Masters Thesis
Optimising Resource Allocation
in Socio-economic Systems
with the Minority Game:
A Case Study on Electric Cars
in Amsterdam
Author: Ramona Roller Supervisor: Dr. Rick Quax First examiner:Prof. Michael Lees
Second examiner:
Dr. D´avid Kop´anyi
A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Computational Science
in the
Section Computational Science Informatics Institute
I, Ramona Roller, declare that this thesis, entitled ‘Optimising Resource Allocation in Socio-economic Systems with the Minority Game: A Case Study on Electric Cars in Amsterdam’ and the work presented in it are my own. I confirm that:
This work was done wholly or mainly while in candidature for a research degree
at the University of Amsterdam.
Where any part of this thesis has previously been submitted for a degree or any
other qualification at this University or any other institution, this has been clearly stated.
Where I have consulted the published work of others, this is always clearly
at-tributed.
Where I have quoted from the work of others, the source is always given. With
the exception of such quotations, this thesis is entirely my own work.
I have acknowledged all main sources of help.
Where the thesis is based on work done by myself jointly with others, I have made
clear exactly what was done by others and what I have contributed myself.
Signed:
Date: 31 October 2017
Abstract
Faculty of Science Informatics Institute
Master of Science in Computational Science
Optimising Resource Allocation in Socio-economic Systems with the Minority Game: A Case Study on Electric Cars in Amsterdam
by Ramona Roller
Electric vehicles (EVs) provide a solution to common modern day problems, such as air pollution, noise, and scarcity of natural resources. In order to recharge batteries EVs require unoccupied charging stations which are often scarce. This classic resource allocation problem in supply- and demand systems has been analysed with the Minority Game (MG) in previous work. In this decentralised model agents repeatedly have to decide between two options and win if they end up in the minority group. The ineffi-ciency of the system systematically depends on the memory size of the agents. Based on the conceptual similarities between the e-mobility system and the MG this study extracts memory and inefficiency metrics from charging data of the EV users. Results reveal that efficient and inefficient charging behaviour can be systematically separated based on the memory value. The specific inefficiency and memory values differ between areas of Amsterdam and also over time. We therefore do not receive a stable pattern of inefficiency regimes that can be compared to the stable one in the original MG. This shows that additional aspects of charging behaviour have to be included in our model, such as the type of information EV users use to decide whether to charge. Furthermore, we present some possibilities of varying memory of the EV users in desired ways in the real world. These practical interventions are discussed with respect to battery size, car sharing schemes, and varying charging fees. This study applies the MG to a new type of supply- and demand system and by using a data-based approach bypasses the limi-tations of theory-based simulation studies. Our approach provides a starting point to achieve efficient charging behaviour in the e-mobility system of Amsterdam.
I would like to thank the following people for their assistance with this project.
My supervisor, Rick Quax, without whose thoughtful advice and constructive criticism this project would not exist in its current form. Thanks for standing my message bom-bardment on Slack.
My housemates, for allowing me to build a small improvised computational cluster with their laptops and tolerating it occupying the table in the living room.
My family for encouragement, unwavering support and exhilarating chitchat from KA-town.
And Pim. For comfort during frustrating periods, for shared enthusiasm when things went brilliantly, for practising presentations a thousand times, and for chocolate when words were useless.
Declaration of Authorship i
Abstract iii
Acknowledgements iv
Contents v
List of Figures viii
List of Tables x
Abbreviations xi
Symbols xii
1 Introduction 1
2 Background 8
2.1 Resource allocation in socio-economic supply- and demand systems . . . . 8
2.1.1 Centralised resource allocation . . . 8
2.1.2 Distributed resource allocation . . . 9
2.1.2.1 Resource allocation in networks . . . 10
2.1.2.2 Resource allocation in games . . . 11
2.1.2.3 Resource allocation in the Minority Game. . . 12
2.1.2.4 Extensions to the original Minority Game . . . 13
2.2 Phase transitions in socio-economic systems . . . 20
2.2.1 Terminology . . . 20
2.2.2 Phase transitions in the real world . . . 22
2.2.3 Phase transitions in the Minority Game . . . 22
3 Methods 25 3.1 General set-up . . . 25
3.2 Pool construction . . . 26
3.3 Pool filtering . . . 27
3.3.1 Competitiveness filter . . . 27
3.3.2 Regularity filter. . . 27
3.3.3 Activity filter . . . 28
3.3.4 User count filter . . . 29
3.4 Parameter measures . . . 29
3.4.1 Constraints of e-mobility system and assumptions . . . 29
3.4.2 Correlation analysis in e-mobility system . . . 29
3.4.3 Memory in e-mobility system . . . 32
3.4.4 Inefficiency in e-mobility system . . . 33
3.5 Memory-inefficiency relationship . . . 34
3.5.1 Validation of relationship between data-driven and fixed memory in original MG . . . 34
3.5.2 Validation of inefficiency change pattern with data-driven memory in original MG . . . 34
3.5.3 Memory-dependent inefficiency clusters in e-mobility system. . . . 34
4 Results 36 4.1 Validation of data-driven memory measures in original MG . . . 36
4.1.1 Correlation analysis in original MG. . . 36
4.1.2 Relationship between measured and fixed memory in original MG 38 4.1.3 Relationship between measured memory and inefficiency in origi-nal MG . . . 39
4.2 Pool filtering . . . 40
4.3 Comparison between inefficiency measures . . . 42
4.4 Memory-inefficiency relationship in e-mobility system. . . 43
4.4.1 Measuring memory and selecting best filtering thresholds . . . 43
4.4.2 Global memory-inefficiency analysis . . . 46
4.4.3 Local spatio-temporal analysis . . . 48
5 Discussion 52 5.1 Interpretation of results in real-world setting . . . 52
5.2 Comparison between global and local analyses in e-mobility system . . . . 55
5.3 Comparison between original MG and e-mobility system . . . 57
6 Conclusion 64 Bibliography 66 Appendices 71 A The original Minority Game . . . 71
A1 Memory-dependent inefficiency changes in original Minority Game 71 A2 Strategies in original Minority Game . . . 72
A3 Different types of agents in original Minority Game. . . 72
B Summary of e-mobility data . . . 76
D Selection of pool dimensions . . . 79
E Preparation of pool filtering . . . 83
E1 Expected number of users . . . 83
E2 Number of targeted stations. . . 84
E3 Filtering thresholds . . . 85
F Results of pool filtering . . . 86
F1 Competition factor . . . 86
F2 Regularity, activity and user count . . . 87
G Properties of Markov Model . . . 89
H Correlation analysis in original Minority Game . . . 94
I Comparison of inefficiency measures . . . 96
J Selection of threshold value combination . . . 97
K Results of memory-inefficiency relationship in e-mobility system . . . 100
L Translating normalised inefficiency measures back to original unnormalised ones . . . 103
1.1 Schematic overview of original MG.. . . 3
2.1 Schematic overview of original MG with extensions. . . 13
3.1 Quantification of relationship between current and past charging behaviour 31 4.1 Association of choice behaviour per sliding window size in original MG for different memory sizes . . . 37
4.2 Relationship between data-driven and fixed memory in original MG. . . . 39
4.3 Relationship between data-driven memory and inefficiency in original MG. 40 4.4 Number of pools plotted in relation to the amount of valid users after successive filtering in city district 3 months. . . 41
4.5 Number of valid pools with at least 80 users for city district 3 months. . . 42
4.6 Frequency of filter threshold combinations with specific values of nor-malised mutual information between the inefficiency measures in the city district 3 month pool dimension. . . 43
4.7 Association of charging behaviour per sliding window size in Amsterdam South Jul - Sep 2016.. . . 44
4.8 Normalised mutual information and pool count for city district 3 months. 45 4.9 Memory-inefficiency relationship for city district 3 months. . . 46
4.10 Comparison of MG metrics between temporal pool components in city district Amsterdam South . . . 50
4.11 Comparison of MG metrics between temporal pool components in resi-dential quarter Buitenveldert . . . 51
A1.1 Memory-dependent change in inefficiency in original MG. . . 71
A2.1 Examples of two strategies in original MG. . . 72
A3.1 Memory-dependent change in inefficiency in MG with differing ratios of flexible and noise agents. . . 73
A3.2 Memory-dependent change in inefficiency in MG with differing ratios of flexible and frozen agents. . . 74
A3.3 Memory-dependent change in inefficiency in MG with differing ratios of flexible, frozen, and noise agents. . . 75
B.1 Mean kWh charged per day. . . 77
B.2 Number of users charged per weekday. . . 77
C.1 Comparison between data-driven approach and our method. . . 78
D.1 Effect of varying agents-rounds ratios on coordination in original MG. . . 80
D.2 City districts in Amsterdam . . . 80
D.3 Residential quarters in Amsterdam . . . 80
E2.1 Fraction of same-named weekdays on which a specific number of charging
stations was used in Amsterdam South within past year of 1st May 2016. 85
F1.1 Numbers of expected users, targeted and existing stations per day in the
pool Amsterdam South May through July 2016.. . . 86
F2.1 Number of pools plotted in relation to the amount of valid users after successive filtering in all pool dimensions. . . 87
F2.2 Number of valid pools with specific user count thresholds for each pool dimension. . . 88
G.1 Schematic representation of 2nd order Markov Model. . . 90
G.2 Association of causal states per sliding window size in Markov Model for different orders with large stationary probability. . . 91
G.3 Association of causal states per sliding window size in Markov Model for different orders with small stationary probability. . . 92
H.1 Association of choice behaviour per sliding window size in original MG for all tested memory sizes. . . 95
I.1 Frequency of filter threshold combinations with specific values of nor-malised mutual information between the inefficiency measures per pool dimension. . . 96
J.1 Normalised mutual information and pool count for each pool dimension. . 98
K.1 Memory-inefficiency relationship per pool dimension . . . 102
L.1 Translating normalised inefficiency to unnormalised values. . . 104
M.1 Inefficiency in residential quarters within 1 month periods . . . 105
M.2 Inefficiency in residential quarters within 3 months periods . . . 106
M.3 Inefficiency in residential quarters within 6 months periods . . . 107
M.4 Inefficiency in city districts within 1 month periods . . . 109
M.5 Inefficiency in city districts within 3 months periods . . . 110
3.1 Translation of original Minority Game to e-mobility system. . . 26
3.2 Schematic contingency table for calculating φ. . . 30
B.1 Features of e-mobility data . . . 76
D.1 Pool dimensions in e-mbolity data . . . 81
E3.1 Threshold values for pool filters. . . 85
J.1 Selected threshold values per pool dimension and inefficiency measure. . . 99
e- electric
EV Electric Vehicle
Kolkata Paise Restaurant Problem
LTM Long-Term Memory
MCMG Multiple Choice Minority Game
MG Minority Game
MM Markov Model
RCG Random Choice Game
STM Short-Term Memory
α Size of alphabet in Markov Model; Control parameter in original MG
ai Activity of user i
c Competitiveness factor
C Total number of available resources in MG
d Day in temporal period t
D Maximum days in temporal period t
∆t Size of sliding window
E[ud] Expected number of users charging on day d
ηm
Inefficiency of charging behaviour from perspective of municipality ηu
Inefficiency of charging behaviour from perspective of user
H Entropy
i Index of user
Inorm Normalised mutual information
j Index of sliding window size
˜
ld Number of targeted stations on day d
κ Total possible number of causal states in Markov Model
m Memory size in original MG
max(u(c)d )
Maximum number of users being able to charge on previous
same-named weekdays as day d
µ m-long history in MG
N Total number of users
Nc Total number of choice options in MG
n Order of Markov Model
n(c)d−∆d&d
Given current day d,
number of same-named weekdays in previous year where user charged
and where user also charged ∆d days before.
n(c)d−∆d
Given last charging day proceeding from day d,
number of same-named weekdays in previous year where user charged
φ Phi coefficient giving the degree of
association between two variables
px,t
Pool including all EV users who charged in spatial component x during period t at least once
Ps Stationary probability in Markov Model
Pt Transition probability in Markov Model
P (i(c)d ) Probability of user i to charge on day d
P (i(c)d−∆d) Probability of user i
to charge ∆d days before day d
R Number of restaurants in
the Kolkata Paise Restaurant Problem
si Number of charging sessions of user i
s(pi x,t) Number of charging sessions user i had in pool p
s(t)i Number of charging sessions user i had in temporal period t
σNmin Standard deviation of minority group size
t Temporal component
θa Activity threshold
θr Regularity threshold
θu User count threshold
ud Number of users on day d
u(c)d Number of users having charged on day d
vi Validity score of user i
x Spatial component in Amsterdam
X N × D − 1 matrix storing x-data per user
and per sliding window size
Y N × D − 1 matrix storing y-data per user
Introduction
Coordination in complex systems is a key driver for emergent properties (Mitchell,2009). In these systems agents interact and adapt on a local level which gives rise to macroscopic behaviour being more than the sum of the individual actions. A common phenomenon in these systems is the competition for scarce resources. Agents have to coordinate their decisions with respect to each other in order to be successful. Usually, this im-plies choosing the option that fewer competitors have chosen. In financial markets for instance, traders will be better off if they outnumber buyers since prices are higher in this situation. Predators increase their chances of finding food if they hunt in areas with fewer competitors. Rush-hour drivers reach their destination faster if they choose the route containing the minority of traffic (Nakar and Hod,2003).
Socio-economic systems are a specific subclass of these competitive complex systems, where social behaviour influences economic properties. For example, social interactions, such as coordinated decision making, affect the productivity, efficiency, and quality of the system’s output. Socio-economic systems therefore meet social needs and provide value (European Environment Agency, 2015). Socio-economic systems are of particu-lar interest, since resources can be related to novel technologies, such as cloud space. Moreover, they link to the prevalent Western idea that societal success is defined on an economic basis. Finally, socio-economic systems allow for advanced coordination mechanisms as humans are able to use advanced communication means.
Game dynamics provide a rich environment to study coordination in these complex socio-economic systems (Fudenberg and Levine,1998). In this theoretical setting each agent has a set of strategies and seeks to maximise her payoff. The game is played successively over many rounds. Winning strategies are rewarded, losing ones are penalised, and the agents maximise their payoff by choosing the strategy that performed best in the past. Agents adapt their behaviour to the other agents’ decisions, whereby the complexity of
the agents and their interaction mechanism can be varied depending on the game (Claus
and Boutilier, 1998; Tan, 1993). Under specific conditions these adaptive responses
may lead to coordination patterns in the system that give rise to economically-desired properties.
The Minority Game (MG) is an example of such a game. In contrast to other resource allocation games, like the ultimatum game (Nowak et al.,2000), the cake cutting problem
(Dubins and Spanier,1961) or the public goods game (McGinty et al.,2012), properties
of the MG match many large-scale systems in the real-world. The number of players in the MG is larger than two, agents are homogenous, meaning that they value resources equally, and resources are not owned by the agents themselves but provided by some external party.
In the original MG, agents face a binary decision between two equally valuable options, such as going to the bar or staying at home (Arthur,1994). The aim of the game is to choose the option that fewer agents have chosen. Those lucky agents form the minority group and are rewarded in the current round. In contrast, the agents in the majority group are penalised. In order for a minority group to form, the MG requires an uneven number of agents. Both options have the same capacity size, that is, they can maximally
host (N −1)2 agents.
The agents base their decisions on strategies which depend on the agents’ memory size. The memory size m is a fixed parameter in the MG and defines the number of previous game rounds for which the agents can remember the outcome of the game. For example, if m = 2 the agents know whether going to the bar or staying at home was the winning option in the last two rounds. In the original MG all agents are assigned the same memory size.
Based on the memory size, the strategy pool is created. A strategy is a look-up table consisting of all possible m-long histories of the game and a decision sequence (see Appendix A2 for an example of a strategy). Each possible history is matched to a decision. All strategies have the same possible histories but differ in the assigned decision sequence. Thus, there are 2m possible histories and 22m possible decision sequences, resulting in 22mpossible strategies. At the beginning of the game each agents is randomly assigned s strategies out of the 22m possible ones. The number of strategies s per agent is a fixed parameter and agents may have the same strategies, as s is sampled with replacement.
At the start of each round the agents choose the strategy that has been most successful in the past. They compare the real history of the game with the possible histories in their chosen strategy. Agents then choose the option that corresponds to the matched
history in the strategy. After all agents have made their decision their choices are evaluated determining whether they won or lost in the current round. The strategies of the winning agents get rewarded so that their use in forthcoming rounds is reinforced. In contrast, strategies of the losing agents are penalised in order to discourage their future use. Figure 1.1 shows the general set-up of the original MG and the actions taken in one game round.
(a) Set-up
(b) One game round
Figure 1.1: Schematic overview of original MG. Several game parameters are specified before the game starts (A). They are used to initialise the game and subsequently used in all game rounds. One round consists of a sequence of several actions (B).
The original MG is most efficient if the capacity of the resource is fully used. This implies that the size of the minority group is maximised, i.e. it is exactly (N −1)2 . In this case the maximum possible number of agents gets rewarded. Since the mean is the optimal value in the original MG, the standard deviation of the minority group sizes σNmincan be used
to quantify the inefficiency of the system (Challet and Zhang, 1998). The larger the standard deviation, the smaller the minority group and the more inefficient the system. In the original MG the standard deviation varies systematically with the memory size
(Savit et al.,1999). For small memory sizes the standard deviation is large, it reaches
a minimum for an intermediate m and finally converges to the standard deviation value of the random choice game (RCG) for large memory sizes. In the RCG agents do not use any strategies but rather choose one of the options at random. These three different inefficiency regimes indicate that agents self-organise to coordinate their decisions in favour of the macroscopic outcome of the game. It is important to note that agents do not communicate directly with each other in the original MG. They only receive temporally limited global information on the best possible option and know about their own choice. Thus, agents coordinate their decisions via indirect communication.
The MG has been used as a model to study coordination in real-world socio-economic systems, such as financial markets, where traders compete for shares (Challet et al.,
2000). In order to better adapt the model to real-world conditions, various extensions of the original MG have been studied, such as the evolutionary MG (Nakar and Hod,
2003) or the grand canonical MG (Challet and Marsili,2003), which are summarised in the literature review (2).
To the best of our knowledge, previous work has exclusively analysed theoretical or simulated MGs. The original MG was usually extended in ways that matched assump-tions about the real world. For example, agents were enabled to decide in which round they enter the game, as not all stockbrokers trade everyday (De Martino and Marsili,
2006). After these extensions were simulated, their results were used as explanations for real-world processes.
In this study we would like to try a different approach that extracts MG-similar dynamics from a real-world data set without simulations. As a subject of study we have chosen an e-mobility system, on which the MG has never been tested before as far as we know. The e-mobility system represents an example of a socio-economic system that is currently transforming the landscape of many cities. E-mobility systems consist of electric vehicles (EVs) and their corresponding infrastructure such as charging stations and high voltage
power lines. EVs are a response to common modern day problems in urban areas,
including air pollution, noise, and scarcity of natural resources (Doucette and McCulloch,
2011).
The municipality of Amsterdam promotes the use of EVs as she aims to become a zero-emission city by 2025 (Iamsterdam, 2017). Specific measures of the municipality include subsidising EV facilities such as parking spots and investing in the infrastructure by setting up public charging stations all over the city (Rijksdienst voor Ondernemend
Nederland, 2017). Whenever EV users connect to these public charging stations, their
charging behaviour is tracked. In this way it is exactly known who is charging at what place and point in time. These data are summarised in AppendixB.
Like other socio-economic systems, resource allocation is a point of concern in the e-mobility system. The supply of charging stations, representing the scarce resource, has to match the demand of the EV users to charge the car. This makes the e-mobility system conceptually very similar to the original MG.
Previous research on the Dutch e-mobility system indicates that the system does not allocate resources efficiently. For example, EV users tend to connect their car much longer than actually required for charging (van den Hoed et al.,2013), EV users with complementary charging patterns (residents charge during the night at home, commuters
charge during the day at work) are not matched so that stations are constantly occupied
(Helmus and van den Hoed, 2015), and the timing of the well-intentioned reservation
schemes for parking spots next to charging stations actually increase parking pressure which decreases efficiency of the system (Wolbertus and van den Hoed,2017).
Although these studies have applied different definitions of efficiency, they all indicate that the matching of EV users to charging stations in the e-mobility system of Amster-dam provides space for improvement.
Since the original MG specifically optimises this process, it might serve as a valid model for resource allocation in the e-mobility system of Amsterdam. A comparison between the original MG and the e-mobility system reveals many similarities.
First, in both system there is competition for a scarce resource. There are fewer charging stations than EV users and the resources in the original MG cannot accommodate all agents in the game.
Second, MG-agents and EV-users adapt their behaviour to the state of the system. If an MG-agent has always lost with a strategy in the past she is very unlikely to choose this strategy in the future again. Similarly, EV-users who find the same charging station always occupied at a particular point in time, are likely to either look for a different station or try the same station at another time.
Third, the previous point implies some form of memory since the agents have to remem-ber previous outcomes of the game to change their behaviour in the present accordingly. Fourth, being in the minority is advantageous in the original MG and in the e-mobility system. The fewer agents choose one of the options the better that option. In the same manner, the fewer EV users decide to charge, the larger their chances to find a free charging station.
Fifth, the original MG and the e-mobility system both represent an nth order Markov Model (MM) since the present behaviour of the agents and users only depends on the previous n rounds.
However, the original MG and the e-mobility system also differ in some important as-pects. First, the resource in the original MG is compact whereas it is spatially divided in the e-mobility system. MG-agents only make a choice between two options. In contrast EV-users not only have to decide if they charge but also where.
Second, whereas MG-agents only communicate indirectly via a global mechanism, EV users use both local- and global information. They organise their charging sessions in local WhatsApp groups and can check a “charging website” to see the global occupation state of each charging station in the system (ecomovement,2017).
Third, all MG-agents regularly play the game every round. EV-users on the other hand do not have to charge their car every day, so there is no need to play the game on a daily basis. Moreover, EV-users cannot observe the game from the sideline and prepare their decision accordingly. This is because agents are unlikely to check the charging website when they do not need to charge. However, they can plan future charging behaviour based on local observations. For example, if an EV-user decided not to play today and coincidentally sees another user charging at her favourite station at her preferred time, the user in question might remember this incident and choose another station for her next charging day.
Fourth, the capacity size of the resource in the original MG remains constant whereas it changes in the e-mobility system. As the municipality sets up new charging stations, existing ones fail, and EV-users enter and leave the system, the station-user ratio changes over time.
Many of these differences arise from a global comparison between the two systems. However, by only considering specific data subsets these differences are likely to vanish. For example by only taking into account regular users or times when no new charging stations were added to the system.
In the following we therefore focus on the similarities between the two systems. Based on these, the current study assumes that EV users play the MG and that MG metrics such as memory and inefficiency can be measured from the data. In contrast to previ-ous work, which has mostly used theory-based simulations of extended MGs to study real-world systems, our study directly uses real-world data. Moreover, it is different from the classic data-driven approach (Solomatine et al., 2009). In the latter, model parameters are estimated from the data and used for simulations so that their results can be quantitatively compared to the real data. In our approach the model remains in its theoretical form and its metrics are qualitatively compared to measured surrogates in the e-mobility data. AppendixCgraphically compares our approach to the data-driven one.
The aim of this study is to quantify efficiency of charging behaviour in the e-mobility system of Amsterdam and to control it via MG dynamics. Based on the similarities between the original MG and the e-mobility data, we formulate two related hypotheses.
1. EV users play the original Minority Game.
2. Data-based memory measures are able to separate efficient from inefficient charging behaviour and therefore allow to control the efficiency of charging behaviour.
This report structures as follows: Section 2 reviews previous work on resource alloca-tionand phase transitions in complex socio-economic systems. Section 3 describes our methodology, Section 4presents the results and Section 5 discusses those.
Background
2.1
Resource allocation in socio-economic supply- and
de-mand systems
Resource allocation has been extensively studied within the multi-agent systems com-munity. The type of resources and of interactions between agents in these systems are important parameters that influence the allocation mechanism (Chevaleyre et al.,2006). Based on the type of resources in the e-mobility system of Amsterdam (i.e. the opportu-nities to charge) and the agent interactions, this literature review is restricted to systems with indirect inter-agent communication and resources that are indivisible, non-sharable, not owned by the agents before the allocation process, and that have to be consumed in the environment itself, i.e. the agents cannot carry the resource away. For a broader survey on issues of resource allocation in multi-agent systems refer to Chevaleyre et al.
(2006).
2.1.1 Centralised resource allocation
Two main types of allocation procedures can be distinguished. Centralised procedures use an overhead institution to distribute resources. Distributed procedures use the self-organised coordination of agents to allocate resources. Typical examples of the centralised approach are auctions (Cramton et al., 2006). They take place between a central authority, the auctioneer, and a collection of agents the bidders. Based on the agents’ bids, the auctioneer rewards the resource to one of the agents, depending on the auction scheme (seeKrishna(2002) for an introduction to auction theory).
For example,Boutilier et al.(1999) have used a variation of a first-price sealed bid auc-tion to allocate resource bundles to agents. Agents desire matching resources from po-tentially different bundles to execute some plan. Prices of different resources are assumed independent. In this scheme individual resources cannot be assigned a well-defined val-uation. This does not allow agents to bid less than the resource’s true valuation, which is normally the best strategy in the classic first-priced sealed bid auction (Wooldridge,
2009).
So rather than assigning value to individual resources agents construct a bidding policy by which their bids for any resource is conditioned on the outcome of events earlier in the round. This is achieved by equipping agents with an initial bias in the form of a subjective probability distribution of the highest bids in the auction. By updating this distribution after each round, agents can overcome their initial bias.
Due to the independence assumption, learning about one price cannot influence an agent’s bidding strategy on other resources. This leads to a slow adaptation in the auc-tion, where agents for example take a long time until realising that no other agents com-pete for their desired resource and that lowering their bids would therefore be efficient. Adding dependencies between resources would require making the process partially ob-servable (open-cry) which can require substantial computational effort.
Despite the simplicity of communication protocols in auctions, a lot of arguments against the centralised approach exist. First, it may be difficult to find an agent that could as-sume the overhead role regarding its trustworthiness and computational abilities (
Cheva-leyre et al.,2006). Second, the system becomes more fragile as a failure of the auctioneer
will cause the breakdown of the whole system (Chevaleyre et al., 2006). Third, in the centralised approach a lot of information about the process is available to the agents who have large information-processing capacities in order to consider a lot of factors when making a decision. This setting bears little resemblance with real-world systems.
2.1.2 Distributed resource allocation
In contrast, real-world systems are usually complex in nature. That is, there is no central control authority but agents self-organise and act on partial information (Mitchell,2009). In this way distributed procedures of resource allocation are more ecologically valid. This section covers network and game approaches, with a specific focus on the Minority Game in the latter.
2.1.2.1 Resource allocation in networks
Resource allocation in networks boils down to optimising the flow between the nodes via capacity-restricted edges. Depending on the context, optimisation may refer to simply maximising the flow in the network but may also be constraint by allocating network capacity in a fair way among agents. Examples of real-world problems include transport networks and power grids. In the former agents are allocated road space. Nodes represent locations which are connected via roads, the edges. Some nodes are starting points or destinations of travel routes. Each road is assigned a capacity value as it can only carry a certain number of vehicles, which are the agents in the system. In the power grid edges are transmission lines which can carry a certain amount of electricity to consumers (nodes).
Carvalho et al.(2012) analysed the effects of network topology on resource allocation in
an artificial transport network with the max-min fairness strategy. This is a common allocation strategy in networks where increasing the system throughput along an edge results in decreasing the allocation to other edges (“the rich get richer while the poor get poorer”).
The authors studied a simple artificial network with uniform degree K, uniform edge capacity, one source-sink pair, and routes only being in placed along the L shortest paths. They found that the K − L space can be partitioned into distinct regions which affect the network throughput between the source-sink pair, as well as the fair allocation of individual path flows. The system throughput decreases as the number of shortest paths L increases. In the limit limL→∞ the system throughput grows with node degree.
Moreover, if congested edges are neighbouring sink or source nodes, the histogram of system throughput followed a power law decay, indicating great inequalities in resource allocation to different paths. In contrast, if congested edges are inside the network, the histogram becomes more scattered implying that the allocation of resources has become more equal. These findings reveal a strong interplay between the structure of transport routes, i.e. the road capacity, path intersections, and network topology.
Max-min fairness solely maximises system throughput without taking into account how fair the allocation is. This protocol only maximises the utility of the network operator but not of the agents. Translated to the transport system this means that vehicles whose destination nodes (sink) are close to their starting point (source) are allowed to travel first. In contrast, roads that lead to locations further away are only opened once the first vehicle fleet has reached their close locations (i.e. the close sinks are saturated). Thus, vehicles which have to travel further are not allocated network capacity, which is unfair in the respect of the agents.
In order to address the limitations in fairness and artificial topology, Carvalho et al.
(2015) studied the effect of the proportional fairness allocation strategy in a network whose topology was derived from a real-world low voltage power grid. Proportional fair-ness maximises the throughput of the system under the constraint that this throughput is spread evenly across edges.
Results revealed that proportional fairness performed equally well as max-min fairness or outperformed it on maximising throughput, that is, the amount of instantaneous electricity sent through the network. Moreover, inequalities of throughput allocation were much smaller for proportional fairness than for max-min fairness, thus maximising the utilities of the network operator and the agents at the same time.
All in all, networks relate the environmental structure of a system (i.e. the network topology) to its efficiency in terms of resource allocation. However, a shortcoming of this approach is that it does not account for cooperation between agents, which is covered by games.
2.1.2.2 Resource allocation in games
Another distributed approach to resource allocation are games. Games analyse the in-teractions between self-interested agents and therefore add competition to the resource allocation process. Whereas in networks agents react to the resource allocation mecha-nism, game agents adapt to and influence this mechanism. Moreover, learning is impor-tant in (but not restricted to) games. Agents coordinate their behaviour which enables them to increase their payoffs. Compared to auctions where the agents have to take many factors into account when deciding how to bid, the intelligence of game systems resides in the payoff mechanism (Galstyan et al.,2003).
Games that deal with resource allocation include the Ultimatum Game (Nowak et al.,
2000), the Cake Cutting Problem (Dubins and Spanier,1961), the Public Goods Game
(McGinty et al., 2012), the Kolkata Paise Restaurant Problem (Ghosh et al., 2010)
the Parking Space Problem (Hanaki et al.,2011), and the Minority Game (Challet and
Zhang,1998). Since the first three do not have the resource characteristics being relevant
for this study the interested reader is referred to the references above and to a general review on resource allocation in game theoretic settings (Fudenberg and Levine,1998). In the Kolkata Paise Restaurant Problem (KPRP) the resource is represented by R restaurants each having a capacity to serve one customer (Chakrabarti et al.,2009). N agents have to repeatedly distribute themselves over the restaurants in every round of the game. Global efficiency is defined as the average fraction of restaurants visited by at least one customer on a given day. Disappointingly, the most efficient solution to
the KPRP is dictatorship: each customer is assigned a restaurant and must eat there. Although agents can learn strategies to exploit the restaurants efficiently, this approach will take much more time to converge to an optimal solution compared to dictatorship
(Chakraborti et al., 2015). Since dictatorship requires a high level of synchronisation
and communication its use for real-world system is suboptimal.
The Parking Space Problem adds space and resource heterogeneity to the KPRP: drivers would like to park as close as possible from their workplace along a linear street and must learn at what distance they are likely to find a vacant space (Hanaki et al.,2011). Results reveal that agents who are initially identical, learn to behave differently and sort themselves into parking at the different spots along the street. This heterogeneity depends on the parameter settings. It emerges if most parking spots are equally valued as parking close to the centre, if parking decisions sufficiently depend on the attractiveness of a specific parking spot, and if the size of the loss in case of failing to park away from the city centre is not too large. In this setting agents who are lucky to find a free spot close to the centre will keep using this strategy since they receive a large payoff. By doing so these agents may reinforce the choices of those who park far away from the centre or fail to find a free spot. Thus, heterogeneity makes the luckier one even luckier. In many complex systems where agents compete for resources this heterogeneity is im-portant since it is advantageous to be as different as possible from other competitors. That is, being in the minority group maximises utility (Nakar and Hod, 2003; Savit
et al.). For example, one’s chances in finding a dating partner increase the fewer other
interested contenders are present. One is also more likely to receive a job offer the fewer candidates apply for the same position.
2.1.2.3 Resource allocation in the Minority Game
Considerable progress in the theoretical understanding of such systems has been gained by studying the simple, yet realistic model of the Minority Game (MG). The MG is a toy agent-based model that captures the mechanics behind competition for a resource in a supply- and demand system (Challet and Zhang, 1998). A remarkable property of this system is its qualitative change in the degree of inefficient resource allocation. This inefficiency is measured by the standard deviation σNmin of the minority group size
and can be described as a function of the agents’ memory size m (see AppendixA1 for reproduced findings).
If m is smaller than a critical value mc, σNmin is large and the exploitation of the
resources is less efficient than in a random choice game (RCG). For m = mc, σNmin
inefficiency value of the RCG for m ≥ mc. Thus, efficiency in this sense refers to the
global outcome of the system and does not relate to the gain of individual agents, some of whom can profit from an inefficient system.
2.1.2.4 Extensions to the original Minority Game
Later studies have extended the original MG in order to account for situations in real-world complex systems, most of them motivated by the financial market. The extensions presented below are not in chronological order. Figure 2.1summarises the set-up of the original MG and the actions taken per round together with the extensions.
(a) Set-up
(b) One game round
Figure 2.1: Schematic overview of original MG (black) with extensions (colour). Several game parameters are specified before the game starts (A). They are used to initialise the game and subsequently used in all game rounds. One round consists of a sequence of several actions (B). An extension can either modify the game set-up or the actions taken during a game round.
Variation of supply-demand ratio (Generalised Minority Game)
Real-world systems differ in their supply-demand ratio whereas the original MG uses a fixed one. For example, highways and city streets both belong to the same transport system but the former can carry more vehicles than the latter. The supply-demand ratio is defined by the relative difference between the number of agents N in the game and the total number of available resources C. The latter is split evenly across the choice options, so that the capacity of an option in the binary stationary MG is C2.
Savit et al. discovered a novel bifurcation point in the efficiency of the game by varying
three different states in a game: the state of limited resources where N = C + 1 cor-responds to the supply-demand ratio in the original Minority Game. For N ≥ C the system is in a state of scarce resources and for N ≤ C in a state of abundant resources. C is distributed by a binary supplier in all cases. Results revealed a phase transition offset in the σNmin− m plot for scarce and abundant resource settings. The minimum of
the standard deviation was reached for smaller m than in the limited resources setting. This is because the strategy space is larger in the scare and abundant resource settings (33m) than in the limited resource setting (22m). Thus, in the former cases the dimen-sion of the strategy space is equal to N for smaller m than in the latter case. σNmin is
therefore reached earlier.
Moreover, the authors found a bifurcation in the standard deviation as a function of N . As N was increased from N = C + 1 onwards, the average standard deviations of the games were found back in one of two phases, with a coexistance region for 207 ≤ N ≤ 210. In the first phase the games are qualitatively similar to the original MG, where one supplier is underloaded and the other is overloaded. As the demand on the system increases, the system switches to the second phase where the system behaves like in the scarce resources setting. Suppliers can either be both overloaded or one is underloaded while the other is overloaded. Whether the system ends up in the first or second phase only depends on the assignment of initial strategies since the control parameters (m, N , C) are kept the same. However, this initial strategy distribution does not always determine the phase in the coexistance region, which makes it harder to precisely control the system.
These different supply-demand ratios were tested in static environments, as they only changed between but not within games. Real-world systems usually are dynamic where resource capacities are subject to spatio-temporal changes. For example, predators have more to feed in spring during the breeding period of their prey and roads in a transport system change their capacity depending on their location (e.g. reduced capacity at con-struction sites).
Dynamic variation of resource capacity size
In another game extension a temporal variation in resource capacity was added to the original MG (Galstyan et al.,2003;Galstyan and Lerman,2002). Findings showed that agents adapt badly to changes in resource capacity. This failure to adapt even deterio-rates over rounds of the game as a growing phase shift between the system response and the changing capacity level shows.
Moreover, large fluctuations in the standard deviation over game rounds revealed inef-ficient resource allocation. So the output of the system varies a lot around the optimal allocation value. This sub-optimal behaviour was observed for the most efficient memory size from the original MG. However, since the authors did not study efficiency changes across memory sizes, the effect of dynamic resource capacities on the systematic m-based change in σNmin from the original MG were not analysed.
Local communication between agents
In a subsequent extension the authors showed that the lack of adaptation and efficiency can be overcome by allowing agents to locally communicate with each other (Galstyan
et al., 2003; Galstyan and Lerman, 2002). In each round an agent gets to know the
choices of its k neighbours in the previous round and accumulates them via a boolean function to determine its personal choice of the current round. The boolean functions are now the strategies, but rather than mapping a global signal (history of winning options) to some choice, like in the original MG, the strategies now map a local signal to the choice. In each round an agent chooses its k neighbours randomly.
Results showed that the system follows the changes in resource capacity very effectively (in phase), providing evidence for adaptive capabilities of the system. For early rounds there are strong fluctuations around the optimal resource allocation. Hence there is poor utilisation of the resource. In later rounds the system as a whole adapts and the strength of fluctuations decreases, providing evidence for efficiency.
The system is most efficient for k = 2, independent of the number of agents in the system. This is different from the original MG, where the position of the inefficiency minimum scales logarithmically with N . For larger k the inefficiency saturates at a value that depends on the amplitude of the perturbation and on the number of agents in the system. These findings imply that neighbourhood size and local communication structures have a larger effect on efficiency than the overall system size.
Multiple choices
In contrast to the original MG, the number of choices in real-world systems is abundant. In order to account for this gap,Chow and Chau(2002) introduced the Multiple Choice Minority Game (MCMG), where the number of choice options Nc is larger than two.
Translating the strategy definitions of the original MG to the MCMG results in an enor-mous strategy space of size NNcm
c . Since this size gets unsuitably large as Ncincreases,
The reduced strategy space comprises all strategies that are significantly different from each other and has been shown to be sufficient for producing the systematic change in σNmin in the original MG (Challet and Zhang, 1998). Whereas the size of the full
strategy space is 22m in the original MG, the reduced space only counts 2m+1 strategies. In the MCMG the reduced strategy space size is Nm+1
c .
The reduced strategy space only contains the ”good” strategies, but agents do not know at the beginning of the game which strategies are good and which are bad. Over the course of the game they learn to use the good strategies as a result of emerging coordination. Thus, providing agents with the reduced rather than the full strategies interferes with the emergence of coordination.
In order to account for this issue, the authors define strategies in a new way that reduces the strategy space and enables coordination at the same time. The choice of agent i using strategy s with history µ is defined as some bias plus the winning option summed over the m previous rounds, where each of the m winning options is weighted by its importance. The resulting full and reduced strategy spaces are of the tractable size Ncm. Analysing the memory-inefficiency relationship in the MCMG reveals the same quali-tative changes as in the original MG. Since the reduced strategy spaces of the original MG and the MCMG differ, the ratio of the reduced strategy space size to the number of strategies at play is used as control parameter α rather than the pure memory size m. For small α the reduced strategy space is much smaller than the number of strategies at play making agents use very similar strategies. The standard deviation of the minority group sizes reaches a minimum atcwhen the reduced strategy space is approximately as
large as the number of strategies agents use. As Ncincreases αcdecreases because more
strategies at play are required such that agents can use all the strategies in the reduced strategy space. As Nc increases, σNmin also increases, making resource allocation less
efficient. This is due to the increasing difficulty for agents to cooperate with each other when there are more choices for them.
Moreover, the attendance of a specific choice option, i.e. the number of agents having chosen that option, in the MCMG has very similar properties with respect to α as in the original MG, no matter how many choices and strategies agents have. The mean attendance fluctuates around the expected value N2 and the smaller m the larger the fluctuations.
Last, the authors investigated the wealth of agents in the MCMG. For small α the mean wealth per agent remains almost constant as α increases. The system is in the over-crowding phase where all choices have the same chance to win. Therefore all the agents, on average, always win the same number of times and make the same amount of profit.
At αcthe mean wealth per agent attains a maximum value. As α increases further some
of the rooms will have a higher chance to win. Consequently, some of the agents will always lose while some of them will always win in the game. As a result, the mean wealth per agent decreases rapidly.
Heterogenous agent types
In a further extension researchers accounted for the fact that agents in real-world systems are heterogenous, i.e. they use different strategies to get to the same resource. Inspired by different types of traders in the financial market Challet et al. (2000) used different strategies for producers and speculators who both compete for market share. Producers are traders that need the market for hedging, financing or any ordinary business. They thus pay less or no attention to “timing the market”. Speculators, on the other hand, join the market with the aim of exploiting the marginal profit pockets. The authors modeled producers as frozen agents who always play the same strategy. In contrast, speculators choose the best out of their s strategies, as in the original MG. Since the producers have a fixed pattern in their market behaviour they put a measurable amount of information into the market, which is exploited by the speculators.
Results revealed that producers and speculators need each other to maximise their re-spective share of the resource. As the amount of one trader type in the system remained constant and the amount of the other type was increased, the average gain per agent of the constant type increased from losing to winning margins until converging to some positive gain value. In contrast, the average gain of the dynamic type decreased. The specific optimal producer-speculator ratios depend on the memory size and the number of agents of the constant type. Thus, producers and speculators live in a symbiosis. Producers introduce systematic biases into the market, and without speculators, their losses would be proportional to these biases. The speculators precisely try to remove this kind of bias, reducing also systematic fluctuations in the market, thus reducing the losses of the producers and their own losses. Unfortunately, the authors did not analyse how these heterogenous agents affect the efficiency of the system. AppendixA3presents the results of our implementation.
In a second experiment the authors tested the effect of noise traders on the system. These are agents who choose randomly between options. Noise traders always decrease the efficiency in the system as an increase in the standard deviation across memory sizes shows. This holds for systems where noise traders are combined with either producers or speculators or both as well as systems where they represent the only agent type. The memory size-dependent phase transition in inefficiency from the original MG therefore becomes less distinct as the amount of noise traders increases.
Remember that efficiency relates to the global outcome of the system. Since noise agents increase the volatility of the market, some smarter agents can exploit it and increase their personal gain, even though the whole system becomes less efficient (Challet et al. 2000; also see AppendixA3). This personal gain is not covered by the efficiency concept used here.
A subsequent third experiment examined the effect of single privileged agents on the system’s dynamics (Challet et al.,2000). The authors increased the number of strategies of one agent while the number of strategies of all other agents remained fixed. Findings showed that the smaller the variance of all assigned strategies the more strategies the privileged agent needed to achieve a positive gain. This is because a small variance means that less information is exploitable from the system, i.e. there is no strategy that is clearly identified as the best. This also explains why the privileged agent uses more strategies as the variance decreases. In the opposite scenario, as variance increases, the agent’s behaviour is packed on the best strategy, so less strategies are used by this agent. Furthermore, the authors also increased memory size of the privileged agent. The aver-age gain of this privileged aver-agent is mirror-reversed compared to that of the other aver-agents. That is, efficiency in the game is proportional to the average gain of each normal agent whereas it is anti-proportional to the average gain of the privileged agent. Thus, the privileged agent optimises its resource share when everyone else does not and vice versa. This is due to the fact that the available information increases for the privileged agent as m decreases. Beyond a critical value of m the information gain grows too slowly in the system and therefore the privileged agent receives the same gain independent of her memory size.
Variation of active game participation (Grand Canonical MG)
Last, agents were allowed to individually decide whether to join the game or watch it from the side (Johnson and Hart,2000). This extension is called the Grand Canonical Minority Game Ensemble after the Grand Canonical Ensemble in thermodynamics. The Grand Canonical Minority Game is motivated by findings from experiments on different agent types where speculators (or producers) only receive a positive gain if a sufficient number of producers (or speculators) is playing the game (Challet and Marsili, 1999). Thus, it would only make sense for agents of one type to join the game once enough agents of the other type are present.
Jefferies et al. (2001) introduced a fixed virtual strategy score threshold under which
agents do not play. The virtual strategy score tracks the performance of the strategy over game rounds and is also updated if agents watch the game from the side line. That
is, if an agent skips a round she still chooses a strategy. If this strategy is successful had the agent participated in this round, the virtual score of this strategy is still updated. Virtual refers to the fact that agents do not bear in mind that the game outcome would have been affected by their participation.
As this threshold is increased, the percentage of active agents in the game suddenly sharply decreases from close to 100% to almost 0% at an intermediate threshold value. At this critical point the standard deviation of active agents peaks due to huge swings in supply and demand. These findings result from alternating moments of illiquidity and rushes to the market.
Furthermore, the authors also introduced a dynamic threshold which depends on an agent’s risk aversion rate and standard deviation of success, i.e. of the agent’s thresholds from previous rounds that resulted in winning. The larger the risk aversion and the larger the standard deviation the larger the chosen threshold for the current round, i.e. the less likely the agent is to play.
Results showed that after a sufficiently large number of rounds the probability distri-bution of the threshold values in the game resembles an exponential or linear decay, meaning that agents apply a great variety of thresholds.
Unfortunately, the authors did not relate the variation in active game participation to measure of efficiency or adaptation. They also did not make the dynamic threshold dependent on the necessity of an agent to participate in the game, which is a common feature in real-world systems. For example, predators still hunt in winter if they are hungry although joining the game at this point in time is not optimal since the number of competitors is large and that of prey small. However, not being able to get prey on one or several days is better than starving, so the predator is forced to join the game by the necessity to eat.
Socio-economic systems may work in the same way. Consider the financial market, where traders who are out of money might still join the market even if winning is highly unlikely given the current game situation. Traders might either assess the game situation in the wrong way by underestimating the risk of losing and/or might consider active behaviour more honourable than passive one (Taleb,2003).
Last, the authors assumed that non-active agents watch the game from the sideline keeping a virtual score of their strategies if they had played. When the agents finally joined the game they base their strategy choice on this virtual score. This scenario might be true for some real-world systems such as financial markets, where passive traders can track how shares have been allocated to active traders via the stock exchange and other
monitoring tools. However, other systems lack this property. For example, in the e-mobility system of this study, users are only informed about their gains if they tried to charge on a given day. If they decided not to charge, i.e. they do not actively participate in this round, they do not know whether they would have been successful if they had tried to charge, since they cannot oversee the charging stations from the side line and are unlikely to check the charging website in this situation..
In conclusion, previous research on resource allocation in complex systems have studied advantageous properties of individual agents and cooperation mechanisms that increase the efficiency of resource allocation. Games represent a theoretical setting to study these phenomena. Research is largely restricted to theoretical and simulated studies and has hardly been related to real data.
2.2
Phase transitions in socio-economic systems
2.2.1 Terminology
Phase transitions are sudden changes in the qualitative behaviour of a system under the continuous change of an external parameter (Li, 2002). They occur in a great variety of complex systems across domains. For example, lakes suddenly shift from a clear to a turbid state if the level of nutrients gradually increases, a magnet changes from a paramagnetic to a ferrormagnetic state once its temperature is sufficiently increased, and traffic conjestions suddenly form if the number of vehicles entering the road reaches a critical point (Chakraborti et al.,2015;Scheffner,2009).
The analysis of phase transitions is invaluable for understanding and sometimes predict-ing and controllpredict-ing the dynamics of systems. In this way we can either prepare to adapt to phase transitions in uncontrollable systems or elicit a desired transition in controllable ones. The aim in both options is to make positive use of the system response.
Let us first review some terminology before dealing with phase transitions in the MG and methods to detect them. The terms phase shift, regime shift, phase transition and critical transition have been used synonymously in the literature but actually describe different aspects of system dynamics (Duffy-Anderson et al.,2005). A regime refers to the state of a system and sudden changes in regimes are then regime shifts. These may refer to any sudden change, including those caused simply by a sudden change to a different set of external conditions, such as an accident causing a traffic jam. Critical transition, phase transition, phase shift, or bifurcation are synonyms for the subset of regime shifts where a system is pushed over a threshold where positive feedback causes
a self-propagating shift to an alternative regime. Positive feedback in traffic systems, for instance, exists between traffic fluidity and free road capacity. Empty roads promote fluid traffic flow and fluid traffic flow increases the free road capacity. If one of these system components decreases their mutual reinforcement decreases as well triggering a phase transition from a fluid to a conjested regime.
Different types of phase transitions can be distinguished (Scheffner,2009): catastrophic transitions are characterised by small changes in the control parameter invoking large changes in the state of the system. Depending on the side from which one approaches the transition, i.e. either increasing or decreasing the control parameter towards the fold between the two regimes, the forward and backward switches between the regimes occur at different values of the control parameter. This property is called hysteresis and implies that a catastrophic transition is not easily reversed. Hopf transitions describe the change from a stable equilibrium to a cyclic attractor. Homoclinic transitions occurs when a cyclic attractor collides with another attractor’s basin of attraction, such as in the case of two interacting predator-prey systems. Transclinical transitions are triggered by two colliding equilibrium points, causing a small change in system state.
Another categorisation based on thermodynamic principles distinguishes between first-and second order phase transitions (Department of Applied Mathematics and
Theoreti-cal Physics,2017;Sole,2006)). First-order phase transitions occur through the formation
of “bubbles” of the new phase in the middle of the old phase. These “bubbles” then expand and collide until the old phase disappears completely and the phase transition is complete. For instance, in boiling water bubbles of steam form and expand in the old liquid phase as they rise to the surface. Second-order phase transitions, on the other hand, proceed smoothly. The old phase transforms itself into the new phase in a contin-uous manner. This is usually the case in ecological, social and economic systems such as the transition from fluid to conjested traffic flow. The category names are derived from the first and second derivative of a thermodynamic variable, respectively (Sole,2006). Last, an important concept related to phase transitions is symmetry breaking. The sym-metry of a system describes a feature that remains unchanged if the system undergoes some transformation. For example, consider a drop of water as system and rotation is the transformation. No matter how we rotate this water drop it always looks the same from all angles. This system is therefore symmetric. In contrast, an ice cube looks very different if we rotate it. It may look like a flat square if directly watched from the top or like an intersection of two walls if looked from the side. The ice cube is therefore less symmetric than the water drop.
2.2.2 Phase transitions in the real world
In socio-economic systems symmetry is associated with basic ideas of social organisation, such as justice in law, equality of people in constitutions, and economic stability. A phase transition in this type of system can be considered as an example of symmetry breaking
(Arthur et al., 1997; T¨onu, 1992), but phase transitions in other types of systems do
not always involve symmetry breaking (Ivancevic and Ivancevic, 2008) 1. Symmetry
breaking and socio-economic transitions are related to critical instabilities and shifts of historical, economic, and political developments (Mainzer,2005).
For instance, two competing firms who initially have equal market shares bifurcate into winner and loser at a certain critical competition value. This value is reached due to random fluctuations resulting in the two firms having unequal market shares. This phase transition may be independent of differences in product quality. Other examples include migrating populations which either shift to a stable mixture, two stable ghettos or a state of restless migration (Haag and Dendrinos,1981), the transition of urban centres from popular attractors to neglected places (White and Engelen, 1993), as well as political revolutions (Mainzer,2005).
Why do these phase transitions occur? According to Scheffner (2009), systems which become increasingly rigid are less flexible to adapt to perturbations and therefore go through a phase transition. Well-known examples include the clinging of Polaroid to the paper photo technology in times of digitisation, or the Vikings continuing to farm when coming to Greenland, even though the climate there does not suit livestock. In both cases both systems underwent a phase transition, as a great loss in market share of Polaroid and the vanishing of the Greenland Vikings showed (Scheffner,2009;Smith,
2009).
2.2.3 Phase transitions in the Minority Game
In the original MG the degree of rigidity is also related to a phase transition. Numerical simulations showed that there exists a second order phase transition between a symmetric and an asymmetric phase (Savit et al.,1999). In the symmetric phase strategies of the agents do not sufficiently differ from each other, leading to a herding effect where a large majority of agents chooses the same option. In contrast, in the asymmetric phase the agents use sufficiently different strategies. Sufficiently different means that strategies differ in at least half of their possible decisions (Challet et al.,2000;Challet and Zhang,
1998).
1
Quantum phase transitions in physics for example do not involve symmetry breaking (Resnick et al.,
The larger m the more likely agents are to play diverse strategies. Rigidity in the original MG therefore corresponds to clinging to the same or similar strategies. This symmetric-asymmetric phase transition is related to the qualitative change in the degree of efficient resource allocation described in section 2.1.2.3. As resource allocation becomes most efficient at mc the system shifts from the symmetric to the asymmetric phase.
Savit et al. (1999) have shown that the predictive power of the information available
to the agents differs between the two phases. For m ≤ mc the history of the minority
groups, (i.e. which was the winning group in the previous game rounds), contains no predictive information about the winning option in the next round. In this respect the authors showed that the conditional probabilities of choosing one of the two options given a possible m-long history were distributed uniformly. In contrast, this histogram was not flat when m ≥ mc, providing evidence that in the second phase a significant
amount of information is available to the strategies.
The phase structure can be explained by two competing effects. First, as m < mc
periodic dynamics control the decision making process. Given that the same m-long history reoccurs in the game, the agents choose the opposite side with respect to the choice in the last round. Thus, the same option does not become the minority group in two consecutive rounds.
Second, as m ≥ mc, the system displays coordination dynamics among the agents’
responses to different m-long histories. The personal ranking of an agent’s strategies must coordinate the agents’ responses to 2m different possible histories. As m increases,
it becomes increasingly difficult for the agents to coordinate all of their responses and the system behaviour will increasingly look random as m increases. In the whole system there can only be as many choices as there are agents, but the latter have to satisfy 2m conditions. Thus, the maximum efficiency of the system can be achieved if the dimension of the strategy space is approximately the same as the total number of agents playing the game. This leads to the finding that as N increases mc increases as well, but decreases
as the number of strategies s per agent increases.
Coordination between agents is possible since the score of an agent’s strategy holds information about other agents’ strategies (Challet and Zhang, 1998). A large score indicates that there is a low probability that the agent will react in the same way as other agents if she decides to play that strategy. The larger the number of different strategies in the game, the larger mc. Thus, identical agents, having the same strategies,
do not contribute to the self-organised coordination of the system and therefore do not help to make resource allocation more efficient.
In summary, there is no predictive information about the next minority group available to agent’s strategies in the symmetric phase, whereas there is predictive information available to agent’s strategies in the asymmetric phase. Detecting phase transitions in complex systems presents a powerful step in understanding and controlling the system. Due to conceptual similarities, the phase transitions in the MG help us to better study these shifts in real-world socio-economic systems.
