in physical world can model an information phenomenon, the latter must be caused by the same motion of information particles in informatical world. Thus, in this study it is essential to find correspondences between a flow and information phenomena. It may be expected that there are many parallels in the way the five sensory systems (eye, ear, nose, tongue and skin) process information. All of the stimuli caused by these systems are considered to be converted into electrochemical signals (or we consider them to be information particles, which flow exactly in the same manner as fluid particles and result in information). This implies that the perceived intensity due to a stimulus is the information. This analogy has been used by Stevens [5] to obtain the relationship between the magnitude of a physical stimulus (e.g. luminance, weight, sound pressure) and the corresponding experienced magnitude (brightness, heaviness, loudness). It is sometimes simply called the power law, which is applied to game-refinement by Iida et al. [6]. In this paper we show that the power law has a limitation to represent how the information on one-sided games varies with increasing the game length near the end. Applications of fluid mechanics encompass a wide range of subjects, including traffic flow [7], green plants [8], and quantum fluid flow [9]. In Computer Science, fluid mechanics has been used to model and simulate network traffic [10]. For example, vehicular traffic flow is concerned with interactions between vehicles, drivers, and infrastructure including highways and signs, with the aim of understanding and developing an optimal road network with efficient flow of traffic and minimal traffic congestion problems. In traffic flow, vehicles are assumed to be the fluid particles. There is a large body of work on dynamic modeling in games (Chadwick and James [11], Shiratori et al. [12], Chaudhuri et al. [13], Tocci et al. [14], Xu et al. [15]). For example, Li et al. [16] have outlined an approach to acquiring complex autonomous agent behavior based on observation of video footage of college football games. They have demonstrated that their cognitive architecturebased approach has agents, which are competitive hand-engineered agents. The main purpose of the present study is to propose two different kinds of novel information dynamic models, for two team (or player) games based on fluid mechanics and to analyze the five Base Ball games in the World Series 2010 using the models.

namic model will be constructed by following the above procedure step by step. 2.1. Let us assume the flow past a flat plate at zero incidence as the information dynamic model (Fig. 1) The simplest example of the application of the boundary-layer equations, which is the simplified Navier–Stokes equations, is afforded by the flow along a very thin flat plate with zero incidence. Historically, this is the first example illustrating the application of Prandtls boundary-layer theory [17]; it has been discussed by Blasius [18] in his Ph.D. thesis at Göttingen. Let the leading edge of the plate be at x = 0, the plate being parallel to the x-axis and infinitely long downstream, as shown in Fig. 1. We shall consider steady flow with a free-stream velocity U1, which is parallel to the x-axis. The boundary-layer equations [3] become

u  @u=@x þ v  @u=@y ¼ 1=q  dp=dx þ m  @ 2 u=@y2 ;

ð1Þ

@u=@x þ @ v =@y ¼ 0;

ð2Þ

y ¼ 0 : u ¼ v ¼ 0;

y ¼ 1 : u ¼ U1 ;

ð3Þ

where u and v are velocity components in the x and y directions, respectively, q the density, p the pressure, and m the kinematic viscosity of the fluid. In the free-stream, U1  dU1/ dx = 1/q  dp/ dx. The free-stream velocity U1 is constant in this case, so that dp/dx = 0, and dp/dy = 0. Since the system under consideration has no preferred length, it is reasonable to suppose that the velocity profiles at varying distances from the leading edge are similar to each other, which means that the velocity curves u(y) for varying distances x can be made identical by selecting suitable scale factors for u and y. The scale factors for u and y appear quite naturally as the free-stream velocity, U1 and the boundary-layer thickness, d(x). Hence the velocity profiles in the boundary-layer can be written as

u=U 1 ¼ f ðy=dÞ;

ð4Þ

Blasius [18] has obtained the solution in the form of a series expansion around y/d = 0 and an asymptotic expansion for y/d very large, the two forms being matched at a suitable value of y/d. The analytical evaluation of the solution is beyond the scope of the present study.

2. Modeling

2.2. Get the solutions

In the present modeling, it is hypothesized that information particles flow exactly the same as fluid particles. Under this hypothesis,1 the procedure for producing an information dynamic model based on fluid mechanics is summarized as follows:

The similarity of velocity profile is here accounted for by assuming that function f depends on y/d only, and contains no additional free parameter. The function f must vanish at the wall (y = 0) and tend to the value of 1 at the outer edge of the boundary-layer (y = d), in view of the boundary conditions for f(y/d) = u/U1. Further boundary conditions for the exact solution might include the continuity of the tangent and curvature at the point, where the two solutions are joined. In other words, we may seek to satisfy the conditions, tangent @u/@y = 0 and curvature @ 2u/@y2 = 0, respectively, at y = d. Another boundary condition on the wall for the exact solution, @ 2u/@y2 = 0 at y = 0 is also of importance, for in (1) u = v = 0 at y = 0.

1. Assume a flow problem as the information dynamic model and solve it. 2. Get the solutions, namely, velocity, pressure, temperature and density, depending on the position and time. 3. Examine whether any solution of the flow problem can be related to information of game outcome, for example, or not. 4. When this is the case, visualize the assumed flow with some means. If not, we must return to the first step. 5. Determine correspondences between the flow solution and information of game outcome. 6. Obtain the mathematical expression of the new information dynamic model.

U∞

y δ

Information dynamics is concerned with the modeling of information using equations from fluid mechanics. The information dy1

A proposition assumed as a premise in the present argument.

0

x Fig. 1. A definition sketch of flow past a flat plate at zero incidence.

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When using the approximate method, it is expedient to place the point at which this transition occurs at a finite distance from the wall, or in other words, to assume a finite boundary-layer thickness d(x) in spite of the fact that all exact solutions of the boundary-layer equations tend asymptotically to the free-stream associated with the particular problem. The approximate method here means all the procedures to find approximate solutions to the exact solution. When writing down an approximate solution of the present flow, it is necessary to satisfy certain boundary conditions for u(y). At least the no-slip condition u = 0 at y = 0 and the condition of the continuity when passing from the boundary-layer profile to the free-stream, u = U1 at y = d, must be satisfied. It will be shown that the following two velocity profiles satisfy all of the boundary conditions as the approximate solutions on the flow past a flat plate at zero incidence,

u=U 1 ¼ ðy=dÞm ;

ð5Þ

2.5. Determine correspondences Proposed are the most plausible correspondences between the flow solution and information of game outcome, which are listed in Table 1. 2.6. Obtain the mathematical expression Considering the correspondences in Table 1, (5) and (6) can be rewritten, respectively, as

I=I0 ¼ ðt=t 0 Þm ;

ð7Þ

and

I=I0 ¼ ½sinðp=2  t=t 0 Þn :

ð8Þ

Introducing the following non-dimensional variables

n ¼ I=I0

and g ¼ t=t 0 ;

(7) and (8) become, respectively,

n ¼ gm ðModel 1Þ;

and

u=U 1 ¼ ½sinðp=2  y=dÞn ;

ð6Þ

in the range 0 6 y/d 6 1, whereas for y/d > 1 we assume simply u/U1 = 1, where m and n are positive real numbers. The function in (5) and (6) vanishes at the wall (y/d = 0) and takes the value of 1 at the outer edge of the boundary-layer (y/d = 1), respectively. That is, starting from 0 on the wall, the velocity u in the boundary-layer should join the free-stream at the finite distance y = d(x) from the wall. Thus, the functions in (5) and (6) are both considered as approximate solutions on the flow past the flat plate at zero incidence. Now, we get the velocities in the x direction, which are the solutions for the assumed flow.

ð9Þ

and

n ¼ ½sinðp=2  gÞn ðModel 2Þ

ð10Þ

where n is the non-dimensional current information of game outcome, and g the non-dimensional current game length. Expressions (9) and (10) will be called Model 1 and Model 2, respectively, hereafter. Note that Model 1 is no more than the power law due to Stevens [5], who has derived this law heuristically. It is also possible to discuss how the information velocity and information acceleration vary with the game length. The information velocities for Model 1 and Model 2 can be expressed, respectively, by

dn=dg ¼ mgm1 ;

ð11Þ

and 2.3. Let us examine whether these solutions are information of game outcome Such an examination immediately provides us that the nondimensional velocities vary from 0 to 1 with increasing the nondimensional vertical distance y/d in many ways as the non-dimensional information, so that these solutions can be information of game outcome. However, the validity of this conjecture will be confirmed by the relevant Base Ball data.

dn=dg ¼ n  p=2½sinðp=2Þn1  cosðp=2  gÞ;

On the one hand, the information accelerations for Model 1 and Model 2 is expressed, respectively, by 2

d n=dg2 ¼ mðm  1Þgm2 ;

ð13Þ

and 2

d n=dg2 ¼ n  ðp=2Þ2  fðn  1Þ½sinðp=2  gÞn2  ½cosðp=2  gÞ2  ½sinðp=2  gÞn g;

2.4. Visualize the flow Imagine that the assumed flow is visualized with neutral buoyant particles, for example. Motion of the visualized particles is detected by the eye almost instantaneously through the light having enormous high speed (3  1010 cm/s) and is mapped on our retina [4], so that during these processes, motion of fluid particles is transformed into information particles. This is why motion of the fluid particles is intact in the physical world, but only the reflected light, or electromagnetic wave consisting of photons can reach the retina. The photons are then converted into electrochemical particles and are passed along the visual cortex for further processing in parts of the cerebral cortex [4]. The photons and/or electrochemical particles can be considered to be information particles. It is, therefore, natural to expect that flow in the physical world is faithfully transformed to that in the informatical world. During this transformation, flow solution in the physical world changes into the information in the informatical world.

ð12Þ

ð14Þ

Fig. 2 shows the relation between the non-dimensional information n and the non-dimensional game length g for Model 1. When m < 1, n-curves are convex, while m > 1, n-curves are concave. When m = 1, n increases linearly with increasing g. Fig. 3 shows the relation between the non-dimensional information n and the non-dimensional game length g for Model 2. When n < 2, n-curves are convex, while n > 2, n-curves are sshaped. This means that all of the information for Model 2 are decelerated with increasing g near the end.

Table 1 Correspondences between flow solution and information. Flow solution

Information

u: flow velocity U1: free stream velocity y: vertical co-ordinate d: boundary layer thickness

I: current information of game outcome I0: full information of game outcome t: current game length or current time t0: total game length or total time

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Non-dimensional Information

1.2 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

-0.2 m=0

m=0.1

Non-dimensional Game m=0.2 m=0.4 m=0.6Lengthm=0.8

m=4

m=6

m=8

m=10

m=20

m=1

m=2

m=50

Fig. 2. Non-dimensional information n against non-dimensional game length g for Model 1.

Non-dimensional Information

1.2 1 0.8 0.6 0.4 0.2 0 0

0.2

-0.2

0.4

0.6

0.8

1

1.2

Non-dimensional Game Length n=0

n=0.1

n=0.2

n=0.4

n=0.6

n=0.8

n=4

n=6

n=8

n=10

n=20

n=50

n=1

n=2

Fig. 3. Non-dimensional information n against non-dimensional game length g for Model 2.

Fig. 4 shows the relation between the non-dimensional information velocity dn/dg and the non-dimensional game length g for Model 1. With increasing m, dn/dg increases more slowly with increasing g in the earlier stage, but it increases more rapidly in the later stage and attains the greater final value. Fig. 5 shows the relation between the non-dimensional information velocity dn/dg and the non-dimensional game length g for Model 2. When n = 1, dn/dg decreases monotonously with increasing g, and finally becomes 0. However, when n > 1, dn/dg increases with increasing g and takes a peak, and then decreases and finally becomes 0. With increasing n, the peak value becomes larger and g at which dn/dg takes the peak value also becomes larger. Fig. 6 shows the relation between the non-dimensional information acceleration d2n/dg2 and non-dimensional game length g for Model 1. With increasing m, d2n/dg2 increases more slowly with increasing g in the earlier stage, but it increases more rapidly in the later stage and attains the greater final value. Fig. 7 shows the relation between the non-dimensional information acceleration d2n/dg2 and non-dimensional game length g

for Model 2. When n = 2, d2n/dg2 decreases monotonously from a positive value to a negative value with increasing g. However, when n > 2, d2n/dg2 increases from 0 to a peak with increasing g, and then decreases and takes a negative value. With increasing n, the peak value becomes larger and g at which d2n/dg2 takes the peak value also becomes larger until n = 6. However, the peak value decreases from n = 6–8 once, and then it becomes larger with increasing n and g at which d2n/dg2 takes the peak value also becomes larger again. It is, however, evident that the final negative value at g = 1 becomes smaller with increasing n. 3. Base Ball games in the World Series 2010 and comparison with the present models In this section, some results of data analyses on the Base Ball games in the World Series 2010 will be presented at first and then a comparison between the present models and the relevant data will be discussed. Base Ball is a game of ball played by two opposing sides of nine players each, on a diamond enclosed by lines

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Non-dimensional Information Velocity

25

20

15

10

5

0 0

0.2

0.4

0.6

0.8

1

1.2

-5

Non-dimensional Game Length m=1

m=2

m=4

m=6

m=8

m=10

m=20

Fig. 4. Non-dimensional information velocity dn/dg against non-dimensional game length g for Model 1.

Non-dimensional Information Velocity

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 -0.5

0

0.2

0.4

0.6

0.8

1

1.2

Non-dimensional Game Length n=1

n=2

n=4

n=6

n=8

n=10

n=20

Fig. 5. Non-dimensional information velocity dn/dg against non-dimensional game length g for Model 2.

connecting four bases, a complete circuit of which must be made by a player after batting, in order to score a run. The proposed models will be applied to the five Base Ball games in the World Series 2010. Some of the relevant information is summarized in Table 2. Fig. 8 shows the relation between the non-dimensional information nB and non-dimensional game length g for the five Base Ball games in the World Series 2010. The non-dimensional information nB in Base Ball is defined as follows: when the total score(s) of the two teams at the end of the game St – 0,

nB ¼ jSf ðgÞ  Ss ðgÞj=St

for 0 6 g < 1;

and,

nB ¼ 1 for g ¼ 1; where Sf(g) is the current score sum for the first batting team, and Ss(g) is the current score sum for the second batting team. At g = 1, nB is assigned the value of 1, for at the end of the game the information must reach the full information of game outcome. On the other hand, when St = 0,

nB ¼ 0 for 0 6 g < 1; and,

nB ¼ 1 for g ¼ 1: Note that in a draw case nB may also take the value of 0 other than 1 at g = 1, depending on the game rules. The game length g is defined as the current number of batting chances for the two teams, and it is normalized by the total game length to obtain the non-dimensional value. There are two batting chances for one inning, for one team makes batting for the first half, while the other team for the second half. Each of the Base Ball games has nine innings except for the extra-inning games, so that there are a total of 18 batting chances for the two teams. This means that the total game length for the present five games is 18. Fig. 8 clearly indicates that non-dimensional information for these five games varies with the non-dimensional game length in a different manner from each other. However, these games can be broadly divided into two groups: The first group consists of the first, third and fifth games. These three games have a common character that the information increases suddenly near the end. It has been suggested by Iida et al. [6] that games of the first group are accounted for by the power law or Model 1. The second group consists of the second and fourth games. These games have a com-

H. Iida et al. / Entertainment Computing 3 (2012) 89–99

Non-dimensional Information Acceleration

94

400 350 300 250 200 150 100 50 0 0

0.2

0.4

0.6

0.8

1

1.2

-50

Non-dimensional Game Length m=1

m=2

m=4

m=6

m=8

m=10

m=20

Non-dimensional Information Acceleration

Fig. 6. Non-dimensional information acceleration d2n/dg2 against non-dimensional game length g for Model 1.

60

40

20

0 0

0.2

0.4

0.6

0.8

1

1.2

-20

-40

-60

Non-dimensional Game Length n=2

n=4

n=6

n=8

n=10

n=20

Fig. 7. Non-dimensional information acceleration d2n/dg2 against non-dimensional game length g for Model 2.

Table 2 Five Base Ball games in the World Series 2010.

First game Second game Third game Fourth game Fifth game

Game number

Score history

Rangers Giants Rangers Giants Giants Rangers Giants Rangers Giants Rangers

110 002 000 000 000 030 002 000 000 000

002 060 000 010 000 010 000 000 000 000

003 03x 000 17x 110 00x 110 000 300 100

7 11 0 9 2 4 4 0 3 1

Date

Place

October 27, 2010

San Francisco

October 28, 2010

San Francisco

October 30, 2010

Arlington

October 31, 2010

Arlington

November 1, 2010

Arlington

Symbol x in this table means that no batting is done in the second half of the 9th inning, for the second batting team leads the score(s) at the end of the first half of the 9th inning.

mon distinctive feature that the information gradually approaches to the full value of game outcome. It is realized that the games of the second group cannot be accounted for by Model 1, so that Model 2 will be used.

Fig. 9 depicts the relation between non-dimensional advantage

a and non-dimensional game length g. The non-dimensional advantage a is here defined as follows: When the total score(s) of the two teams at the end of game St – 0,

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Non - dimensional Information

1.2

1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

-0.2

Non - dimemsional Game Length First Game

Second Game

Third Game

Fourth Game

Fifth Game]

Fig. 8. Non-dimensional information nB against non-dimensional game length g for the five Base Ball games in the World Series 2010.

a ¼ ½SR ðgÞ þ SG ðgÞ=St for 0 6 g 6 1; where SR(g) is the current score sum for Rangers and positive h+i sign is assigned for Rangers score(s), and SG(g) is the current core sum for Giants and negative hi sign is assigned for Giants score(s). On the other hand, when St = 0,

a ¼ 0 for 0 6 g 6 1: This means that when a > 0, Rangers get the advantage in the game, when a < 0, Giants get the advantage, and when a = 0 the game is balancing. This figure, therefore, illustrates how the non-dimensional advantage a of each game for two teams changes with the nondimensional game length g: in the case of the first game, Rangers get the advantage during the early stage, but Giants keep the advantage during the second half, though the game is balancing in the middle. In the case of the second game, it is balancing until the middle, but Giants then obtain the advantage and increases it until the end. The third game is balancing at first, but Rangers take the advantage and keep it until the end, though the advantage

attains the peak value in the middle. In the case of the fourth game, it is balancing at first, but Giants then get the advantage which increases with the game length until the end. The fifth game is balancing until the non-dimensional game length of 2/3, but Giants then take the advantage and hold it until the end. The game sinuosity S is defined as follows:

S ¼ L=gt ; where L is the total length of non-dimensional advantage locus for the game, and gt the non-dimensional total game length. When we obtain the total length of non-dimensional advantage locus, it is plotted on the graph, on which the abscissa and ordinate are scaled by the same non-dimensional unit length, and then length of the locus is measured directly by string, following its locus faithfully. The results are shown in Table 3. Fig. 10 shows the relation between the non-dimensional information n and the non-dimensional game length g. In this figure, non-dimensional information for the first group games has been plotted and is compared with seven curves for Model 1. It is evident

Non-dimensional advantage

1.2

0.7

0.2

0

0.2

0.4

0.6

0.8

1

1.2

-0.3

-0.8

-1.3

Non-dimensional Game Length First Game

Second Game

Third Game

Fourth Game

Fifth Game

Fig. 9. Non-dimensional advantage a against non-dimensional game length g for the five World Series Base Ball games.

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that none of the information for these games fits to any model curve through the total non-dimensional game length, but near the end the information for all of these games follows the model curve at m = 20 approximately. As being already shown in Fig. 4, when m = 20, the non-dimensional information velocity dn/dg increases exponentially with increasing non-dimensional game length g, and takes the maximum value of 20 at g = 1. On the other hand, as being already shown in Fig. 6, when m = 20, non-dimensional information acceleration d2n/dg2 increases exponentially with increasing the non-dimensional game length g, and takes the maximum value of 380 at g = 1. It is considered that Model 1 accounts for the games with a sudden increase of information near the end. Fig. 11 shows the relation between the non-dimensional information n and the non-dimensional game length g. In this figure, the non-dimensional information for the second and fourth games, respectively, has been plotted and is compared with eight curves for Model 2. It may be clear that although the non-dimensional information for the second and fourth games, respectively, proceeds in a zigzag line, on the whole, the non-dimensional information for the second game follows the model curve at n = 20, while the non-dimensional information for the fourth game follows the model curve at n = 2. As being already shown in Fig. 5, when n = 20, the non-dimensional information velocity dn/dg increases exponentially with increasing non-dimensional game length g, and takes the peak value of about 4.3 at g = 0.85. Then, dn/dg decreases with increasing g and becomes 0 at g = 1. When n = 2, dn/dg increases from 0 with increasing g and takes the peak value of about 1.6 at g = 0.5. Then, dn/dg decreases with increasing g and becomes 0 at g = 1. As being already shown in Fig. 7, when n = 20, non-dimensional information acceleration d2n/dg2 increases exponentially with increasing the non-dimensional game length g, and takes the peak value of about 23 at g = 0.75 and then decreases and finally becomes about 49 at g = 1. However, when n = 2, d2n/dg2 decreases monotonously with increasing g from about 4.9 to 4.9, crossing the abscissa at g ’ 0.5. Hence, it is considered that

Model 2 can properly account for the games of the second group, or one-sided games, where the information gradually approaches to the value of game outcome near the end. 4. Discussion 4.1. Information momentum and information force If the information particles having mass in our informatical world flow exactly the same as fluid particles in the physical world, it is possible to discuss how the information momentum and information force vary depending on the game length during the game. Let us now consider the information momentum and information force that may affect observers (including the players) during the game. We assume that information particles flow in our informatical world either as photons or electrochemical particles. Define the density of information fluid q as

q ¼ m=V;

ð15Þ

where m is the mass of the information particles contained in the infinitesimal volume V, which can be small to the extent that only one particle is contained in it. Introducing a reference density q0 as

q0 ¼ m0 =V;

ð16Þ

where m0 is the reference mass of information particles contained in the volume V, we have the non-dimensional density or nondimensional mass / as

/ ¼ q=q0 ¼ m=m0 :

ð17Þ

By multiplying the non-dimensional information velocity dn/dg given by either (11) or (12) with non-dimensional mass (17), we have the non-dimensional momentum M as

M ¼ /dn=dg:

ð18Þ

Table 3 Sinuosity of Base Ball games in the World Series 2010. First game

Second game

Third game

Fourth game

Fifth game

1.32

1.67

1.61

1.65

1.69

1.2

1

Non-dimensional Information

Sinuosity

0.8 0.6

0.4 0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

-0.2

Non-dimensional Game Length First Game

Third Game

Fifth Game

m=0.6

m=1

m=2

m=4

m=6

m=10

m=20

Fig. 10. Non-dimensional information n against non-dimensional game length g: a comparison between the first group games and Model 1.

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Non-dimensional Information

1.2

1 0.8 0.6 0.4 0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

-0.2

Non-dimensional Game Length Second Game

Fourth Game

n=1

n=2

n=4

n=6

n=8

n=10

n=20

n=50

Fig. 11. Non-dimensional information n against non-dimensional game length g: a comparison between the second group games and Model 2.

On one hand, by multiplying the information acceleration d2n/dg2 given by either (13) or (14) with non-dimensional mass (17), we obtain the non-dimensional information force F as 2

F ¼ /d n=dg2 :

ð19Þ

In particular, when the mass m is equal to the reference mass m0, or / = 1 in (17), the non-dimensional information momentum (18) and non-dimensional information force (19) become equal to the nondimensional information velocity dn/dg given by either (11) or (12), and non-dimensional information acceleration d2n/dg2 given by either (13) or (14), respectively. Thus, when the non-dimensional mass / = 1, Figs. 4 and 5 show the relation between the non-dimensional information momentum M and non-dimensional game length g, while Figs. 6 and 7 show the relation between the nondimensional information force F and non-dimensional game length g. This infers that the present models make it possible to discuss the information dynamics on games and/or real world problems, such as projects starting from zero information, and ending at full information.

angle is the angle of contact between the water and the solid surface for each transverse cross section of the water rivulet. The transverse unbalanced force between the centrifugal force and the surface tension force to either right or left with respect to the local main flow direction causes the transverse oscillation of water rivulets, and thus their meandering is maintained [21]. Fig. 12 shows a stable meandering stream system on the smooth surface, with grid lines drawn on the underside of the plate. Each of the regular squares has side length of 10 mm. The meander lops are invariably smoothly curved, swinging gradually from left-handed to right-handed deflection, and vice versa, with an almost-constant amplitude and wavelength of the meanders. The flow within the stream has been visualized using a dye streak of methylene blue aqueous solution, released into the flow 10 mm downstream from the tube mouth via a hypodermic needle (0.3 mm diameter). Fig. 13 shows the visualization of the spiral

4.2. Game-advantage meandering The transversal changes from positive to negative advantage in the first game (Fig. 9), have some similarity to those in meandering phenomena in rivers [19], ocean currents [20] and/or water rivulets on the smooth hydrophobic surface [21]. This is because both of the forward force and transverse force exist in Base Ball, where the forward force is the increase of inning with respect to the game length, while the transverse force is positive and negative advantages. The forward force is generated by the rule that the teams switch every time the defending team gets three players of the batting team out. In the case of a water rivulet on an inclined smooth hydrophobic surface, for example, the forward force is the component of the gravity force acting to the water particles along the line of maximum slope of the surface, and the transverse force is the centrifugal force acting to the water particles and the counter balancing surface tension force acting to the contact lines in the direction of the thinner part of the stream transverse cross sectiontowards the side with the smaller contact angle, where the contact line of the water rivulet is the locus of the points in which the three phases, viz., water, air and solid surface coexist, and the contact

Fig. 12. A stable meandering water rivulet on the smooth hydrophobic Plexiglas surface. Discharge = 2.0 cm3/s; surface slope = 20°. The flow direction is from top to bottom of the surface.

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there is a balanced state in the middle. But since the positive and negative values are relatively small, the sinuosity of the first game is the minimum value of 1.32 among the five games. It may be worth classifying the game-progress patterns. When both of the teams have no score in a game, it is called balanced game, in which the game is completely balanced. When one team leads score(s), then the other team leads score(s), and this is repeated alternately in a game, it is called seesaw game. When only one team gets score(s), but the other team no score, it is called onesided game. Fig. 9 shows that all the games are balanced partially. Only the first game is a seesaw game, the second and fourth games are each an example of a one-sided game, and the third and fifth games are each an incomplete one-sided game. 5. Conclusion

Fig. 13. A stable meandering water rivulet on the smooth hydrophobic Plexiglas surface. Discharge = 4.5 cm3/s; surface slope = 10°. The flow direction is from top to bottom of the surface (after Nakagawa and Scott [21]).

secondary flows. The dye release point is near the center of the stream, close to the surface of the plate. The dye filament flows initially on the right-hand side of the stream facing to the downstream direction, and then moves to the left-hand side as the stream swings right. At a point near the end of the straight portion of the right-moving stream the dye streak bifurcates, the branch that stays close to the plate surface continuing its course on the left of the stream, but the branch nearer the stream surface crossing over to the right-hand side due to the centrifugal force to the right. This results in a righthanded spiral secondary flow along the main flow direction. Nakagawa and Scott [21] reported a study of a meandering water stream upon an inclined smooth hydrophobic surface. They have shown experimentally that the sinuosity (ratio of the total stream length to the length of the projection of the stream on the line of maximum slope of the surface) increases with both increasing discharge rate and surface slope in general and takes the values ranging from 1.05 to 1.80. It is worth noting that the sinuosity of the advantage for the five games as listed in Table 3 is comparable with that of the meandering water rivulets [21]. This result may suggest that the sinuosity of the advantage in the game can become an indicator for the degree of game-advantage meandering: The greater the sinuosity is, the higher the degree of gameadvantage meandering is, and vice versa. However, it must be noted that the game with larger sinuosity does not always repeat cyclic transversal changes from positive to negative advantage, and vice versa: though the fifth game is incomplete one sided game, the sinuosity takes the maximum value of 1.69 among the five games. In an incomplete one sided game, major score(s) and minor score(s) are taken by one team and the other team, respectively. On one hand, the first game changes the advantage from positive to negative value, though

The new knowledge and insights obtained through the present investigation are summarized as follows. Two information dynamic models based on fluid mechanics have been proposed. These models are a series of approximate solutions for the flow past a flat plate at zero incidence. The modeling procedure of information dynamics based on fluid mechanics has been described. The five Base Ball games in the World Series 2010 have been analyzed using the present models. The first model (Model 1) is expressed by n = gm, where n is nondimensional game information, g the non-dimensional game length, and m a positive real number. It is found that Model 1 represents one game group where the information increases very rapidly with increasing the game length near the end and takes the full value at the end. It is shown that the first, third and fifth games belong to this first group. The second model (Model 2) is expressed by n = [sin(p/2  g)]n, where n is a positive real number. It is found that Model 2 represents another game group where the information gradually approaches to the full value at the end. It is shown that the second and fourth games belong to this second group, which consists of one-sided games. Three game-progress patterns are identified in the five Base Ball games in the World Series 2010, viz. balanced, seesaw, and onesided games. In a balanced game, both of the teams have no score during the game. In a seesaw game, one team leads score(s), then the other team leads score(s) and this is repeated alternately. In a one-sided game, only one team gets score(s), but the other team no score. It is realized that all the games are balanced partially, but it can be said that the first game is a seesaw game, the second and fourth games are each a one-sided game, and the third and fifth games are each an incomplete one-sided game. It is realized that the transversal changes from positive to negative advantage in the first game, have some similarity to those in meandering phenomena in rivers, ocean currents and/or water rivulets on the inclined smooth surface. It is, therefore, inferred that the sinuosity of the advantage for the game can be an indicator for characterizing game meandering. The proposed models make it possible to discuss the information dynamics on games and/or practical problems such as projects starting from zero information and ending at the full information. References [1] M. Rauterberg, Information System Concepts – Towards a Consolidation of Views, Chapman & Hall, London, 1995. Chapter about A Framework for Information and Information Processing of Learning Systems, pp. 54–69. [2] C.E. Shannon, A mathematical theory of communication, The Bell Syst. Tech. J. 27 (1948) 379–423. 623–656. [3] H. Schlichting, Boundary-Layer Theory, sixth ed., McGraw-Hill, New York, 1968. [4] R. Solso, Cognition and the Visual Arts, MIT Press, Cambridge, 1994. [5] S. Stevens, On the psychophysical law, Psychol. Rev. 64 (1957) 153–181.

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