Effect of Information Exchange in a Social Network on Investment: a study of Herd Effect in Group Parrondo Games

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1 Effect of Information Exchange in a Social Network on Investment: a study of Herd Effect in Group Parrondo Games Ho Fai MA, Ka Wai CHEUNG, Ga Ching LUI, Degang Wu, Kwok Yip Szeto 1 Department of Phyiscs, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, HKSAR, China 1 address: phszeto@ust.hk Abstract. The problem of playing investment games with information exchange is studied using a model of Parrondo players linked in a social network so that herd effects can be implemented. Here the players play Parrondo games which can be considered as an investment into two slot machines, C and D, so that playing continuously on one slot machine will lose, but by suitable switching the play on these two slot machines a player can win in the long run statistically. The Parrondo effect can be explained in terms of Markov chain and has its roots in the non-equilibrium physics of Brownian ratchets. The original Parrondo Game contains two independent Games, which called Game A and Game B. Game A can be thought as a coin-tossing game, which has probability a bit less than half to win. On the other hand, game B is a complicated game with two biased coins: a good coin and a bad coin. The good coin's winning probability is 0.75, and the bad coin's winning probability is only 0.1. If the player's capital is multiple of 3, the bad coin is used, otherwise the good coin is used. One can also show that continuous playing of game B leads to loss. Here the individual players can win or lose depending on their own strategies of sequences of game they play. Numerical and analytical results show that there are special sequences composed of motif ABABB will lead to winning. When we couple these players with information exchange in a social network, we can study the impact of information on the collective behavior of the players. We investigate herd effect via a simple model where the players can adopt one of two strategies of game: (I) Follow the winners or (II) Avoid the loser. We find interesting behavior of the resulted average gain of the population, depending on the susceptibility of the players being influenced by others. We observe that players using either follow the winner alone, or avoid the loser alone will lead to loss for the entire population. This particular feature of herd effect is explained and verified by numerical experiment. We then provide two distinct, though similar strategy of evolution for the population of players, communicating with the nearest neighbors in a ring type social network, so that the population can actually win with herd effect. The first strategy of evolution is to randomly mix the game plan follow the winner with avoid the loser. The second strategy of evolution is to give each player a preset probability to switch between follow the winner and avoid the loser. Despite the similarity of these two strategies of evolution in our study of herd effect, we observe difference in the gain of the population. We also provide a heuristic explanation for this difference in gain based on the probability distribution of the resident time for the player in his choice.

2 1 Introduction In 1996, a game was invented by J.M.R. Parrondo based on the physics of the Brownian rachet [1]. The original Parrondo s game consists of two games called game A and game B. Game A can be regarded as tossing a coin, player has probability p A = 1/2 - ɛ to win. Game B is a little complex: if the current capital of the player is divisible by a parameter M (usually set to be 3), then player will have winning probability p b (usually set to be 1/10 - ɛ); if the current capital is not divisible by M, the winning probability will be p g (usually set to be 3/4 - ɛ). Here ɛ is a small parameter which can be interpreted as some kind of commission or transaction cost charged by the casino when A and B are two slot machines played by the gambler. Game A and game B are winning if ɛ is negative; fair if ɛ is zero; losing if ɛ is positive. Both games can be considered as a coin flipping game. Game A involves a fair coin, while game B is also a coin flipping game that involves two bias coins: a good coin (it is played when the current capital is not a multiple of M with a winning probability p g ), and a bad coin (it is played when the current capital is multiple of M, with a winning probability p b ). In case of ɛ is positive, either playing game A or game B when played alone, the long term average capital is decreasing. However, alternating playing game A and game B can lead to an increasing average capital, corresponding to a winning situation. This strange phenomenon is called Parrondo s effect: two losing strategies combined to form a winning strategy [2,4,6]. The optimal deterministic sequence for Parrondo s game is discovered by L. Dinis [10]. Other form of games which have Parrondo s effect are also develop, such as history-dependent game [3], cooperative Parrondo s games [5], quantum Parrondo s game [7-8], etc In this paper, the original Parrondo s game is used to study the herd effect on a social network. 2 Oringinal Parrondo s game The original Parrondo s game can be analyzed using the discrete time Markov chain. Let π(t) = (π 0 (t), π 1 (t) π M-1 (t)) be the probability distribution, with the entry π i (t) being the probability that the capital is i (mod M) at time t. For the case M=3, the transition matrix for game A is: 0 pa 1 pa A 1 pa 0 p pa 1 pa 0 and the transition matrix for game B 0 pb 1 p b B 1 pg 0 pg pg 1 pg 0

3 Here the left hand notation is used. For example, if game A is played at time t, the probability distribution at time t+1 can be calculated by matrix multiplication using π(t+1) = π(t) Π A. To calculate the capital gain, let us suppose we have probability π i to be at state i, and the corresponding winning probability is p i. For the winning situation, we will get capital π i p i ; while for losing situation, we will lose capital -π i (1 - p i ). So we will obtain capital on the average π i p i - π i (1 - p i )= π i (2p i - 1). Summing all possible states, we can calculate the expectation value of the capital gain g. We can also accumulate the expected capital gain to find the expected capital g=σπ i (2p i - 1). For instance, if only game A is played, we know the winning probability is p A for all possible states, thus the capital gain is g A =2p A Method At the beginning, all players are assigned into a communicating social network with the topology of a ring, as shown in Fig.1. The information that player receives is minimal. Players are supposed not to know which one is game A, or game B (The games are labelled as game C, and game D respectively if they player do not know the intrinsic winning probabilities of game A and B, but only know game C can either be A or B and correspondingly game D be B or A). Furthermore, we assume that the players do not know their current capital state. In fact, if they do, then player can do data mining of the result of their past games and try to figure out the nature of the game C (whether it is A or B) as well as the parameters (p A, p g, p b, and M). If the player obtains the knowledge of game C and D, such as the case that C is game A and D is game B, then he can win by simply following the optimal sequence for Parrondo s game from Dinis s work [10] and win. In our model, the players are forming a communicating social network and they simply play game C and D without knowing which one is A or B, and has no knowledge of the parameters (p A, p g, p b, and M). However, at each time step, each player receives the information from his nearest neighbors. Player will know the nearest neighbors last game (game C/game D), and the nearest neighbor s winning situation in last game (win/lose). After collecting these information from his neighbors, the player can adopt one of two evolution strategies (i) Avoid loser (AL), or (ii) Follow winner (FW). The AL strategy is that, player will use the game not played by the loser(s) when there is no ambiguity, otherwise, player will continue playing his last game. For examples, if both neighbors of player i have lost in the last game and both used game C, then player i will use game D since there is a clear loser; if there is no clear loser to avoid, then player i will keep playing the game he played last time. The FW strategy is similar, player will use the game played by the winner(s) when the situation is unambiguous, otherwise, player i will keep playing the game he played last time his last game. We compare the performance of pure AL, pure FW, stochastic mixture of AL and FW, and randomly switching between AL and FW.

4 Figure 1: The Network of Ring Structure In principle, we can analyze the behavior of the network by using discrete time Markov chain. For each player, we need to consider his present capital, winning situation and the type of game he played last time. Therefore the dimension of the state of one player is 32 2=12. (Here 3 stands for the three possible states of the capital, 0, 1, and 2; and 2 for the two possible types of game, C or D; and two possible outcomes: win or lose). If we have N players, the dimension will be =12 N. Thus, even for the smallest cluster of three players, the dimensionality of the vector describing the state of the population is The evolution of the population of players will involve a Markov chain of a large matrix. Thus, we will use numerical simulation to study herd effect instead of exact calculation of the Markov Chain. 4 Result of Numerical Simulation If only the strategy Follow the winner (FW) is applied, the expected capital is decreasing as shown by Fig.2. This initially counter-intuitive result can be explained. It is because under the FW strategy, there are two absorption states: CCC and DDD corresponding to the fact that when all players played the same game last time, no player can change their game. The FW strategy is reminiscent of ferromagnetism in statistical physics, when all players play either game C or game D after a sufficient long period, resulting in loss since continuous playing one game will be losing in long term, thus Follow the winner is a losing strategy. The population of players using follow the winner becomes one single minded player playing only one type of game. There is no switching and the result is continuous loss, as dictated by the setting of the parameter of the original Parrondo game where playing one type of game alone will lead to loss.

5 Expected Capital Expected Capital time Figure 2: Simulated result of the protocol Follow the winner. The expected capital is the ensemble average over configurations. The population size is We see that the expected capital is decreasing with time. When the strategy Avoid the loser (AL) is applied, the above situation of the entire population following playing one game will not happen. However, the expected capital is still decreasing. The simulated result is plotted in Fig.3. This time, the absorption state is CDCDCD This time the population is switching regularly from C to D back to C. The Avoid the loser strategy is reminiscent of anti-ferromagnetism in statistical physics, so that when player i plays game C, the nearest neighbors player i- 1 and player i+1 plays game D. Under this strategy, players will not change his/her game choice, so Avoid the loser (AL) is also a losing strategy time Figure 3: Simulated result of the protocol Avoid the loser. The expected capital is the ensemble average over configurations. The population size is We see that the expected capital is also decreasing with time.

6 Expected Capital Expected Capital Gain We can see that either Avoid the loser or Follow the winner alone, the performance of the population of communicating players is not good. We therefore explore the situation when the evolution strategy is not purely Avoid the loser or purely Follow the winner. Let us consider a network which evolves under strategies Avoid the loser and Follow the winner randomly, the randomized process is controlled by a parameter γ so that at each turn, there will be a probability γ to adopt the Avoid the loser strategy and a probability 1 - γ to adopt the Follow the winner strategy. The stochastic mixture of AL and FL can be expressed as γal+(1 - γ)fw. Numerical result shows that in some regime of the parameter space (p A, p g, p b, and M), the expected capital gain is positive, corresponding to a winning situation time (a) (b) Figure 4: (a) The stochastic mixture of AL and FL, γal+(1 - γ)fw. Here the value of γ is 0.5. The expected capital is an ensemble average result over 5000 configurations. We see that the expected capital is increasing. (b) The expected capital gain for different value of γ at long time. The randomized strategies is controlled by the mixing parameter γ. We can see that the return is positive in some region of this mixing parameter, showing that by appropriately mixing the AL and FW strategies, the population of communicating players in the ring type social network can win. In Fig.4a, we see that for a given γ=0.5, there is a positive expected capital increase at long time even initially the mixed strategy is losing. On the other hand, from Fig.4b, we observe that the expected capital gain at long time is negative for γ close to zero or γ close to one, so that too much of one type (either AL or FW) of strategy leads to loss at long time. This result for a population of players is similar to a single player playing the original classical Parrondo games where the player mix game A and game B stochastically with γa+(1 - γ)b, similar to our mixed evolution strategies for the population in a ring type social network, the expected capital gain is negative when γ close to zero or γ close to one [5]. The reason is also similar: when γ is close to zero, almost only game B is played; by contrast when γ close to one, almost only game A is

7 Expected Capital Gain played. Either playing game A or game B alone, the player will lose in the long run, with negative expected capital gain. In our analysis of herd effect in a communicating social network, the result is similar: when γ is close to zero, almost only Follow the winner FL is adopted; and when γ is close to one, almost only Avoid the loser is adopted. Either these two strategy is losing, so the expected capital gain is negative. Noticing the similarity between herd effect in a population of players and single player in Parrondo game, we can explore other combination of AL and FW strategies. Let s first consider the slightest modification from random mixing. We consider switching probabilistically between the two strategies AL and FW. Let α be the probability of changing the evolution strategy in the next step. The numerical result shows that if we switch the strategy more frequently, the performance will be better Figure 5: The expected capital gain versus α. The switching between Avoid the loser and Follow the loser is controlled by a parameter α. The capital gain is increasing monotonically with α. From Fig.5, we can see that the expected capital gain is negative if α is close to zero. For the case of small α, there is a large chance to adopt the same evolution strategy continuously. The network may go to the absorbing state of corresponding strategy. Consequently, players will keep playing the same game, it lead to negative capital gain. The higher value of α will have the smaller chance to go to the absorbing state, result in higher expected capital gain. Now if we compare Fig.5 with Fig.4b, the result for the small value of α in switching probability is similar to the case of random mixing with small value of γ. This is understandable since in these situations the social network of players will go to the absorbing state of playing only one game, making the entire population similar to a player playing always one game. However, the highest value of expected capital gain in Fig.4b is less than the highest value in Fig.5. For the switching strategy with probability α, the highest expected capital gain occurs when α=1, which means that the evolution strategy must be changed each time.

8 In order to understand the difference between the result of random mixing (Fig.4b) and probabilistically switching (Fig.5), we extract the histogram of the number of steps that the player does not change his game. We run numerical experiments for each strategy of evolution N(=10,000) times, and collect the number of times N T that a player does not change his game (A or B) for T steps. A player stays unchanged in his selection of game (A or B) for T steps is the resident time the player spend continuously in a particular game. The probability distribution p(t) of the resident time T is obtained from the histogram after normalization, so that p(t) = N T /N with Σp(T) = 1. Since we have two strategies of evolution, random mixing or probabilistically switching, we collect the data for each strategy and plot the corresponding histogram N T vs T in Fig.6. (a) (b) Figure 6: (a) The stochastic mixture of AL and FL with γ = 0.4 (b): The probabilistic switching of AL and FL for α=1. The x-axis is the duration T (number of games or steps) of playing the same game. The y-axis is N T. These results are obtained by numerical simulation, with total N=10,000 experiments We see that there is a characteristic long tail for random mixing evolution shown in Fig.6a, implying that this random mixing cannot prevent players keep on either AL or FW for a long time. On the other hand, there is no such long tail in the histogram in Fig.6b, which is the case for probabilistic switching. In fact, in Fig.6b, there is a significant peak in the histogram at T=2, implying that most of the players in probabilistic switching like to switch more frequently. Since we know that players stay playing in either A or B will lead to loss, the histogram distribution in Fig.6 provides the reason behind the higher expected gain (0.015 at α=1) for probabilistic switching than random mixing (0.012 at γ 0.4).

9 5 Conclusion We have studied the herd effect on a ring network by using Parrondo s game. We find that the two strategies Avoid the loser and Follow the winner are losing. Adopting the two strategies randomly can lead to winning in the long term. This phenomenon is similar to the Parrondo effect for a single player. We also see similar effect of positive gain for the population of players when we allow a probabilistic switching of strategy between avoid the loser and follow the winner. We understand qualitatively the higher gain in probabilistic switching compared to random mixing from an analysis of the histogram distribution of resident time of the players. Since our social network is very simple, being a ring type network, we expect that more complex herd effect behavior can arise from the topological properties of the social network, and the resident time distribution may exhibit features linked to the topology of the social network for the communication of the players. For future work, we will investigate the importance of the social network on the gain of the population, which will then provide more insight into herd effect in more realistic social network. 6 Acknowledgment K.Y. Szeto acknowledges the support of grant FSGRF13SC25 and FSGRF14SC28 References 1. Parrondo, J. M. R. "How to cheat a bad mathematician." EEC HC&M Network on Complexity and Chaos (1996). 2. Abbott, D., & Harmer, G. (1999). Parrondo's paradox. Statist. Sci., 14(2), Parrondo, J., Harmer, G., & Abbott, D. (2000). New Paradoxical Games Based on Brownian Ratchets. Phys. Rev. Lett., 85(24), Harmer, G., Abbott, D., & Taylor, P. (2000). The paradox of Parrondo's games. Proceedings Of The Royal Society A: Mathematical, Physical And Engineering Sciences, 456(1994), TORAL, R. (2001). COOPERATIVE PARRONDO'S GAMES. Fluct. Noise Lett., 01(01), L7-L HARMER, G., & ABBOTT, D. (2002). A REVIEW OF PARRONDO'S PARADOX. Fluct. Noise Lett., 02(02), R71-R Flitney, A., Ng, J., & Abbott, D. (2002). Quantum Parrondo's games. Physica A: Statistical Mechanics And Its Applications, 314(1-4), Flitney, A., & Abbott, D. (2003). Quantum models of Parrondo's games. Physica A: Statistical Mechanics And Its Applications, 324(1-2), Dinís, L., & Parrondo, J. (2004). Inefficiency of voting in Parrondo games. Physica A: Statistical Mechanics And Its Applications, 343, Dinis, L(2008). Optimal sequence for Parrondo games. Physical Review E, 77(2).

10 11. ETHIER, S., & LEE, J. (2012). PARRONDO GAMES WITH SPATIAL DEPENDENCE. Fluct. Noise Lett., 11(02), Wu, D., & Szeto, K. (2014). Extended Parrondo's game and Brownian ratchets: Strong and weak Parrondo effect. Physical Review E, 89(2).

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