Minority Game
The Impact of Information in a Multi-Round Strategy
Experiment
Master Thesis
In Partial Fulfillment of the Requirements of the Degree Master of Science Economics
University of Amsterdam
Handed in by: Daniel Gietl Author: Daniel Gietl Student ID: 10603271
Supervisor: Prof. Dr. J.H. (Joep) Sonnemans
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Outline
1. Introduction ... 2
2. Literature Review ... 3
2.1 Origin, Definition and Relevance of the Minority Game ... 3
2.2 Research on Minority Games ... 5
3. Experimental Design ... 8
3.1 Formulating Strategies ... 9
3.2 Practice Simulations by Participants ... 10
3.3 Computer Tournament, Feedback and Earnings ... 11
4. Results and Analysis ... 12
4.1 Aggregate Outcomes and Performance of Participants ... 12
4.2 Simulation Analysis ... 16
4.3 Imitation Analysis ... 19
4.4 Cluster Analysis ... 20
4.4.1 Setup of the Cluster Analysis & Cluster Descriptions ... 21
4.4.2 General Interdependencies of Cluster Performances ... 23
4.4.3 Round-specific Cluster Sizes and Performances ... 27
4.4.4 Reasons for the Weak Performance in Round 2 ... 29
4.4.5 Reasons for the Bad Performance in Round 3 ... 30
4.5 Analysis in a Nutshell ... 32
5. Discussion about the Selection Effect ... 33
6. Conclusion ... 36
7. Appendix ... 38
7.1 Appendix A: Instructions of the Experiment (translated from Dutch) ... 38
7.2 Appendix B: Cluster Changes across Rounds ... 42
7.3 Appendix C: Two Cluster Simulations ... 43
7.4 Appendix D: Average Rank of Clusters in each Round ... 45
7.5 Appendix E: Performance in Round 2 with all Strategies and when excluding single Clusters . 45 8. References ... 46
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1. Introduction
The repeated minority game stylizes many strategic economic situations that involve both competition and coordination. For example, it can be seen as an abstract (and simplified) version of financial markets, road-choice games, market entry games, cobweb markets and Cournot oligopolies. Although there are different methods to analyze the repeated minority game, they all have drawbacks. Specifically, all equilibria provided by game theory are non-strict and/or prone to suboptimal outcomes, simulation models suffer from the fact that researchers somewhat arbitrarily have to choose the strategies that are used, and laboratory experiments have the disadvantage of making it very hard to elicit exact strategies. Linde, Sonnemans and Tuinstra (2014) recently extended and improved the literature with an internet-based strategy method minority game that enabled participants to program their own strategies, thereby eliciting strategies humans actually choose. In their five player minority game with five rounds, participants programmed a strategy for each round (consisting of 100 periods each) based on the history of the last five periods. Before handing in a strategy for a round, the participants could try out strategies and run simulations against random strategies of the previous round to improve their performance. However, one of the main results of Linde, Sonnemans and Tuinstra (2014) was that the participants showed very little, if any, learning. The average points basically remained constant and similar to the average points of the symmetric mixed strategy Nash equilibrium across all rounds. Therefore, the question arose of how learning in minority games can be improved.
Thus, Linde, Sonnemans and Tuinstra ran a second very similar strategy method experiment, which is analyzed in this master thesis. This experiment has two additional features compared to Linde, Sonnemans and Tuinstra (2014). Firstly, participants are shown the rank and performance of all submitted strategies of the previous round. And secondly, the subjects have the possibility to run practice simulations against any strategies to their liking and not just against random previous round strategies. These two additional features initially lead to a strong decrease in efficiency in round 2 and round 3 of the five round minority game. This is intuitively surprising and in contrast with the (rather scarce) experimental literature on the repeated minority game, which up to this point has found that information has either no significant effects or a positive impact on efficiency.1 The focus of this thesis is therefore on why and how the additional information leads to the initial decline in performance. In the thesis, the simulation and imitation behavior of participants is assessed. Moreover, the strategies chosen by participants are analyzed via cluster analysis and simulations are run in order to find out how the different clusters perform against (dis)similar clusters. The main result of the analysis is that the information about the previous round strategies of all other participants and the performance of these strategies in the previous round leads many participants to imitate the previous round winner(s). Therefore participants chose strategies that are too similar to each other. Clearly, this is detrimental for
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all the players who chose similar strategies as, by definition, one can only be successful in the minority game if one behaves differently than most other players. Moreover, the imitation behavior shown by participants in round 2 and 3 of the experiment is also detrimental on an aggregate level, as only a few different strategies benefit from the many similar strategies, while the many similar strategies suffer from their resemblance.
The paper is structured as follows. Section 2 presents a literature review on the minority game. In Section 3, the detailed experimental design is presented. Section 4, the centerpiece of this thesis, presents detailed results of the experiment and provides an explanation for these results. More precisely, the section looks at the simulation behavior of participants and at whether and what previous round strategies were imitated. For a more insightful look into how strategies influence one another, strategies are allocated to clusters and many php simulations with different cluster compositions are run. A short discussion of the selection effect in the strategy method minority game is given in Section 5, before Section 6 concludes.
2. Literature Review
This part presents an overview over the minority game. Section 2.1 deals with the development of the El Farol bar game to the minority game, defines the minority game and shows its applications. In Section 2.2, different approaches (game theory, econophysics, laboratory experiments and strategy method experiments) for analyzing the minority game are presented and discussed, including short overviews on the main findings in each field.
2.1 Origin, Definition and Relevance of the Minority Game
The minority game is based on the El Farol Bar game, which was introduced by Arthur (1994). In this game, there are 100 music lovers who like to attend a weekly music show at the El Farol Bar, if the bar is not overcrowded (i.e. if fewer than 60 people show up). If 60 or more people show up, they would prefer to stay home. Each of the music lovers has to decide individually and simultaneously in each week whether or not to go to the bar. The main point of Arthur (1994) is that agents behave inductively rather than deductively. The deductive, game-theoretic approach would assume that people form beliefs about every other player’s behavior. Arthur (1994) argues, however, that agents only possess bounded rationality and that agents in his model show inductive behavior. More precisely, each agent considers a distinct set of only a few strategies and each agent applies the strategy from their set of strategies that has proven best so far. It is shown that even with these agents of very limited rationality, bar attendance evolves to the optimal value.
The econophycisists Challet and Zhang (1997) then defined the original minority game, which has two main modifications compared to Arthur’s El Farol Bar game. First, in Challet and Zhang (1997), agents’ strategies only consider whether or not it was worthwhile to go to the bar in the previous
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weeks and not the precise weekly attendance, which limits the number of possible strategies drastically.2 And second, their model is made symmetric. Transferred to the El Farol Bar Game, this means that the bar only has seats for half of all players and being in an uncrowded bar is just as good as not going when the bar is overcrowded. Hence, a player always wins equally many points whenever he is in the minority. After its introduction in 1997, the minority game was studied extensively by econophysicists (who generally assume that people behave inductively), publishing over one hundred papers in just five years (Challet, Marsili and Zhang, 2013). Meanwhile, the minority game has also caught the attention of other fields such as game theory and laboratory and internet-based experiments. The minority game has been examined with varying features such as the memory length of players, the factors participants can base their strategies on or the availability of certain information. However, the basic rules of the minority game are always defined as follows. The minority game is played by an odd number of players. In each period, each of these N players simultaneously and independently chooses between two sides/colours (say A and B/red and blue). Each player that chooses the side which was chosen by fewer players earns a fixed payoff in that period (one point in this thesis’ experiment), while players that chose the majority side receive zero points for that period.3
Due to its simplicity, the minority game stylizes many situations in which people aim to behave in opposition to the majority in a population. For example, the minority game is often seen as a stylized model of financial markets, especially speculative financial trading (e.g. see Challet et al. 2000, 2001). More precisely, one of the two sides of the minority game can be interpreted as buying and the other as selling. Correspondingly, buyers earn money when there are few buyers, because this drives down the price, and sellers make a profit when there are few sellers, as they will be able to demand a higher price. However, this basic model is often seen as too simplistic, because it does not feature transaction costs (among other things) and also because it is sometimes better to belong to the majority in financial trading. In fact, Challet, Marsili and Zhang (2013) themselves note that “(t)he real financial trading probably requires a mixed minority-majority strategy, in which timing is essential” (p.13). It can be very profitable to go along with the majority during an explosive boom, but a financial trader should aim to be one of the few who sell just before the bubble bursts.
In addition to financial markets, the minority game stylizes many other economic situations. For example, drivers tend to try to choose the least crowded lane on a busy multilane highway in order to reach their destination faster (these situations are modeled by road-choice games).4 A similar situation occurs when travelers decide on when, within a certain timeframe, to go on a holiday at a popular destination near the beach. Assuming they rather want to enjoy a less crowded beach, they should choose a time slot chosen by the minority. These decisions do not only impact travelers, but also tour
2 In the minority game of this thesis, participants are able to condition their strategy on past group sizes. 3
Obviously the minority side therefore always consists of maximally (N-1)/2 players and the majority side always consists of at least (N+1)/2 players.
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operators and hotels, because they have to figure out in advance the distribution of travellers over time in order to optimize their pricing and capacity (Challet, Marsili and Zhang, 2013). The minority game also stylizes a firm’s decision about whether or not to enter a new market (these situations are modelled by market-entry games).5 For example, if a certain business opportunity emerges, firms generally want to get in on this opportunity only if a few other firms do so, because otherwise competition would be too fierce. Moreover, Nobel laureate Thomas Schelling describes and models many situations where people try to belong to minorities (Schelling, 1978), such as housing migration. Linde, Sonnemans and Tunistra (2014) also remark that “the minority game is an abstract version of games where actions are strategic substitutes” (p.79), such as Cournot oligopolies and cobweb markets (Ezekiel, 1938). In a Cournot oligopoly, it is beneficial to produce a lot, if other firms, on average, have low production levels (and vice versa). Similarly, if the majority of producers in a cobweb market expect the next period’s price to be lower (higher) than the rational expectations equilibrium, the market clearing price in the next period will actually be higher (lower) than the rational expectations equilibrium. Therefore, it is optimal for producers in a cobweb market to have predictions that deviate from the majority prediction.
2.2 Research on Minority Games
The minority game can be analyzed with a variety of approaches. From a deductive, game-theoretic point of view, the finitely repeated minority game has a very unstable equilibrium structure. The Nash equilibria in pure strategies, where exactly (N-1)/2 players choose one side and (N+1)/2 choose the other side, are non-strict (players that are in the majority would not be hurt by switching sides) and payoff-asymmetric (players in the majority earn no points while players in the minority earn the maximum amount of points) and therefore unlikely to persist. Note that the Nash equilibria in pure strategies are the most efficient allocations possible in a minority game, because they ensure that the minority group is always as large as possible. Moreover, there is also a symmetric mixed Nash equilibrium (where each player chooses each side with probability 0.5 in each round) and infinitely many asymmetric mixed strategy Nash equilibria. For example, any situation with just as many participants always choosing A as participants always choosing B while all other players randomize between the two sides in each round is an asymmetric mixed strategy Nash equilibrium. All mixed Nash equilibria are non-strict and almost all of them are less efficient than the pure strategy Nash equilibrium, because they do not always achieve a minority containing (N-1)/2 players due to randomization.6 Due to the non-strictness, the general inefficiency and the quantity of mixed strategy Nash equilibria, it is unclear if players coordinate on a specific mixed strategy Nash equilibrium and if
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The market-entry game was introduced and defined by Selten and Güth (1982).
6 The asymmetric mixed strategy Nash equilibrium where (N-1)/2 players always choose side A and (N-1)/2 players always choose B while the remaining player randomizes between A and B is an exception and achieves the optimal efficiency. Note however that the one remaining player loses in each period and can therefore not do worse by changing his strategy. He might therefore decide to change his strategy in order to get other players to change theirs as well.
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so, which equilibrium they would choose. All in all, game theory therefore cannot unambiguously predict how participants behave in the finitely repeated minority game.
Another approach to analyze the minority game is the use of econophysics. Unlike game theorists, econophysicists, in the spirit of Arthur (1994), assume that agents behave inductively, possess only bounded rationality and are heterogeneous. In order to model these features, they run computer simulations using statistical physics of disordered systems. In these simulations, every agent is randomly drawn with a particular distribution of strategies and only chooses among these strategies. Every agent counts how often each of their strategies would have belonged to the winning side to that point and chooses their strategy that would have performed best so far.7 Simulations in which the sets of strategies evolve under evolutionary pressure especially show that agents coordinate on aggregate outcomes that are more efficient than the symmetric mixed Nash equilibrium (e.g. Li et al. 2000a, 2000b; Challet and Zhang 1997, 1998; Sysi-Aho et al. 2005).8 The downside of econophysicists’ computer simulations is that arbitrary assumptions have to be made, for example about the set of strategies chosen by agents, which, as argued by Linde, Sonnemans and Tuinstra (2014), may not represent the strategies actually chosen by humans.
Laboratory experiments, however, show how humans behave in minority games. In these experiments humans play the minority game and earn money depending on their performance. In Platkowski and Ramsza (2003) subjects participate in a 15-player minority game. The subjects’ only public information is the information about which side won in each of the previous M periods (varying from three to eleven). The authors find that this history length M does not have a significant impact on performance and that subjects on aggregate manage to coordinate more efficiently than in the symmetric mixed strategy Nash equilibrium. Chmura and Pitz (2006) performe a nine-player minority game with two treatments. In the first treatment, participants only have the information as to whether or not they belonged to the minority in the previous period, while in the second treatment they also receive the information about how many players chose each side in the last period. The authors find that the additional information in the second treatment improves aggregate efficiency. Bottazzi and Devetag (2007) have participants play the finitely repeated minority game with stationary groups of five players under different information conditions. They also find that aggregate efficiency is generally higher than in the symmetric mixed strategy Nash equilibrium and that individually participants show non-equilibrium behavior. Moreover, they state that information on individual
7 Agents generally do not consider the impact of a strategy on the aggregate outcome in a round. For example, assume that side A consisted of (N-1)/2 players in round i, and an agent chose a strategy S that chose the losing side B in that round. In round i, all strategies of the agent’s strategy set that would have chosen side A in round i would be counted as winners, even though their inclusion instead of strategy S would mean that side A would become the majority side.
8 The econophysics literature on the minority game is very rich. Challet et al. (2013) even consider the minority game as “more or less solved’” (Challet et al. (2013), p. 13). For much more detailed summaries of the minority game in econophysics see Challet et al. (2013) and Moro (2004). Also, http://www3.unifr.ch/econophysics/ (opened on 17/08/2014) contains a lot of econophysic analysis, discussions and extensions of the minority game.
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choices within the groups of five has no effect on aggregate performance, but revealing information about more than just the previous round does. Chmura and Güth (2011) investigate a three-player minority game, where groups are re-matched after each period and players can explicitly use a mixing device that enables them to randomize behavior. They find very heterogeneous decision rules with only about 25% of participants using the symmetric mixed Nash equilibrium. Finally, Devetag, Pancotto and Brenner (2011) conduct a three-player minority game where each player is a team consisting of three participants. The analysis of the teams’ discussions provides no evidence in support of the mixed strategy Nash equilibrium. They also find that when forming decisions in a minority game, teams are backward-looking and show low-rationality. All in all, the experimental literature on the minority game indicates that participants have heterogeneous behavioral rules and generally don’t show equilibrium behavior at the individual level. Moreover, the experimental literature on the impact of information in minority games is relatively scarce, while the literature that does exist either finds no effect or a positive effect of information on efficiency.
More recently, Linde, Sonnemans and Tuinstra (2014) add to the literature with a strategy method minority game. In experiments using the strategy method, participants formulate strategies in advance that contain their action for each possible decision node. This is in contrast to the laboratory experiments mentioned in the previous paragraph which use a direct-response method (i.e. players always choose an action whenever one of their decision nodes is reached).
Brandts and Charness (2011) provide a good overview on possible advantages and disadvantages of the strategy method. The major potential point of criticism of strategy method experiments is the possibility that the strategy method could lead to different decisions than the direct-respond method. From a game-theoretic point of view, the methods should not make a difference. Behaviorally, however, they possibly could. For example, Roth (1995) states that “having to submit entire strategies forces subjects to think about each information set in a different way than if they could primarily concentrate on those information sets that arise in the course of the game” (p.323). In order to shed light on this issue, Brandts and Charness (2011) analyze experiments that compare the two methods and conclude that the strategy method and the direct-response method lead to qualitatively similar results. Moreover, the strategy method has the advantage that it collects more observations as it also shows how participants behave at decision nodes that are not actually reached. Unlike the direct-response method, the strategy method therefore elicits the exact strategies of participants.
The five-player strategy method minority game of Linde, Sonnemans and Tuinstra (2014) consists of five rounds of 100 periods. In each round, participants formulate their strategy for each of the 100 periods based on the history of the previous five periods. Before submitting their strategy, the participants can try out strategies and run simulations against random strategies of the previous round to improve their performance. The main results are that participants use very heterogeneous strategies and that, somewhat surprisingly, the participants show very little, if any, learning, with the aggregate
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outcomes basically remaining constant and similar to the symmetric mixed Nash equilibrium across all rounds. The experiment analyzed in this thesis is a follow-up to Linde, Sonnemans and Tuinstra (2014), again designed and executed by Linde, Sonnemans and Tuinstra. The experimental design is modified in two ways (see Section 3), in order to find whether aggregate efficiency can be improved when participants have more information. It turns out that, contrary to the expected result, aggregate outcomes even drop initially, leading to the main question of this thesis about how the additional information causes the deterioration in efficiency. This question will be answered in Section 4, after the design of the experiment is explained in Section 3.
3. Experimental Design
The experiment is a five-player minority game in which participants have to submit one strategy for 100 periods for each of the five rounds. The experiment started in April 2010 and each round lasted one week. All participants are students from the very challenging “beta-gamma” bachelor program at the University of Amsterdam.9 A total of 43 students (16 women and 27 men) with an average age of 19.5 years (ranging from 18 to 24 years) participated in the experiment. The experimental design resembles the experimental design that Linde, Sonnemans and Tuinstra (2014) used in 2009 except for two main differences. Firstly, the participants in 2010 cannot only test a strategy against random previous round strategies, but they can also try a strategy against any strategies to their liking (see Section 3.2). Secondly, participants in 2010 are informed about the previous round strategies of all other participants and the performance of these strategies (see Section 3.3).
The first round of the experiment differs from the following four. It takes place at the CREED laboratory at the University of Amsterdam. After the minority game is explained to the participants, they play the game twice for ten periods in two different groups of players. Subsequently, participants are instructed on a handout and via a computer screen on how to formulate a strategy. After programming two verbal strategies, they formulate, test and submit their first strategy.
A few days later, participants receive the results of the first round including each participant’s strategy and performance. Then they can login on the website whenever they want and try out strategies of their own making against strategies of their own choice and/or simply against random previous round strategies. Their definitive strategy for the next round (which can also be the same as in the previous round) has to be submitted within a week after the laboratory experiment. Two days after the deadline the participants receive the results of the second round (again including everyone’s strategy and performance). The same procedure is repeated for rounds 3 to 5. After the fifth round the goals of the
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Students of this program follow courses in natural and social sciences and are generally well above average in motivation and capabilities. Especially their programming experience is markedly higher than that of average undergraduate students.
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experiment are explained, the results of the final round are announced and all earnings of the experiment are paid out.
Figure 1: Computer screen as seen by the participants when they formulate a strategy (translated from Dutch)
3.1 Formulating Strategies
The participants program their condition to change colors (sides) using a list of IF-statements.10 If the IF-statement is met, the strategy changes colors with the chosen return probability [0,1] for that condition. All IF-statements but the first one are in fact else-if-statements, which means they are only considered if none of the previous IF-statements are met. All strategies automatically end with the statement “ELSE {RETURN 0;}”, meaning that the strategy won’t change sides if no IF-statement is met. The strategies are based on the history of the last five periods: the strategy’s conditions can be based on the group size in any specific period of the last five periods and on whether the strategy has
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changed in any specific period of the last five periods.11 Participants can further refine their strategies by using logical expressions (e.g. AND, OR, negation, (in)equality) for their conditions, thereby allowing participants to base each condition on any combination of group sizes in any of the last five periods and/or past change behavior in any of the last five periods. The instructions contain many examples of possible strategies (see Appendix A for an English translation of the instructions).
The strategy space was restricted in two ways. Firstly, the history was restricted to five periods.12 Secondly, strategies could only decide on changing colors (sides) but not on choosing a specific color. While participants might have preferences for one color over the other, the color has generally no meaning in stylized applications of the minority game. Furthermore, the decision to focus only on changing colors halves the number of variables.
3.2 Practice Simulations by Participants
Before handing in their final strategy, participants are allowed to run as many practice simulations as they want. In each practice simulation they can try a simulation of a strategy of their own making against four opponent strategies. Participants could also choose each opponent strategy. They could type or copy and paste their own strategy into the left-hand box and the four opponent strategies in the right-hand boxes of Figure 1. The interface enabled them to easily copy and paste their own strategies from the previous rounds, their practice strategies from this round and the previous round strategies of all participants (they were also shown the ranking of these strategies in the previous round) in any of these five boxes. If participants did not enter a strategy into a right-hand box, a random previous round strategy was used automatically. Note that in Linde, Sonnemans and Tuinstra (2014) opponent strategies were always random previous round strategies and participants did not have the possibility to choose opponent strategies to their own liking.
For the simulation of the five strategies, the history for the first five periods is randomly drawn. Then, the next 100 periods are simulated according to the five strategies. After each simulation of five strategies the results are shown to the participant. The website shows what color each strategy uses in each of the 105 periods and gives summary statistics about the last 100 periods. These summary statistics include the number of points earned and the frequency of winning in a group of one (W1), winning in a group of two (W2) and losing in a group of three (L3), four (L4) and five (L5). You can see the results page of a practice simulation (against random previous round strategies) in Figure 2 below.
11 There are thus 55 * 25 = 100,000 possible histories. 12
This seems reasonable as 90.6% of participants in Linde, Sonnemans and Tuinstra (2014) (where participants could use exactly the same history length to program strategies) reported that they were able to program the strategy they wanted to program.
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Figure 2: Example of a screen after a strategy is tried out by the participants (translated from Dutch).
3.3 Computer Tournament, Feedback and Earnings
For each round we run a computer tournament with all final strategies. A simulation is run for each possible combination of five strategies. One simulation consists of randomly selected outcomes in the first five periods and 100 consecutive periods where the five strategies are executed. These 100 periods are then used to calculate the points achieved by each of the five strategies in this round. We then calculate the average points earned by each strategy in all simulations and use these to rank the strategies. After each round, all participants receive an email containing the rank and the average points of each participant. Additionally, they could also see each participants strategy (and each strategies’ performance) when they logged in. Thus, in this experiment, participants are informed about everyone’s previous round strategy and these strategies’ performances. This is a main difference to Linde, Sonnemans and Tuinstra (2014), where it was not possible to observe the strategies used by
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other participants. After the experiment, only the top five strategies of each round are paid out and receive €75, €60, €45, €30 and €15.13
4. Results and Analysis
In each of the 5 rounds we run one simulation of 100 periods for each possible combination of five submitted strategies to determine the results. For N submitted strategies in round i, we therefore run a total of ( ) simulations of 100 periods in round i and each submitted strategy is simulated in ( ) different groups of five submitted strategies. For each round between 30 (in round 5) and 43 (round 1) participants submitted a strategy. Average earnings totaled €58.37 with individual earnings ranging from €0 to €200.14 In Section 4.1 aggregate outcomes are presented and compared to game theoretic predictions and the experiment of Linde, Sonnemans and Tuinstra (2014). Section 4.2 deals with the simulation behavior of participants as well as its effects on performance, and Section 4.3 analyzes the imitation behavior of individuals with respect to the previous round. Moreover, we perform a cluster analysis in Section 4.4 and run several php simulations to see how these clusters affect another. This sheds further light on the cause for the bad aggregate performance in round 2 and especially round 3 of the experiment. Finally, Section 4.5 summarizes the findings of Section 4.
4.1 Aggregate Outcomes and Performance of Participants
Firstly, we compare the aggregate outcome of the experiment to the pure strategy Nash equilibria (PSNE) and the symmetric mixed strategy Nash equilibrium (MSNE). Note that each PSNE leads to the most efficient outcome possible, as they guarantee that the minority is as large as possible (just one player smaller than the majority).15 In our repeated five-player minority game, a PSNE consists of 2 players in the minority and 3 players in the majority for each period. Therefore, players in the PSNE earn on average 0.4 points per period and 40 points per simulation of 100 periods. The MSNE, however, is not as efficient as the PSNE, because its randomization sometimes leads to smaller minorities. More precisely, in a five-player minority game, the probability of an outcome where the minority consists of 0 (1) players is 6.25% (31.25%).16 Therefore, the expected value of points in the MSNE of a five-player minority game in a simulation of 100 periods equals only 31.25 points.17
13 In Linde, Sonnemans and Tuinstra (2014), the prizes for round 1 to 4 were identical. In round 5 of that experiment, prizes were twice as high (€150, €120, €90, €60 and €30). Also, in the thesis’ experiment, participants,could also fill out a questionnaire to earn more money. This questionnaire, however, will not be analyzed in this thesis.
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On average, women earned almost €10 more than men (€64.38 vs. €54.81).
15 An allocation where the difference between minority and majority is larger than one can never be a pure strategy Nash equilibrium, because (ceteris paribus) a player in the majority could benefit from unilaterally switching sides.
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0.54=6.25% and (5*0.54=31.25%).
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Table 1 shows that the aggregate performance of strategies in round 1 (31.43 points) is quite similar to the number of points expected in a MSNE (31.25 points). However, there is a sharp deterioration in round 2 and especially round 3, where the participants perform distinctly worse than the MSNE. Finally, participants’ performance improves markedly in round 4, reaching the level of the MSNE, and eventually outperforms the MSNE in round 5, although it still remains far below the points earned in a PSNE (40 points). Note also that the unequal distribution of points as indicated by the standard deviation, the minimum and the maximum, suggests that there is a substantial heterogeneity in performances of individual strategies. Therefore, it is clear that (many) individual strategies also do not conform to the MSNE. Performing a similar amount of simulations of 100 periods with only MSNE strategies would eliminate almost all randomization and lead to a standard deviation close to zero as well as a negligible difference between the minimum and the maximum.18
Table 1: Performance of participants over the rounds (2010)
Round 1 2 3 4 5 Number of participants 43 40 32 35 30 Points Average 31.43 29.57 26.32 31.41 32.84 Stand. Dev. 4.54 8.39 12.44 7.12 4.08 Minimum 21.35 13.74 8.47 20.64 26.82 Maximum 39.66 47.61 48.66 47.98 41.92 Average change propensity 46.75 37.23 50.49 38.24 35.35 Pearson correlation change and points
-0.59 0.22 -0.69 -0.22 0.21
Secondly, we compare the outcome of the thesis’ experiment (often referred to as “2010”) to the results of the experiment of Linde, Sonnemans and Tuinstra (2014) (often referred to as “2009”). As explained in the design section, the main difference between the experimental design of Linde, Sonnemans and Tuinstra (2014) and the one of this thesis is that in the latter, participants additionally are informed, from round 2 on, of all previous round’s strategies and the score of these strategies in the previous round and could choose opponent strategies to their liking. Therefore, the comparison of these two experiments provides interesting insights regarding the effect of this additional information.
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Each strategy was involved in 23,751 (round 5 strategies) to 111,930 (round 1 strategies) simulations of 100 periods.
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Table 2: Performance of participants over the rounds (2009)
Round 1 2 3 4 5 Number of participants 42 36 34 36 32 Points Average 31.69 31.12 31.88 32.29 31.85 Stand. Dev. 1.49 5.17 3.10 3.20 6.36 Minimum 29.31 21.65 24.98 27.89 18.93 Maximum 34.68 41.65 36.96 39.87 43.06 Average change propensity 47.61 38.15 45.99 38.51 40.29 Pearson correlation change and points
-0.531 0.028 -0.690 -0.493 -0.878
Table 2 shows that the aggregate first round performance in Linde, Sonnemans and Tuinstra (2014) was quite similar to the aggregate first round performance of 2010 (Table 1). This is not surprising, as the experimental design for the first round was exactly the same in both experiments. In the following rounds, the aggregate performance in 2009 remains quite constant as opposed to the drop (round 2 and 3) and resurgence (round 4 and 5) in 2010. Also, the heterogeneity of performances in 2009 is generally much smaller than in 2010, especially in round 2 and 3, where participants in 2010 perform relatively badly. Finally, both in 2009 and 2010 average change propensity is mostly negatively correlated with performance.19 In line with this, it is noteworthy that the change propensity in 2010 increases (decreases) heavily from round 2 to 3 (round 3 to 4), while the performance plunges (improved heavily).
Table 3: Performance of old and new strategies in old and new environment (2010)
Experiment 2010: The performance of the old and the new strategies in the old (column 2) and the new (column 3) environment. Strategies from participants who handed in the same strategy as in the previous round are dropped from the analysis.
Round The new strategy would have done better than the old strategy in the old environment
The new strategy does better than the old strategy would have done in the new environment
2 80.0% 30.0%
3 87.5% 6.3%
4 69.0% 20.7%
5 59.3% 51.9%
Total 75.0% 26.6%
In order to see how participants adapt their strategies over the rounds, we run simulations with each participant’s new strategy against all other old strategies and vice versa. Table 3 shows that in 2010 in 75% of all cases the new strategy would have done better against previous round strategies than the old strategy (column 2). On average, especially the round 2 (3) strategy of a participant would have
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This is in line with results from laboratory experiments and computational models on the minority game (e.g. Chmura and Pitz (2006) and Challet and Zhang (1997) respectively).
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outperformed the first (second) round strategy of the same participant in the first (second) round environment. At first glance, column 2 might indicate learning. However, the environment of strategies changes from round to round and therefore strategies that would have performed superbly in the previous round might perform very badly in the current one. Therefore, whether their new strategy performs better than the old one would have done in the new environment (column 3) is more important for participants. In only 26.6% of all cases the current round strategy did better than the previous round strategy against all current round strategies. Remarkably, 93.7% of all participants that changed their strategy from round 2 to round 3 would have done better in round 3 (ceteris paribus) if they had simply stuck with their round 2 strategy instead. All in all, Table 3 indicates that participants focus on doing well against previous round strategies. In the 2010 experiment, this could be done by testing the new strategy against the strategies of the previous round and/or by imitating the best strategies of the previous round and/or by shifting away from strategies that performed poorly in the previous round. However, participants do not seem to take into account that other participants might change their strategy as well, leading to the bad performances in round 2 and 3.
Table 4: Performance of old and new strategies in old and new environment (2009)
Experiment 2009: The performance of the old and the new strategies in the old (column 2) and the new (column 3) environment. Strategies from participants who handed in the same strategy as in the previous round are dropped from the analysis.
Round The new strategy would have done better than the old strategy in the old environment
The new strategy does better than the old strategy would have done in the new environment
2 42.9% 50.0%
3 77.8% 38.9%
4 66.7% 80.0%
5 38.9% 50.0%
Total 54.4% 53.1%
Table 4 shows the same analysis as the previous paragraph but for the experiment in 2009. Like in 2010, participants in 2009 are able to try new strategies in a simulation against random previous round strategies, which can lead participants to submit new strategies that do well in the old environment. However, in 2009 they do not know the previous round strategies of others and the performance of these strategies and are thus not able to imitate the best strategies of the previous round, which might explain why new strategies in 2009 only do better in their old environment compared to their old strategies in 54.4% of all cases (as compared to 75% in 2010). Also, in 2009 the new strategy performs better than the old strategy in the new environment in 53.1% of cases (as compared to 26.6% in 2010). As the further analysis will show, this higher percentage in 2009 is probably caused by the fact that imitation is not possible in 2009: in 2010 too many participants imitate the same strategies of the previous round (the winners), and these former winners then do very badly in the new environment, because most strategies perform badly if they play against themselves.
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All in all, the results of the 2010 experiment show that participants perform markedly worse in round 2 and especially round 3 than participants in 2009 and the mixed strategy Nash equilibrium. This is surprising given the fact that participants of 2010 are given additional information compared to 2009. Moreover, performances in 2010 are generally more heterogeneous than in 2009, especially so in round 2 and 3. Finally, in 2010, new strategies perform better (worse) in the old (new) environment than old strategies. There are two main factors that could have caused these sharp differences between 2009 and 2010. Firstly, it is possible that strategies of participants that simulated against the best strategies of the previous round (only possible in 2010) are different than strategies of participants that only simulated against random strategies (also possible in 2009). And secondly, it is possible that the most successful strategies are simply imitated by participants (only possible in 2010). These two possibilities will be analyzed in Section 4.2 and Section 4.3 respectively, which show that imitation had a stronger impact on the strategies chosen than simulation behavior.
4.2 Simulation Analysis
As in 2009, participants in 2010 can run practice simulations with a strategy of their own making against four opponent strategies before submitting their final strategy for a round. In 2009, their own practice strategy is always run against four opponent strategies that are randomly drawn strategies (without replacement) from the previous round. This is also possible in 2010. However, in 2010, participants can also replace one or more of the four randomly drawn strategies from the previous round with arbitrary strategies (either of their own making or any specific strategy from the previous rounds). This section analyzes whether there are significant differences in strategies and strategy performance depending on simulation behavior, in order to see whether the new simulation possibilities in 2010 could have caused the bad aggregate performances in round 2 and especially 3. All practice simulations were automatically recorded in a mysql database, giving us ample opportunities for analysis. All in all, participants run a total of 2080 practice simulations, starting with 909 simulations for round 1, following with 661, 242 and 116 for rounds 2 to 4, and finishing with 152 for the last round. For round 1, participants can only run strategies against preprogrammed strategies or strategies of their own making, just like in 2009. Therefore, our analysis focuses on simulation behavior for rounds 2 to 5, where participants in 2010 also had the possibility of trying strategies against strategies of their own liking (including previous round’s strategies).
Generally, participants make use of practice simulations. However, the number of participants running practice simulations decreases from 35 in round 2 to only 19 in round 5 (Table 5). Moreover, among the participants running simulations, the share of participants trying a strategy at least once against at least one of the “Top5” strategies of the previous round (“Top5 Simulators”) declines from 40% in round 2 to 26% in round 5. Among the Top5 Simulators, many participants only simulate against the best (and sometimes second best strategies) of the previous round.
17 Table 5: Simulation Behavior
Round 2 Round 3 Round 4 Round 5
All Participants 40 32 30 35
Not Simulating 5 10 8 11
Simulating 35 22 22 19
against at least one of the Top5 of previous round*
14 5 5 5
against 1. of previous round* 13 5 4 3 against 2. of previous round* 12 4 4 1 against 3. of previous round* 9 2 2 1 against 4. of previous round* 7 3 1 1 against 5. of previous round* 2 1 2 1
* Participants are counted if they simulate at least once against a strategy of that category, even if they did not do so in other practice simulations for that round.
Table 6 shows the shares of certain categories of strategies on all simulated opponent strategies in practice simulations. Of all opponent strategies used in practice simulations in the experiment, 78.6% are random previous round strategies. In comparison, few opponent strategies are Top5 strategies of the previous round (14.8%), especially so in the last two rounds. Only 6.6% of all opponent strategies are neither random previous round nor Top5 of previous round strategies. These “Other” strategies do not show any consistent pattern.
Table 6: Shares on all simulated opponent strategies
Round 2 Round 3 Round 4 Round 5 Total
Random previous round Strategy
74.9% 73.6% 90.3% 93.9% 78.6%
Top 5 of previous round 17.2% 18.4% 7.3% 4.4% 14.8%
1. of previous round 6.5% 7.2% 3.7% 2.3% 5.8% 2. of previous round 3.9% 4.9% 2.2% 0.2% 3.4% 3. of previous round 3.4% 2.4% 0.4% 1.2% 2.6% 4. of previous round 2.5% 2.9% 0.4% 0.7% 2.1% 5. of previous round 1.0% 1.0% 0.6% 0.2% 0.9% Other 7.9% 8.1% 2.4% 1.6% 6.6%
Furthermore, we look at differences in performance depending on simulation behavior (Table 7). At first glance, participants that run at least one simulation against a Top5 strategy of the last round (“Top5 Simulators”) perform slightly better in round 2, 3, and 4 and slightly worse in round 5 than all other participants. Also, all participants that ran at least one practice simulation in the corresponding
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round (“Simulators”) perform a bit better in rounds 2, 4, and 5 than participants that did not (“Non-Simulators”). However, we run Mann-Whitney U tests for each possible combination of two categories (e.g. Top 5 Simulators and Non-Simulators) in each round and none of the categories’ performances are different at the 5% significance level in any round.
Table 7: Simulation Behavior and Average Rank Average Rank Round 2 Average Rank Round 3 Average Rank Round 4 Average Rank Round 5 All Participants 20.50 (11.54) 16.50 (9.23) 18.00 (10.10) 15.50 (8.66) Simulators 20.17 (12.03) 17.50 (9.95) 17.05 (9.85) 15.11 (8.56) Non-Simulators 22.80 (6.88) 14.30 (6.91) 19.62 (10.31) 16.18 (8.77) Top 5 Simulators 17.43 (11.24) 13.80 (10.26) 16.20 (8.91) 20.75 (9.04) All Participants without
Top5 Simulators 22.15 (11.37) 17.00 (8.94) 18.48 (10.14) 14.69 (8.42) Simulators without Top5 Simulators 22.00 (12.19) 18.59 (9.59) 17.67 (9.93) 13.60 (7.97) * Standard Deviation in brackets
Finally, we look at the distance of final strategies of Top5 Simulators strategies to other Top 5 Simulator strategies and to all other participants, respectively (Table 8). Distances between strategies in this master thesis are calculated as in Linde, Sonnemans and Tuinstra (2014, p.86f.): “Strategies can use the history of the last 5 periods (outcome and whether they had changed colors in that period), which gives 100,000 possible histories. For every strategy the probability of changing color is calculated for each possible history. The distance between two strategies is then defined as the weighted average absolute difference between these probabilities.20 Because not all histories are equally likely (e.g. a 5-0 outcome is less likely than a 3-2 outcome) the weights are based upon the distribution that would result from the symmetric mixed strategy Nash equilibrium.” For both rows of Table 8, we take the Top5 simulators. For each Top5 simulator in each round, we calculate the average distance to all other Top5 simulators in the same round (row 1) and the average distance to all strategies from participants that do not simulate against the Top5 in the same round (row 2). Table 8 shows that strategies of Top5 simulators have a similar distance to other strategies of other Top5 simulators than to strategies of non-Top5 Simulators, indicating that these two kinds of strategies are indeed quite similar to another.
19 Table 8: Distances of Top5 Simulators
Average distances of strategies of participants that at least once simulated against the Top5 of the previous round to other such strategies (row 1) and to all strategies of participants that never simulated against the Top5 of the previous round (row 2).
Average Distance Round 2 Average Distance Round 3 Average Distance Round 4 Average Distance Round 5 Other Top5 Simulators 46.48 (4.77) 38.07 (4.60) 46.77 (3.17) 42.18 (5.78) All Participants without Top5 Simulators 43.89 (9.91) 36.66 (5.93) 43.52 (4.61) 43.32 (9.19)
* Standard Deviation between average distances of each strategy in brackets. ** The lower the distance the more similar the strategies.
All in all, the simulation analysis shows four main results. Firstly, relatively few people simulate against the top strategies of the last round. Secondly, these strategies only make up a small share of all opponent strategies. Thirdly, simulation behavior is not significantly related to performance. And finally, the strategies of participants that simulate against top strategies of the previous round are about as similar to other strategies of that kind as to strategies of participants that do not simulate against top strategies of the previous round. These four results all indicate that the possibility of trying strategies of one’s own making against strategies that performed well in the previous round does not majorly change behavior and thus cannot substantially contribute to the bad aggregate performance in round 2 and especially 3 in 2010. Another potential explanation for the bad aggregate performance in round 2 and 3 is that successful previous round strategies are imitated, which will be analyzed in the next section.
4.3 Imitation Analysis
In this section, the distance measure explained at the end of Section 4.2 is used to analyze whether certain strategies from the previous round x-1 are imitated in the current round x. More precisely, we compute the average distance of each single strategy in round x-1 to all strategies in round x-1 and, separately, to all strategies in round x. Then, for each round x-1 strategy, we subtract the average distance to all round x-1 strategies from the average distance to all round x strategies. This gives the change in average distance from round x-1 to round x for each strategy in round x-1 which is shown on the vertical axes of Figure 3. A positive value for the change in average distance implies that a strategy in round x-1 is less similar (i.e. the distance is bigger) to round x strategies than to round x-1 strategies. This means that participants shifted away from this round x-1 strategy. Finally, on the horizontal axes of Figure 3 we rank each strategy in round x-1 by their achieved points in round x-1.
The graphs in Figure 3 show two main characteristics. Firstly, round x-1 strategies that ranked poorly in round x-1 are less similar to round x than to round x-1 strategies as indicated by the high changes in average distance for high ranks in all but the last graph. This means participants shift away from strategies that performed poorly in the previous round. Secondly, round x-1 strategies that ranked well
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in round x-1 are more similar to round x than to round x-1 strategies. That means subjects imitate strategies that did well in the previous round. However, this is only consistently the case for the two best strategies and strongly so only for the best strategy of the previous round. The two best strategies of round x-1 are always more similar to the strategies in round x than to the strategies in round x-1. The best strategy was generally more heavily imitated than the second best one. The two main characteristics (imitating the best strategy and shifting away from the worst strategies) are less strong regarding the change in average distances from round 4 to round 5 than in other rounds.21 The next section will investigate how the two main characteristics influence aggregate outcomes.
Figure 3: Change in Average Distance to previous round
4.4 Cluster Analysis
Section 4.3 indicates that strategies that ranked first in the previous round are heavily imitated and that participants shift away from strategies that performed poorly in the previous round. However, it is so far unclear how (and if) this leads to the bad aggregate performances in round 2 and especially 3. In
21
For the analysis about how participants changed between clusters from round to round see Appendix B. An additional main trend that can be derived from Appendix B is that strategies of the cluster of the winner tend to stay within the same cluster in the next round.
-30 -20 -10 0 10 20 30 1 5 9 13 17 21 25 29 33 37 41 Ch an ge in Av e rag e D istan ce (R o u n d 2 - R o u n d 1)
Rank of Strategy in Round 1
Round 1 Strategies
-30 -20 -10 0 10 20 30 1 5 9 13 17 21 25 29 33 37 41 Ch an ge in Av e rag e D istan ce (R o u n d 3 - R o u n d 2)Rank of Strategy in Round 2
Round 2 Strategies
-30 -20 -10 0 10 20 30 1 5 9 13 17 21 25 29 33 37 41 Ch an ge in Av e rag e D istan ce (R o u n d 4 -R o u n d 3)Rank of Strategy in Round 3
Round 3 Strategies
-30 -20 -10 0 10 20 30 1 5 9 13 17 21 25 29 33 37 41 Ch an ge in Av e rag e D istan ce (R o u n d 5 - R o u n d 4)Rank of Strategy in Round 4
21
order to shed light on this, a cluster analysis is performed in this section, which will show that, in the experiment, imitation behavior leads to strategies being too similar to one another and thereby causes bad aggregate performance. Part 4.4.1 deals with the setup of the cluster analysis and describes the individual clusters. In Part 4.4.2 it is shown how clusters generally affect performances of the own and other clusters. Part 4.4.3 shows cluster sizes and performances in each round. Finally, using the analysis in Part 4.4.2 and 4.4.3 as well as additional round-specific simulations, Part 4.4.4 explains the weak performance in round 2 and Part 4.4.5 analyzes the causes for the bad performance in round 3.
4.4.1 Setup of the Cluster Analysis & Cluster Descriptions
We use a matrix of weighted distances between all 124 unique strategies of the experiment for the cluster analysis. The results are shown in the dendrograms (Figure 3), which are drawn with the program “multidendrograms”22
using the clustering algorithm “WARD”. The x-axis lists all unique strategies and the y-axis shows a measure of distances between the strategies within a cluster. For the analysis in this paper, we use the five clusters highlighted in Figure 4.
Figure 4: Cluster analysis of all 124 unique strategies
Tables 9 and 10 give a first impression of each cluster. Strategies in Cluster 1 generally change with a relatively high probability (but not certainty) after losing in a period. While more than half of Cluster 1 strategies also consider winning situations, these strategies generally only change with a very low (or zero) probability when winning in one of the previous periods. Cluster 2 strategies change sides very
22
http://deim.urv.cat/~sgomez/multidendrograms.php (opened on 17/08/2014). See also Fernández and Gómez (2008). Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5
22
often. Some of the strategies change (almost) every period. Notably, strategies that always change after losing in the previous period are in cluster 2, as they change often as well. Strategies in Cluster 3 roughly resemble the symmetric mixed strategy Nash Equilibrium. These strategies seldom take into account past change behavior. Cluster 4 contains strategies that rarely change. They often do consider a lot of periods, winning and losing situations a well as whether they had previously changed. However, they generally either assign a small change probability if a change condition (i.e. an IF-statement) is met or conditions for changing are relatively seldom met. Cluster 5 strategies generally change with a high probability if they won in the last period (or sometimes in one of the last two periods). Loses and changes in previous periods barely lead to changes. It is the smallest cluster by a large margin.
Table 9: The most central strategy
The most central strategy in each of the five clusters (the strategy with the minimum average distance to the other strategies in that cluster).
Cluster 1 If you lost the last period change with a probability of 0.6
Cluster 2 If you won in a group of 1 in the last period or if you lost in a group of 3 in the last period always change; else if change with probability 0.75 Cluster 3 If you won in a group of 2 in both the last and the second last period change
with probability 0.6; else if change with probability 0.5
Cluster 4 Never Change
Cluster 5 If you won in the last period change with probability 0.8
Table 10: Characteristics of clusters and of all unique strategies
Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5 Unique Strat. # of strategies 22 36 21 36 9 124 Randomization 91% 67% 100% 83% 67% 81% # of periods considered 2.32 2.58 2.33 3.08 1.44 2.56 Consider winning situations 55% 92% 95% 78% 100% 82% Consider losing situations 100% 94% 86% 92% 22% 88%
Consider whether you changed 27% 61% 19% 64% 22% 46% Average Change Propensity (Standard Deviation) 45.30 (10.12) 65.45 (15.89) 50.09 (8.01) 22.65 (11.32) 25.00 (6.51) 41.74 (23.00) Rough Description Lose=>Probably
(but not certainly) shift Change a lot. Including: Lose=>Certainly Shift Similar to MSNE Hardly Change Win=>Shift with high probability or certainty
23
4.4.2 General Interdependencies of Cluster Performances
In order to investigate the reasons for the uneven and weak performances in round 2 and 3, we run several simulations with all unique strategies of the experiment that show how the performance of clusters is generally influenced by strategies of the own (Table 11) or other clusters (Table 12a to 12e). At first, we look at the performance of clusters against strategies from the same cluster (Within Simulations) and identical strategies (Homogenous Simulations) respectively. For the Within Simulations, we use all unique strategies of that cluster and run one simulation of 100 periods for each possible combination of five strategies.23 For the Homogenous Simulation, we run 10,000 simulations of 100 periods with 5 identical strategies for each unique strategy and then calculate the average points of strategies in each cluster.24
Table 11: Within Cluster Simulations and simulations against identical strategies
Within Simulation Standard Deviation Homogenous Simulation Standard Deviation
Cluster 1 26.77 6.04 23.89 9.02
Cluster 2 31.92 7.01 22.43 11.49
Cluster 3 30.45 2.21 22.56 11.84
Cluster 4 35.57 6.37 26.74 10.85
Cluster 5 0.59 1.16 0.71 1.54
The Within Simulations indicate that cluster 4 strategies do quite well against strategies of the same cluster. Cluster 2 and 3 perform about as well as the MSNE would do against itself. Notably, cluster 1 and especially cluster 5 strategies do very badly against strategies from their own cluster. The remarkably bad performance of cluster 5 strategies in the Within Simulation is caused by the fact that the strategies that win in the first period very soon or even immediately switch to the losing side. Then, all the cluster 5 strategies get stuck in a 5 player majority as they basically never switch after losing. All clusters in the Homogenous Simulations perform poorly, showing that strategies suffer from meeting identical strategies and indicating that strategies are rather designed to exploit different strategies. Again, cluster 4 strategies gain the most points and cluster 5 strategies do not even gain one point on average. The reason for the latter is, like in the Within Simulation, that cluster 5 strategies get stuck in a five player majority after very few periods.
We run several simulations with unique strategies to see how strategies from each cluster influence the performance of strategies from other clusters as well as their own cluster (Table 12a -12e). Table 12a
23 With N being the number of unique strategies in a cluster (see Table 10, row 2), this gives us ( ) simulations of 100 periods for that cluster and each strategy in that cluster is simulated in ( ) different groups of five submitted strategies.
24
We use 10,000 simulations per strategy (instead of just 1) in order to eliminate (most of) the randomness. With N being the number of unique strategies in a cluster this gives a total of N*10000 simulations of 100 periods for a cluster.
24
focuses on the effect of cluster 1 strategies on all clusters. For each column in Table 12a, we draw 100,000 combinations of five strategies using a Mersenne Twister random number generator 25 and run one simulation of 100 periods for each combination. The difference between the columns is the way the combinations of five strategies are drawn. In the second column of Table 12a, we exclude all cluster 1 strategies and randomly draw five strategies from all other unique strategies. In the third column, for each combination, exactly one strategy is randomly drawn from all unique cluster 1 strategies and the other four strategies are randomly drawn from all other unique strategies. Similarly, a combination of five strategies in column 4 (5, 6) consists of two (three, four) randomly drawn strategies from all unique cluster 1 strategies and three (two, one) randomly drawn strategies from all other unique strategies.26 Tables 12b to 12e are constructed the same way as Table 12a but focus on the effect of cluster 2, 3, 4 and 5 on all clusters.For each combination of five strategies in Table 12a to 12e the strategies are drawn with replacement.Note that the last columns, which analyze the performances when four strategies from one cluster face one strategy from all other strategies, are of lesser importance as this situation happens relatively infrequently in the experiment.
Table 12a: Effect of Cluster 1
Performance of clusters in simulations with zero, one, two, three and four cluster 1 strategies. Points in bold, standard deviations below.
No Cluster 1 Strategy One Cluster 1 Strategy Two Cluster 1 Strategies Three Cluster 1 Strategies Four Cluster 1 Strategies Cluster 1 x 38.23 31.38 27.03 26.23 x 2.48 2.26 3.10 4.52 Cluster 2 30.95 28.88 28.45 30.53 33.59 2.64 1.81 5.55 9.79 13.85 Cluster 3 30.48 30.43 32.61 37.11 41.51 1.98 0.99 2.90 5.23 7.72 Cluster 4 33.64 33.32 35.15 38.73 40.56 3.62 2.26 2.84 3.89 4.83 Cluster 5 21.89 30.14 40.06 50.77 57.51 0.68 0.46 1.31 2.96 5.28 Total 31.00 32.34 32.16 30.86 28.94 4.22 3.81 4.99 8.89 11.82
Table 12a shows that strategies from cluster 1 (lose=>probably shift) perform very well if they only face strategies from other clusters (38.23 points) and their performance deteriorates with the number of cluster 1 strategies in the simulation. Cluster 2 is relatively unaffected by cluster 1, while cluster 3 and 4 slightly benefit if they meet many cluster 1 strategies. Importantly, cluster 5 performs remarkably
25 http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html (opened on 17/08/2014).
26 Whenever one or more strategies are drawn from all other clusters, each strategy has the same likelihood to be chosen. For example, in Table12a there are 36 unique strategies from cluster 2 (22 from cluster 3, 36 from cluster 4 and 9 from cluster 5). It is therefore much more likely that a cluster 2 strategy is drawn from all other clusters than a cluster 5 strategy, simply because there are more unique cluster 2 strategies. This feature has the advantage that the environment in Table12 is quite similar to the environment of the experiment, because clusters that were more frequently used in the experiment are generally also more often used as a strategy from all other clusters in the simulations for Table 12a-12e.
