ORIGINAL ARTICLE

T. Ueda · Y. Ishida (*)Department of Knowledge-Based Information Engineering, Toyohashi University of Technology,Toyohashi, Aichi 441-8580, Japane-mail: ishida@tutkie.tut.ac.jp

This work was presented in part at the 14th International Symposium on Artifi cial Life and Robotics, Oita, Japan, February 5–7, 2009

Artif Life Robotics (2009) 14:498–501 © ISAROB 2009DOI 10.1007/s10015-009-0736-4

Takuya Ueda · Yoshiteru Ishida

Reverse engineering of spatiotemporal patterns in the spatial prisoner’s dilemma

2 Spatial prisoner’s dilemma and spatiotemporal patterns

2.1 SPD

The iterated prisoner’s dilemma (IPD) iterates the prison-er’s dilemma for N steps. The spatial prisoner’s dilemma (SPD)1,3 is an extended IPD with a spatial axis, i.e., the SPD is played by multiple players placed in neighboring cells in a lattice space. Here, we restrict ourselves to the case of a one-dimensional square lattice. In our SPD, a player has a strategy and decides an action at each step based on their strategy. Each player plays with neighboring players (includ-ing themselves). The score of each play will be calculated from a payoff matrix (Table 1) for neighboring players within a radius r. For example, when r = 1 and the space is a one-dimensional square lattice, the number of neighbor-ing players is three. After every interaction, the strategy of the highest score will be given to the neighboring players.

All-D and All-C are the base strategies.1 All-D (All-C) chooses D (C) every time. In addition to All-D (All-C), we will study spatial strategies such as k-D (k-C).3 The k-D (k-C) strategy chooses D (C) when the number of players with D (C) in the neighborhood exceeds k. We will also use the strategy code. The 2-D strategy, for example, can be expressed by a strategy code CCD (Table 2).

2.2 Spatiotemporal patterns generated by SPD

The spatiotemporal patterns found in SPD can involve many interesting ones, including fractal patterns. Figure 1 shows the spatiotemporal patterns obtained with parameter b = 1.5 in Table 1.

3 SPD with a single strategy as a DCA

The SPD can be identifi ed as deterministic cellular autom-ata (DCA)4 with D as 1, C as 0, and neighborhood radius

Abstract Many complex patterns are produced by the spatial prisoner’s dilemma, such as spatial games (Nature 1992;359:826–829) and spatial strategies (Artif Life Robot-ics 2004;9:139–143). We have studied the inverse problem of identifying a game by estimating the parameters in the payoff of the game from spatiotemporal patterns.

Key words Spatial prisoner’s dilemma · Reverse engineer-ing · Cellular automata · Fractal

1 Introduction

Many spatiotemporal patterns can be found in nature, such as ice crystals, a coastline, and shells. A spatial game1 can produce spatiotemporal patterns. This is an extension of the well-known prisoner’s dilemma (PD) in a space dimension.

While in general a game is played by two players, a spatial game is played by multiple players. Each player is placed at a cell in a lattice and interacts with the neighbor-ing players. Figure 1 shows the spatiotemporal patterns generated by the one-dimensional spatial prisoner’s dilemma.

This article deals with the inverse problem of estimating the parameters of a spatial game from spatiotemporal pat-terns (generated by the spatial game). When the space is a k-dimensional lattice (k is a natural number), a spatial game can be identifi ed as cellular automata. A spatial game process can be estimated as a transition rule of cellular automata from spatial patterns by the rule identifi cation algorithm.2 Hence, the strategies of each player can be inferred by analyzing the spatiotemporal patterns.

Received and accepted: June 8, 2009

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rCA. Therefore we can apply the rule identifi cation algo-rithm2 to the spatiotemporal patterns of SPD. Table 3 is the result of rule identifi cation. We use six patterns (Fig. 2) generated by a single strategy of the SPD (without inter-action and with action update only) with the payoff matrix in Table 1 and parameter b = 1.5. Each strategy has been tested by assuming DCA and rCA = 1. When two strategies are involved, the rules cannot be identifi ed as DCA with a binary state and rCA = 1. Only in the case of All-C vs All-D can rules be identifi ed as DCA with a binary state and rCA = 2.

4 Estimating the payoff matrix with All-C vs All-D

Figure 3 shows the spatiotemporal patterns from All-C vs All-D when b is varied. The lattice size is 30 with a periodic

boundary condition, and the simulations were carried out in 30 steps. Cs and Ds are initially seeded at random with an equal probability. Table 4 lists the rules identifi ed from the spatiotemporal patterns of Fig. 3. When 1.0 < b ≤ 1.5, the rule is identifi ed as (ECF4EE30)16, and when 1.5 < b < 2.0 as (ECF4EF34)16. Thus the rules of the SPD with All-C vs All-D can be identifi ed as either (ECF4EE30)16 or (ECF4EF34)16 when rCA = 1. In other words, we can esti-

Fig. 1. Spatiotemporal patterns generated by a one-dimensional spatial prisoner’s dilemma of All-D vs All-C. White cells cooperate and black cells defect. The simulation is carried out in 50 steps with a periodic boundary condition. The space size is 300 (a one-dimensional square lattice) and the neighborhood radius is r = 1. The pattern above is the result of All-D vs All-C, the pattern in the middle is from All-D vs DCD, and the pattern at the bottom is from 1-D vs DCD. The top pattern is monotonous, but the middle and bottom patterns involve fractals

Table 3. Results of rule identifi cation with a single strategy is used. Rules are represented by Wolfram’s numbering5

Fig. 2. Spatiotemporal patterns generated by a single strategy of SPD with initial random seeds

Fig. 3. The spatiotemporal patterns generated by All-C vs All-D in a one-dimensional lattice with a periodic boundary condition

Table 4. Rule identifi cation in the case of All-C vs All-D when param-eter b is varied

Table 2. Strategy codes when r = 1 and the space is a one-dimensional square

Table 1. Payoff matrix (T > R > P > S, 2R > T + S, and 1.0 < b < 2.0)2

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mate the range of parameter b from the spatiotemporal patterns generated by All-C vs All-D.

5 SPD with multiple strategies as PCA

When the SPD is analyzed by the probabilistic rule identi-fi cation algorithm,2 the SPD with multiple strategies can be identifi ed as probabilistic cellular automata (PCA) with a binary state and rCA = 2 (Table 5). Table 6 shows the rules of the SPD identifi ed as a rule of PCA with All-D vs 2-D.

6 Estimating the payoff matrix with All-D vs other strategies using PCA

Since the SPD with All-C vs All-D can be identifi ed as DCA, we can estimate the range of parameter b, but other cases (such as All-D vs 1-D) cannot be identifi ed as DCA. However, these cases can be identifi ed as PCA, and we can also estimate the range of parameter b (except for the case All-D vs 1-D). Figures 4–6 are binary trees showing the estimated range of parameter b in each game. We can fi nd

Table 5. Rule of PCA. pl indicates the probability of being D

Table 6. SPD identifi ed as a rule of PCA with All-D vs 2-D with the following parameters: lattice size, 500; time steps, 500; b, 1.5. These results are the average of one thousand trials

Fig. 4. A binary tree showing the estimated range of parameter b with All-D vs 2-D or DCD

Fig. 5. A binary tree showing the estimated range of parameter b with All-D vs CDC or DDC

Fig. 6. A binary tree showing the estimated range of parameter b with All-D vs DCC

the range of parameter b by tracing the tree with condi-tional branches of probability pi. For example, parameter bl in the case of All-D vs 2D with p9 ≠ 0, p18 ≠ 0, p2 = 0, p8 = 0 is estimated to be within the range 1.0 < b ≤ 1.5.

7 Estimating strategies from spatiotemporal patterns

Since the SPD with a single strategy can be identifi ed as DCA, we can infer the strategies of each player. Figure 7 is a spatiotemporal pattern generated by 1-D vs DCD. Both the bottom left and bottom right patterns in Fig. 7 are part

Fig. 7. The spatiotemporal pattern generated by 1-D vs DCD (above) and its enlarged clusters (below)

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of the pattern above. Table 7 lists the result of the rule identifi cation from strategy cluster-A, and Table 8 lists the result from strategy cluster-B. The upper row of each table indicates the neighborhood confi guration (e.g., CCC), and the lower row indicates the next state of the center. The symbol u (unknown) means that the state cannot be identi-fi ed from the spatiotemporal patterns. The strategy of cluster-A is the 1-D strategy (Table 7), and that of cluster-B is the DCD strategy (Table 8). By analyzing a cluster of spatiotemporal patterns, we can infer possible strategies that could generate the patterns.

8 Conclusions

The spatial prisoner’s dilemma (SPD) may be considered as cellular automata (CA). Since reverse engineering on the

spatiotemporal patterns generated by CA allows us to rec-ognize the possible rule generating the patterns, we can likewise infer the possible strategy underlying the game by reverse engineering on the spatiotemporal patterns gener-ated by the SPD. We regarded the SPD with a single spatial strategy as being deterministic cellular automata (DCA), and the SPD with multiple strategies as being probabilistic cellular automata (PCA). The spatiotemporal patterns gen-erated by the SPD include suffi cient information to estimate the range of parameters of the SPD (and hence identifying the game played when an appropriate space–time frame is used).

Acknowledgments This work was supported in part by Grants-in-Aid from Toyohashi University of Technology and CASIO Science Promo-tion Foundation.

References

1. Nowak MA, May RM (1992) Evolutionary games and spatial chaos. Nature 359:826–829

2. Ishida Y, Mori T (2004) Spatial strategies on a generalized spatial prisoner’s dilemma. Artif Life Robotics 9:139–143

3. Ichise Y, Ishida Y (2008) Reverse engineering of spatial patterns in cellular automata. Artif Life Robotics 13:172–175

4. von Neumann JJ (1966) Theory of self-reproducing automata. Burks A (ed) University of Illinois Press

5. Wolfram S (1984) Universality and complexity in cellular automata. Physica D 10:1–35

Table 7. Identifi ed rule of cluster-A

Table 8. Identifi ed rule of cluster-B